Duffing adaptive oscillator physical reservoir computer

Abstract

Various examples are provided related to adaptive oscillator physical reservoir computers. In one example, a physical reservoir computer (PRC) includes a Duffing adaptive oscillator (AO) for adaptive stochastic resonance. The PRC can utilize the adaptive stochastic resonance to perform reconfigurable tasks. In another example, a PRC includes a Helmholtz adaptive oscillator (AO). In another example, a vibratory energy harvester or a signal booster includes a Duffing adaptive oscillator including an adjustable rail and a controller that can adjust a potential energy curve, electrostatic force, tension, length, or voltage. In another example, a spline adaptive oscillator includes a flexible rail, actuators that can modify the shape of the rail, and a controller that can adjust the actuators to create the desired frequency of a movable cart or linear bearing. In another example, a cybersecure authentication key includes a Duffing adaptive oscillator (AO) and a challenge-response pair.

Description

Docket: 511607-2030 DUFFING ADAPTIVE OSCILLATOR PHYSICAL RESERVOIR COMPUTER CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to, and the benefit of, U.S. provisional applications entitled “Duffing Adaptive Oscillator Physical Reservoir Computer” having serial no. 63 / 625,144, filed January 25, 2024; having serial no.63 / 701,887, filed October 1, 2024; and having serial no.63 / 730,791, filed December 11, 2024, all of which are hereby incorporated by reference in their entireties. BACKGROUND

[0002] Neuromorphic computing is a computing paradigm that takes inspiration from biological neurons. Physical reservoir computers (PRCs) repurpose the nonlinear dynamics of a physical system for computation. PRCs were a natural progression of non-physical reservoir computers, which are deep neural networks that have hidden layers with untrained weights. Growing from echo state networks and liquid state machines, reservoir computing has several advantages, including lower training costs and robustness to overfitting. Notably, PRCs differ from traditional Turing machines, as the information storage is not static. Physical reservoir computers have been employed for a wide variety of tasks, including digit recognition (written and spoken), image recognition, prediction of time series, and logic operations. Physical reservoir computers have been created from optoelectronics, soft robots, tensegrities, quantum systems, and limit cycle oscillators. SUMMARY

[0003] Aspects of the present disclosure are related to adaptive oscillator physical reservoir computers. In one embodiment, among others, a physical reservoir computer (PRC) comprises a Duffing adaptive oscillator (AO) configured for adaptive stochastic resonance, the PRC configured to utilize the adaptive stochastic resonance to perform reconfigurable tasks. In one or more aspects, the Duffing AO is not divergent. The DuffingAO can be monostable hardening (^^^ ^ 0, ^^ଷ ^ 0). The Duffing AO can be monostablesoftening (^^^ ^ 0, ^^ଷ ^ 0). The Duffing AO can be bistable (^^^ ^ 0, ^^ଷ ^ 0). The Duffing AOcan be implemented as an electrical circuit. The electrical circuit can be implemented using a FPAA, a FPGA, a PCB, or a VLSI. The Duffing AO can be implemented as a mechanical system. The mechanical system can comprise a mechanical beam, clamped-clamped beams, large deformation cantilevered beams, or string in tension. The mechanical systemDocket: 511607-2030 can comprise a quartz crystal. The Duffing AO can be implemented as an electromechanical system. The electromechanical system can comprise a comb drive tuning fork (commonly used for MEMS gyroscopes). The Duffing AO can be implemented as a flexible rail with a linear bearing, cart, or rolamite. The Duffing AO can be implemented as a quantum system. The quantum system can comprise a Josephson junction. The quantum system can comprise a tunnel diode oscillator.

[0004] In various aspects, the PRC can or may not utilize time multiplexing. The PRC can or may not utilize spatial multiplexing. The Duffing AO can be configured as a morphable logic gate. The PRC can perform different tasks in parallel with different vibratory modes. Various signals can be encoded with frequencies that correspond to the different vibratory modes.

[0005] In another aspect, a physical reservoir computer (PRC) comprises a Helmholtz adaptive oscillator (AO). The Helmholtz AO can be implemented as a mechanical system. The mechanical system can comprise a plate or membrane. The Helmholtz AO can be implemented as an electrical system. The electrical circuit can be implemented using a FPAA, a FPGA, a PCB, or a VLSI.

[0006] In another aspect, a vibratory energy harvester comprises a Duffing adaptive oscillator comprising an adjustable rail that is shaped like a potential energy curve, comb drive tuning fork, clamped-clamped beam, large deformation cantilever beam, string, plate, or Josephson junction; a controller configured to adjust the potential energy curve, electrostatic force, tension, length, or voltage; and circuitry to convert and store energy in a battery or capacitor.

[0007] In another aspect, a signal booster for sensors comprises a Duffing adaptive oscillator comprising an adjustable rail that is shaped like a potential energy curve, comb drive tuning fork, clamped-clamped beam, large deformation cantilever beam, string, plate, or Josephson junction; and a controller configured to adjust the potential energy curve, electrostatic force, tension, length, or voltage. Adaptive stochastic resonance can use noise to boost the signal-to-noise ratio of a weak signal. Adaptive stochastic resonance can boost a weak electromagnetic or optical signal. Adaptive stochastic resonance can boost a weak acoustic signal. Adaptive stochastic resonance can boost a weak quantum signal. Adaptive stochastic resonance can boost a weak mechanical signal. Adaptive stochastic resonance can boost a weak electrical signal. Noise can enhance the learning rate.

[0008] In another aspect, a spline adaptive oscillator comprises a flexible rail, actuators placed at one or more locations to modify the shape of the rail, and a controller configured to adjust the actuators to create the desired frequency of a movable cart or linear bearing.Docket: 511607-2030

[0009] In another aspect, a cybersecure authentication key comprises a Duffing adaptive oscillator (AO) and a challenge-response pair. The challenge-response pair can be a bifurcation diagram. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

[0011] FIG.1 illustrates an example of displacement vs. potential energy for a bistable Duffing oscillator, in accordance with various embodiments of the present disclosure.

[0012] FIGS.2A-2D illustrates examples of hardening monostable Duffing adaptive oscillator low amplitude responses, in accordance with various embodiments of the present disclosure.

[0013] FIGS.3A-3D illustrates examples of softening monostable Duffing adaptive oscillator large amplitude responses, in accordance with various embodiments of the present disclosure.

[0014] FIGS.4A-4D illustrates examples of hardening monostable Duffing adaptive oscillator large amplitude responses, in accordance with various embodiments of the present disclosure.

[0015] FIGS.5A-5B illustrate examples of a nonadaptive bistable Duffing oscillator response, in accordance with various embodiments of the present disclosure.

[0016] FIGS.6A-6B and 7A-7C illustrate an example of a bistable Duffing adaptive oscillator response, in accordance with various embodiments of the present disclosure.

[0017] FIGS.8A-8B illustrate examples comparing the frequency-amplitude relationship of a non-adaptive Duffing oscillator and a bistable Duffing oscillator, in accordance with various embodiments of the present disclosure.

[0018] FIG.9 illustrates an FPAA circuit schematic of a monostable Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0019] FIG.10 illustrates an example of the response of the FPAA circuit of FIG.9, in accordance with various embodiments of the present disclosure.

[0020] FIG.11 illustrates an example of a frequency-amplitude relationship of the Duffing adaptive oscillator of FIG.9, in accordance with various embodiments of the present disclosure.Docket: 511607-2030

[0021] FIGS.12A-12B illustrate FPAA circuit schematics of bistable Duffing adaptive oscillators, in accordance with various embodiments of the present disclosure.

[0022] FIGS.13A and 13B illustrate examples of experimental results of the bistable Duffing adaptive oscillators of FIGS.12A and 12B, respectively, in accordance with various embodiments of the present disclosure.

[0023] FIG.14 illustrates an example of stochastic resonance of a non-adaptive Duffing oscillator, in accordance with various embodiments of the present disclosure.

[0024] FIG.15 illustrates an example of stochastic resonance of an adaptive Duffing oscillator, in accordance with various embodiments of the present disclosure.

[0025] FIG.16 illustrates an example of a circuit schematic of a Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0026] FIG.17 illustrates an example of frequency-amplitude response curves for a Duffing oscillator and Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0027] FIG.18 illustrates an example of stochastic resonance of a non-adaptive Duffing oscillator circuit, in accordance with various embodiments of the present disclosure.

[0028] FIG.19 illustrates an example of adaptive stochastic resonance of a Duffing adaptive oscillator circuit, in accordance with various embodiments of the present disclosure.

[0029] FIG.20 includes images illustrating an example of a Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0030] FIG.21 illustrates an example of a frequency-amplitude response of several non- adaptive Duffing oscillators with different static linear stiffnesses, in accordance with various embodiments of the present disclosure.

[0031] FIG.22 illustrates an example of a frequency-amplitude response of the Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure, in accordance with various embodiments of the present disclosure.

[0032] FIG.23 illustrates an example of an amplitude of the oscillations as a function of time, as the Duffing adaptive oscillator learns a resonance condition, in accordance with various embodiments of the present disclosure.

[0033] FIG.24 illustrates an example of the input function, h(t), to the Duffing adaptive oscillator and the response of one of the states of the Duffing adaptive oscillator, x(t).

[0034] FIG.25 illustrates examples of the Duffing adaptive oscillator calculating different tasks, such as the NARMA task, in accordance with various embodiments of the present disclosure.

[0035] FIG.26 illustrates an example of states of the Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.Docket: 511607-2030

[0036] FIG.27 illustrates an example of a trained prediction of a Duffing adaptive oscillator physical reservoir computer, in accordance with various embodiments of the present disclosure.

[0037] FIG.28 illustrates an example of a response of a Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0038] FIG.29 illustrates an example of a physical reservoir computer trained for an XOR task, in accordance with various embodiments of the present disclosure.

[0039] FIG.30 illustrates an example of performance of a physical reservoir computer without and with noise, in accordance with various embodiments of the present disclosure.

[0040] FIG.31 illustrates an example of potential energy of a Duffing adaptive oscillator, monostable and bistable, in accordance with various embodiments of the present disclosure.

[0041] FIG.32 illustrates an example of a Helmholtz oscillator, in accordance with various embodiments of the present disclosure.

[0042] FIG.33 illustrates examples of predictions from the bistable Duffing adaptive oscillator physical reservoir computer for different orders of NARMA tasks, in accordance with various embodiments of the present disclosure.

[0043] FIG.34 illustrates an example of a bifurcation diagram of the original bistable Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0044] FIG.35 illustrates an example of a period-3 orbit for the bistable Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0045] FIG.36 illustrates an example of a histogram of clones' NMSE values for a 5thorder NARMA task, in accordance with various embodiments of the present disclosure.

[0046] FIG.37 illustrates an example of a 2D histogram of the bifurcation diagrams of 100 simulations of the original oscillator with random initial conditions, in accordance with various embodiments of the present disclosure.

[0047] FIG.38 illustrates an example of a 2D histogram of the bifurcation diagrams of the 100 clone oscillators, in accordance with various embodiments of the present disclosure.

[0048] FIG.39 illustrates an example of the average frequency state for an ensemble of stochastic differential equation simulations of the Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.

[0049] FIG.40 illustrates an example of the average frequency state for the Fokker- Planck equation of the Duffing adaptive oscillator, in accordance with various embodiments of the present disclosure.Docket: 511607-2030 DETAILED DESCRIPTION

[0050] Disclosed herein are various examples related to adaptive oscillator physical reservoir computers. Reference will now be made in detail to the description of the embodiments as illustrated in the drawings, wherein like reference numbers indicate like parts throughout the several views.

[0051] As a subset of nonlinear oscillators, adaptive oscillators are often constructed by appending additional dynamic states to a nonlinear oscillator. Adaptive oscillators have dynamic states that can learn and store information. This type of dynamic learning has received relatively little attention, but it is an attractive method for many types of systems that require data processing in real-world applications. As an example, adaptive oscillators provide a unique solution for energy harvesting, which is both high quality factor and broadband. Adaptive oscillators can function as physical reservoir computers, in which adaptation can be used for both self-learning and multiplex-free morphable logic gates. In this disclosure, an adaptive oscillator is constructed from the Duffing oscillator for the first time. The dynamics of the hardening monostable, softening monostable, and bistable Duffing oscillator is considered. As the Duffing oscillator is a common model for a wide range of physical systems, it is likely that adaptation could be extended to many of these systems for enhanced functionality.

[0052] In addition to a base oscillator, adaptive oscillators (AOs) have plastic dynamic states that both learn and store information from an external stimuli. This form of dynamic intelligence is notably dissimilar to other types of vibratory control or phase locked loops. Since frequency adaptation can be directly embedded into the oscillator’s dynamics, adaptive oscillators are an attractive option for many edge computing applications. For these applications, the adaptive oscillator is capable of processing fairly complex tasks without needing a microcontroller. For instance, adaption can be leveraged to create a high-quality factor broadband energy harvester. As another example, dynamic learning can be repurposed to make the adaptive oscillator into a physical reservoir computer with reconfigurable sampling rates. Adaption can be repurposed to make the adaptive oscillator into a physical reservoir computer that exhibits self-learning for signals with different sampling rates. By further leveraging only the plastic states, an adaptive oscillator physical reservoir computer architecture can even be created that does not need time multiplexing, which substantially increases its processing speed.

[0053] Adaptive oscillators have been proposed for other applications, such as central pattern generators for locomotion, increasing the efficiency of gait, and analog frequency analyzers. Although frequency adaptation is a common plastic state for adaptive oscillators, other plastic states can also be used, such as a plastic amplitude state and even theDocket: 511607-2030 learning rate for the adaptive oscillator itself. Using the Fokker-Planck equation and experiments, the effects of noise have been studied for the adaptive frequency oscillator.

[0054] The non-adaptive Duffing oscillator is a prolific model for a wide range of physical systems. It exhibits many interesting phenomena, such as hysteresis, chaotic motion, and stochastic resonance. When Duffing oscillators are coupled together in an array, they can exhibit intrinsic localized modes, which are persistent vibratory modes that are localized to a small number of oscillators, and they can also be used as a physical reservoir computer.

[0055] Here, the Duffing oscillator is constructed as an adaptive oscillator. The softening monostable case, the hardening monostable case, and the bistable case are all considered. Interestingly, the plastic frequency state is capable of learning resonance conditions, rather than simply learning the external forcing frequency. Of relevance, this resonance-tracking effect was also observed in a pendulum adaptive oscillator, while the pendulum adaptive oscillator was also shown to be chaotic for some parameter combinations. Both the monostable Duffing adaptive oscillator and the bistable Duffing adaptive oscillator are extremely nonlinear. The dynamic memory of the frequency state greatly increases the complexity of this oscillator. The following disclosure includes equations of motion for the Duffing adaptive oscillators for each case, parametric studies including simulation results, a local stability analysis for the adaptive oscillator, description of an experimental setup and discussion of experimental results. Equations of Motion

[0056] Monostable Duffing Adaptive Oscillator. The equation of motion for the monostable Duffing oscillator is given by: (1) The ^^ termand ^^ଷis the nonlinear stiffness term. In this version of the equation, the mass has been normalized. To simplify these Duffing equations further, the ^^^term present in the damping will be removed. Thus, rewriting Eq. (1) in state space: (2) When theterm is negative, the monostable Duffing equation is softening. For both cases, the monostable Duffing oscillator has multiple dynamic solutions and hysteresis.

[0057] By augmenting Eq. (2) with an additional frequency state, the monostable Duffing adaptive oscillator is given by:Docket: 511607-2030 (3) In Eq. (3),stateof Hebbian learning works by shifting the ^^ state up or down based on a net bias over a period. To better explain this, the base oscillator will entrain to the external force, such that the response will have a specific phase. When the base oscillator is not at resonance, there is a bias that pushes the ^^ state towards resonance. This bias is produced by the multiplication of the ^^ state and the external force; over a period, there is a net positive ornegative amount that acts to shift the ^^ state. Since the phase is 90° at resonance, ^^^ hasonly small oscillations without a net shift over a period when the system is at resonance.(This phase can be seen in FIG. 10.) When the phase is too low, the ^^^ equation will act toshift the ^^ state higher. When the phase is too high, the ^^^ equation will act to shift the ^^state lower. Through this learning procedure, the ^^ state will align near the resonance value. It should be noted that the resonance value is amplitude-dependent for the Duffing oscillator, and this is why the external forcing frequency is not necessarily the same as the learned frequency. It should also be noted that, since the Duffing oscillator can become chaotic, there are likely parameter combinations for which the Duffing adaptive oscillator may become chaotic as well.

[0058] Bistable Duffing Adaptive Oscillator. The equation of motion for the bistable Duffing oscillator is given by: (4) Foroscillator is unstable if ^^ ^ ^^^^ଶ^⁄ ^^ଷ ^, which is not considered here. The bistable Duffingequation has three fixedേ൫^^^⁄ ^^^ଷ ൯ and ^^ ൌ 0. The first two are stable, while the^^ ൌ 0 fixed point is unstable. The displacement versus the potential energy plot for thebistable Duffing oscillator is shown with the two stable and one unstable equilibrium points in FIG.1.

[0059] Notably, the bistable Duffing oscillator has different resonance frequencies for intrawell and interwell responses. Rewriting Eq. (4) in state space and adding a frequency state, the bistable Duffing adaptive oscillator is given by:Docket: 511607-2030 (5) The ^^^௧^it is resetoscillator’s motion begins moving in the negative direction). This ^^^௧^value is left as a constant until the next instantaneous period is completed. By including the ^^^௧^value in Eq. (5), the ^^ state learns an intrawell resonance condition. Note that without this term, the offset caused by the bistability is effectively a zero frequency component, which prevents the bistable Duffing adaptive oscillator from learning the resonance condition. This will be further discussed in the next section.

[0060] Consider the equation of motion for the bistable Duffing oscillator is given by Eq.(4) where ^^^ ൌ 1:(4a) Foroscillator is unstable if |^^| ^ ^^^ଶ^⁄ ^^ଷ , which is not considered here. The bistable Duffingequation has three fixed points: ^^ ൌ േ൫^^^⁄ ^^^ଷ ൯ and ^^ ൌ 0. The first two are stable, while the^^ ൌ 0 fixed point is unstable. The displacement versus the potential energy plot for thebistable Duffing oscillator is shown with the two stable and one unstable equilibrium points in FIG.1.

[0061] This type of Hebbian learning works by shifting the ^^ state up or down based on a net bias over a period. To better explain this, the base oscillator can entrain to the external force, such that the response will have a specific phase. When the base oscillator is not at resonance, there is a bias that pushes the ^^ state towards resonance. This bias is produced by the multiplication of the x state and the external force; over a period, there is a net positive or negative amount that acts to shift the ^^ state. Since the phase is 90 degrees at resonance, derivative of the ^^ has only small oscillations without a net shift over a period when the system is at resonance. When the phase is too low, the derivative of the ^^ equation will act to shift the ^^ state higher. When the phase is too high, the derivative of the ^^ equation will act to shift the ω state lower. Through this learning procedure, the ^^ state will align near the resonance value. It should be noted that the resonance value is amplitude- dependent for the Duffing oscillator, and this is why the external forcing frequency is not necessarily the same as the learned frequency. It should also be noted that, since theDocket: 511607-2030 Duffing oscillator can become chaotic, there are likely parameter combinations for which the Duffing adaptive oscillator may become chaotic as well.

[0062] Notably, the bistable Duffing oscillator has different resonance frequencies for intrawell and interwell responses. Rewriting Eq. (4a) in state space and adding a frequency state, the bistable Duffing adaptive oscillator is given by: As in Eq. (3), anet bias over omega up or on the phase between the external force and the ^^ state. However, there is an added complication with the bistable oscillator: when the bistable Duffing oscillator is oscillating in awell, its resonance is not the same as it would be if it were traversing both wells. Thus, the ^^^quantity must be “corrected” to create oscillations around zero rather than within the center of a well’s orbit. The ^^ state oscillates about a non-zero quantity that is approximated by ^^^௧^, because of the wells caused by the bistability. To further complicate this, the orbit’s center changes as a function of the system parameters and of the amplitude. The orbit’s center is approximated by taking a moving average during one period of the oscillator’s ^^ state oscillations. The moving average starts when the oscillator’s ^^ state begins moving in the negative direction (i.e., the oscillation has peaked) and continues averaging until the next peak. The ^^^௧^term is calculated for each period in this way, and it is held constant until the next moving average is completed. This ^^^௧^term is subtracted from the ^^ state to push the ^^ state to a value to promote well-hopping through resonance. Once well-hopping starts, the ^^^௧^term approaches zero, and so the ^^ state learns the intrawell frequency. The ^^ state learns an intrawell resonance, which causes much larger oscillations within the well. For a larger amplitude of the external sinusoid, the non-adaptive bistable Duffing oscillator is still stuck in one well. However, the larger forcing does cause a period-doubling bifurcation to occur.

[0063] The bistable Duffing adaptive oscillator has the added complication that the intrawell and interwell frequencies can be different; moreover, the oscillations are centered about different locations during the intrawell and interwell cases. An ^^^௧^term was introduced to approximate the center of the oscillations for both cases. The ^^ state helps to induce well-hopping. It should be noted that since Ω is inside of a sinusoid, its value is not directly “seen” by the adaptive oscillator. Thus, the ability of the oscillator to correctly learn Ω is only achieved by shifting the adaptive frequency state to maximize the resonance of theDocket: 511607-2030 base oscillator. This ability to learn the resonance frequency without a priori knowledge of it is important, as it allows the adaptive oscillator to function as an analog frequency analyzer.

[0064] This adaptive procedure can also be implemented for other types of physical oscillators. For instance, tunnel diode oscillators are capable of oscillation frequencies above 1 terahertz, even at room temperature. For this reason, tunnel diode oscillators have applications to wireless communications, spectroscopy, and imaging. Chaotic tunnel diode oscillators have been synchronized using an active adaptive feedback coupling scheme. The current adaptive scheme is somewhat different: rather than synchronizing the oscillator with another signal, it instead learns a key piece of information about the signal. In the current example, the Duffing adaptive oscillator does not exactly synchronize with the noisy signal, but it instead learns information about the noisy signal to maximize its own resonance. By appending adaptive states to the tunnel diode oscillator, an adaptive oscillator with frequencies above 1 terahertz could be used as both an analog frequency analyzer and a physical reservoir computer. Simulation Results

[0065] In this section, Eqs. (3), (5) and (5a) were simulated in MATLAB using a fixed step Euler’s method solver. This fixed step solver was chosen so that ^^^௧^could be calculated. The responses of non-adaptive monostable and bistable Duffing oscillators are also presented for comparison.

[0066] Monostable Duffing Adaptive Oscillator. For low amplitude responses, the monostable Duffing adaptive oscillator works as expected, as the ^^ state converges to the external forcing frequency Ω. This type of response for Eq. (5) is illustrated in FIGS.2A and 2B and for Eq. (5a) is illustrated in FIGS.2C and 2D. In FIGS.2A and 2C, for a low amplitude response, the hardening monostable Duffing adaptive oscillator is compared with several non-adaptive Duffing oscillators for several different values of ^^^. As the amplitudes are small, the non-adaptive Duffing oscillators have an almost linear response. The monostable Duffing adaptive oscillator has a response that follows the maximum amplitude for each non-adaptive oscillator. In FIGS.2B and 2D, due to the low amplitude, the ^^ statelearns the external forcing frequency, Ω. The ^^ ൌ Ω line is plotted for reference. Here, ^^ ൌ0.1, ^^ ൌ 0.1, ^^ଷ ൌ െ100, and ^^ఠ ൌ 1. For large amplitude responses, the monostable Duffingadaptive oscillator does not work as expected. Rather than ^^ learning the Ω value, ^^ converges to a value that produces a resonance. This behavior for Eq. (5) is shown in FIGS. 3A-3B and 4A-4B and for Eq. (5a) is shown in FIGS.3C-3D and 4C-4D.

[0067] In FIGS.3A and 3C, for a large amplitude response, the softening monostable Duffing adaptive oscillator is compared with several non-adaptive Duffing oscillators for several different values of ^^^. While the non-adaptive Duffing oscillators have the expectedDocket: 511607-2030 hysteresis, the monostable Duffing adaptive oscillator has a response that follows the maximum amplitude for each non-adaptive oscillator. In FIGS.3B and 3D, due to the hysteresis, the ^^ state learns a resonance condition rather than the external forcingfrequency, Ω. The ^^ ൌ Ω line is plotted for reference. Here, ^^ ൌ 0.1, ^^ ൌ 1, ^^ଷ ൌ െ100, and^^ఠ ൌ 1.

[0068] In FIGS.4A and 4C, for a large amplitude response, the hardening monostable adaptive oscillator is compared with several non-adaptive Duffing oscillators for several different values of ^^^. While the non-adaptive Duffing oscillators have the expected hysteresis, the monostable Duffing adaptive oscillator has a response that follows the maximum amplitude for each non-adaptive oscillator. In FIGS.4B and 4D, due to the hysteresis, the ^^ state learns a resonance condition rather than the external forcingfrequency, Ω. The ^^ ൌ Ω line is plotted for reference. Here, ^^ ൌ 0.1, ^^ ൌ 1, ^^ଷ ൌ െ100, and^^ఠ ൌ 1.

[0069] In FIGS.3A and 3C, the response of the softening monostable Duffing adaptive oscillator is shown. Rather than learn the external forcing frequency, the ^^ state learns a resonance condition such that the amplitude of the ^^ state follows the maximal amplitude of the backbone curve formed by non-adaptive Duffing oscillators of varying ^^^. Similarly, in FIGS.4A and 4C, the response of the hardening monostable Duffing adaptive oscillator is shown. The response for the amplitude and ^^ state is similar to the softening case, exceptthat the ^^ state is reflected about the ^^ ൌ Ω line in FIG. 4B.

[0070] Bistable Duffing Adaptive Oscillator. The response of the bistable Duffing adaptive oscillator is more complex. To highlight this complexity and the role of the ^^^௧^term, the response of the non-adaptive bistable Duffing oscillator is shown in FIGS.5A and 5B. The response of the non-adaptive bistable Duffing oscillator is provided for comparison atthe top. In the case of FIG. 5A, ^^^ ൌ 1, ^^ ൌ 0.01, ^^ଷ ൌ 1, ^^ ൌ 1, and Ω ൌ 0.5, while the initialconditions are ^^^0^ ൌ ^^ଶ^⁄ ^^ଷ and ^^^0^ ൌ 0. In the case of FIG. 5B, ^^^ ൌ 5, ^^ ൌ 0.1, ^^ଷ ൌ 1,^^ ൌ 0.7, and Ω ൌ 3.5, while the initial conditions are ^^^0^ ൌ ^^^ଶ^⁄ ^^ଷ ^ ^ .1 and ^^^0^ ൌ 0. Inthese cases, the oscillator’s response remains trapped in one well with a relatively smallamplitude response. The state space is shown in the middle of FIG. 5B, in which the ^^ െ ^^trajectory oscillates inside of a well. The fast Fourier transform is shown on the bottom, which has a large peak at the driving frequency. For the chosen parameter values and initial conditions, the non-adaptive oscillator is trapped in one well for this case.

[0071] For the same parameter values and comparable initial conditions (i.e., ^^^0^ was chosen as ^^^), the response of the bistable Duffing adaptive oscillator is shown in FIGS.6Aand 6B. In the case of FIG. 6A, ^^ ൌ 0.01, ^^ଷ ൌ 1, ^^ ൌ 1, and Ω ൌ 0.5, while the initialconditions are ^^^0^ ൌ ^^ଶ^⁄ ^^ଷ and ^^^0^ ൌ 0, and ^^^0^ ൌ 1. Even though the oscillator has theDocket: 511607-2030 same starting conditions as those used for FIG.5A, the ^^ state causes the oscillator toapproach an intrawell resonance, which causes it to jump the barrier at ^^ ൌ 0. When thishappens, ^^^௧^becomes zero, so the ^^ state begins to approach an interwell resonance. However, when the response gets too large, it momentarily undergoes a chaotic response. As shown at the bottom, the ^^ state can learn both intrawell and interwell resonances to increase the amplitude of the ^^ state. First, the ^^ state learns an intrawell resonance, whichcauses the oscillator to jump the barrier at ^^ ൌ 0. Next, the ^^ state starts to learn an interwellresonance. As the amplitude of ^^ increases, the oscillator occasionally exhibits a chaotic response due to large oscillations. A zoomed portion of FIG.6A is shown in FIG.7A. The state space of the ^^ and ^^ states is shown, along with the ^^^௧^value. The “X” in the center of the plot denotes the values of ^^^௧^^25^. The ^^ state allows the bistable Duffing adaptive oscillator to have considerably larger amplitude oscillations.

[0072] The response of the bistable Duffing adaptive oscillator to a small amplitudeforce is shown in FIG. 6B. In this case, ^^ ൌ 0.1, ^^ଷ ൌ 1, ^^ ൌ 0.7, ^^ఠ ൌ 0.1 and Ω ൌ 3.5, whilethe initial conditions are ^^^0^ ൌ ^^^^0^ଶ⁄ ^^ଷ ^ ^ .1 and ^^^0^ ൌ 0, and ^^^0^ ൌ 3.7. The statespace is shown (second from top), in which the ^^ െ ^^ trajectory oscillates inside of a well.The adaptive frequency shifts the well’s location and also creates an intrawell resonance. In the third from the top, the ^^ state learns a resonance condition. The fast Fourier transform is shown on the bottom, which has a large peak at the driving frequency. The response of the non-adaptive bistable Duffing adaptive oscillator to a large amplitude force is shown in FIG.7B. In this case, ^^ ൌ 0.1, ^^ଷ ൌ 1, ^^ ൌ 0.7, ^^ఠ ൌ 0.1 and Ω ൌ 3.5, while the initial conditionsare ^^^0^ ൌ ^^^^0^ଶ⁄ ^^ଷ ^ ^ .1 and ^^^0^ ൌ 0, and ^^^0^ ൌ 3.7. The state space is shown (secondfrom top), in which the ^^ െ ^^ trajectory oscillates inside of a well. The adaptive frequencyshifts the well’s location and also creates an intrawell resonance. In the third from the top, the ^^ state learns a resonance condition. The fast Fourier transform is shown on the bottom, which has a large peak at the driving frequency and another at twice the driving frequency (because of a period doubling bifurcation). The response of the non-adaptive bistable Duffing adaptive oscillator to a large amplitude force is also shown in FIG.7C. In this case,^^ ൌ 0.1, ^^ଷ ൌ 1, ^^ ൌ 5.0, ^^ఠ ൌ 0.1 and Ω ൌ 3.5, while the initial conditions are ^^^0^ ൌ^^^^0^ଶ⁄ ^^ଷ ^ ^ .1 and ^^^0^ ൌ 0, and ^^^0^ ൌ 3.7. The state space is shown (second from top),in which the ^^ െ ^^ trajectory oscillates inside of a well. The adaptive frequency shifts thewell’s location and also creates an intrawell resonance. In the third from the top, the ^^ state learns a resonance condition. The fast Fourier transform is shown on the bottom, which has a large peak at the driving frequency.

[0073] This is the first time that a bistable system has been constructed as an adaptive oscillator. The formulation of the ^^^௧^term allows the ^^ state to learn either an intrawell orDocket: 511607-2030 interwell resonance. Unlike traditional adaptive oscillators, the ^^ state does not learn the external forcing frequency, but it instead learns a resonance that increases the amplitude of the oscillations.

[0074] The amplitudes of the non-adaptive Duffing oscillator are considerably smaller than the bistable Duffing adaptive oscillator, as shown in FIGS.8A and 8B. In FIG.8A, the frequency versus amplitude relationship of the two oscillators (frequency-amplitude relationship of the non-adaptive Duffing oscillator and the bistable Duffing adaptive oscillator) is shown for comparison. As the bistable Duffing adaptive oscillator can sometimes produce a chaotic result, the last half of each simulation was used, and the amplitude of the response of the x state was averaged to compute ^^^^^^^^. As the system is bistable, the offset value, ^^^^, was subtracted. Thus, if there are ^^ peaks in the ^^ state from the last half of a simulation, then: (6) In FIG.8B, thesimulation was any a Duffing oscillator’s highest amplitude is only equal to the bistable Duffing adaptive oscillator’s amplitude for one value. As shown on the bottom, the ^^ state learns the external forcing frequency, except there is a correction based on the√2 that is found from the local analysis.The parameter values were chosen as ^^ ൌ 0.1, ^^ଷ ൌ 1.0, ^^ ൌ 0.7, ^^ఠ ൌ 1.0 and Ω ൌ 5.0. Bycapitalizing on both bistability and intrawell and interwell the bistable Duffingadaptive oscillator has large amplitude oscillations. This system an excellent candidate as a vibratory energy harvester for this reason. Local Analysis

[0075] Local stability analyses of the Duffing adaptive oscillator is now examined. This provides information about the behavior of the dynamics of the ^^ state. To do this, the Eqs. (2) and (5) are first converted to autonomous sets of equations. This is done by replacing the external force, cos^Ω^^^, with an oscillator that exhibits a supercritical Andronov-Hopf bifurcation: (7) Next, thenonlinear system.

[0076] Monostable Duffing Adaptive Oscillator. By modifying the monostable Duffing adaptive oscillator with Eq. (7), Eq. (2) can be written as an autonomous set of equations:Docket: 511607-2030 (8)the fixed point ^^^,^^,^^, ^^^ ൌ , the eigenvalues of ^^^^^^ are 1 േ ^^Ω, െ^^ േ ^^^^^ଶ െ ^^ଶ, and 0.The conjugate pair of 1 േ ^^Ω corresponds to the oscillator used for theexternal forcing, which has a resonance frequency of Ω. Forconsidered here, ^^ ≪ ^^; thus, it is possible to write െ^^ േ ^^^^^ଶ െ ^^ଶ ^ േ^^^^. With thisapproximation, the conjugate pair േ^^^^ Duffing oscillator witha fundamental frequency equal to ^^. Finally, the equal to zero corresponds to the ^^ state. This state is plastically deformable, so it is not stable nor unstable. This analysis applies to both the softening and hardening monostable Duffing oscillator.

[0078] Bistable Duffing Adaptive Oscillator. By modifying the monostable Duffing adaptive oscillator with Eq. (7), Eq. (5) can be written as an autonomous set of equations: (9) and Eq. (5a) as:Docket: 511607-2030.To^^ ൌ 1. For thefixed point ^^^,^^,^^, ^^^ ൌ ^ఠ, 0,0 ଶ ଶ^^య ,0^, the eigenvalues of ^^^^ are 1 േ ^^Ω, െ^^ േ ^^^2^^ െ ^^ , and0.

[0079] The 1 േ ^^Ω corresponds to the oscillator represented by Eq. (7)used for the external forcing, which has a resonance frequency of Ω. For the underdampedcase considered here, ^^ ≪ ^^; thus, it is possible to write െ^^ േ ^^^2^^ଶ െ ^^ଶ ^ േ√2^^^^. Withthis approximation, the conjugate pair േ√2^^^^ corresponds to the bistable Duffing oscillator with a fundamental frequency equal to√2^^. Finally, the eigenvalue that is equal to zero corresponds to the ^^ state. Again, this state is plastically deformable, so it is not stable nor unstable. It should be noted that this analysis is only valid when the oscillator is oscillatingabout one side of the potential barrier at ^^ ൌ 0. As the oscillator begins to cross the potentialbarrier, the resonance frequency of the oscillator experiences a bifurcation. Note that in FIGS.6A and 6B, the oscillator begins jumping the potential barrier when ^^ is approximately√2Ω.Experimental Results

[0080] A field-programmable analog array (FPAA) is used to experimentally validate the adaptive oscillator. FPAAs are highly reconfigurable analog circuits that utilize switched-Docket: 511607-2030 capacitor technology. Due to this reconfigurability, many nonlinear oscillators have been experimentally realized using FPAAs, including the 4-state Hopf adaptive oscillator, the chaotic adaptive pendulum, the van der Pol oscillator, and the Lorenz system.

[0081] Schematics for the monostable and bistable Duffing adaptive oscillators are shown in FIGS.9 and 12A-12B, respectively. These FPAA circuits utilize severalconfigurable analog modules (CAMs), including integration (^ ), multiplication (ൈ), sample-and-hold (^^ି^), summation (Σ), inverse gain (െG), divider (^^^^^^), square root (√), boxcar integrator (consisting of an integrator, comparator, and sample-and-hold), zero-crossing detector, and differentiator (^^). The Anadigm QuadApex board (Anadigm, Paso Robles, CA) has four FPAA AN231E04 chips. The monostable Duffing adaptive oscillator was implemented with two of these chips, while the bistable Duffing adaptive oscillator was implemented with all four. National Instruments 9263 and 9201 modules were used to send a forcing function, ^^^^^^, from MATLAB and collect the states’ voltages back to MATLAB, respectively. Compared to the monostable Duffing adaptive oscillator circuit, the bistable circuit requires considerably more components, but it can still be implemented on a single Anadigm QuadApex board.

[0082] Monostable Duffing Adaptive Oscillator Experiment. An example of the response of the FPAA circuit is shown in FIG.10 for a monostable hardening Duffing adaptive oscillator. For the chosen forcing amplitude and forcing frequency, the ω state learns the forcing frequency, Ω. The state space of ^^ and ^^ are shown in the right portion ofthe figure. For ^^ ൌ 0.06 and Ω ൌ 750 Hz, the ^^ state learns the forcing frequency. Thebolder lines correspond to the forcing function, ^^ sin^Ωt^, and the thinner lines correspond tothe adaptive oscillator’s respective states. FIG.11 illustrates examples of the frequency- amplitude relationship of the FPAA circuit experiment for the Duffing adaptive oscillator. For small forcing amplitudes, the ^^ state learns the forcing frequency; for larger forcing amplitudes, the ^^ state learns the resonance condition instead. The hardening Duffing AO isshown on the left. This can be compared to FIG. 4A. Here, ^^ ൌ 0.25, ^^ଷ ൌ 1, and ^^ఠ ൌ 0.75.The softening Duffing AO is shown on the right. This can be compared to FIG. 3A. Here, ^^ ൌ0.1, ^^ଷ ൌ െ0.1, and ^^ఠ ൌ 0.5.

[0083] Bistable Duffing Adaptive Oscillator Experiment. The bistable Duffing adaptive oscillator can also be constructed using an FPAA circuit as well, which is depicted in FIG.12A. This circuit needs twice as many FPAA chips on the Anadigm QuadApex board, as the ^^ state is more complex for this case. The zero-crossing detector and boxcar integrator are used to capture the ^^^௧^value in the experiment, but other electrical components would work similarly. Another example of a bistable Duffing adaptive oscillator is depicted in FIG.12B.Docket: 511607-2030

[0084] Discussion. The monostable and bistable Duffing adaptive oscillators were proposed and studied. Both the hardening and softening monostable Duffing adaptive oscillators were considered. For both these cases, the monostable Duffing adaptive oscillator did not exhibit the characteristic hysteresis of the non-adaptive Duffing oscillator. The amplitude of the monostable Duffing oscillator follows the maximal value of the backbone curve of the non-adaptive monostable Duffing oscillator. Rather than learning the external forcing frequency, Ω, the monostable Duffing adaptive oscillator learns the resonance condition that provides the maximal amplitude. This behavior is comparable to the pendulum, which has a softening Duffing-like form when the method of multiple scales is used to approximate its equation: (10)only on as affects both the linear and nonlinear terms simultaneously.

[0085] The bistable Duffing adaptive oscillator has the added complication that the intrawell and interwell frequencies are different; moreover, the oscillations are centered about different locations during the intrawell and interwell cases. An ^^^௧^term was introduced to approximate the center of the oscillations for both cases, and it was found that the bistable Duffing adaptive oscillator first learns the intrawell resonance frequency to induce well-hopping. After well-hopping begins, the ^^ state then learns an interwell resonance frequency. When ^^ oscillates too far from the Ω, it can cause the ^^ state to become briefly “stuck” in a well. This results in a bursting phenomenon. On the top of FIG. 13A, the ^^ state is plotted together with the ^^^௧^for reference. The ^^ state is initially stuck inone well, but the ^^ state eventually drives it to a resonance condition when ^^ ^ 0.18seconds. On the bottom of FIG.13A, the ^^ state is plotted with Ω for reference. Instead of converging to Ω, the ^^ state oscillates near Ω. A bursting phenomenon occurs when ^^ gets too far away from Ω. On the top of FIG.13B, the ^^ state is initially oscillating within the twowells, but the ^^ state eventually drives it to a resonance condition just after ^^ ^ 0.4 seconds.On the bottom, the ^^ state is plotted with Ω for reference. Instead of converging to Ω, the ^^state oscillates near Ω⁄ √2.

[0086] These Duffing adaptive oscillators exhibit large oscillations over a wide range of frequencies. For this reason, these Duffing adaptive oscillators are ideal as energy harvesters. Further, many common engineering structures can be modeled with the Duffing oscillator, so it is somewhat straightforward to convert many of these systems into an adaptive form. In hardware, the ^^^௧^term can be easily approximated with a runningDocket: 511607-2030 average, which was demonstrated in hardware using a boxcar integrator and zero-crossing detector.

[0087] More generally, an example of physical computing is presented. Adaptive oscillators can be used as physical reservoir computers, which have preferential behaviors. For instance, they possess the ability to self-learn (which allows them to be much more robust than non-adaptive oscillators), and the adaptive state can be directly tapped into to be used as a morphable logic gate. Thus, the Duffing adaptive oscillator can also be a powerful physical reservoir computer as well. Adaptive Stochastic Resonance

[0088] Stochastic resonance is a phenomenon possessed by some nonlinear oscillators, in which a weak signal is boosted by noise. Both the monostable and bistable Duffing oscillators can exhibit this property. Due to the prevalence of stochastic resonance in biological systems, it is theorized that it provides an evolutionary advantage in processing signals in an inherently noisy environment. However, stochastic resonance has a strong frequency-dependence, as only a band of frequencies may be boosted by noise. On the other hand, adaptive oscillators are a subset of nonlinear oscillators that can learn features of an external force. Here, an adaptive state is added to a Duffing oscillator. This adaptive state enables the Duffing adaptive oscillator to exhibit stochastic resonance over a wide range of frequencies by learning a resonance condition.

[0089] Introduction. Stochastic resonance is a counterintuitive phenomenon in which noise can amplify a weak signal by exploiting a nonlinear system. As both noise and nonlinearity are ubiquitous, it has been found in many seemingly disparate fields. Importantly, stochastic resonance has been shown to play an active role in biological sensors, such as mechanoreceptors in crayfish and tactile sensation in humans, as well as in biological neural networks, such as the cricket cercal sensory system and the human brain. Pointing to biomimetic sensors, stereocilia have been shown to exhibit stochastic resonance as well.

[0090] Stochastic resonance has been exhibited in an optomechanical system and bistable nanomechanical oscillators. Stochastic resonance is also an important phenomenon in quantum system, climate modeling, chemical systems, and many others, which even includes ant foraging models. Noise also modifies the relationship of the phase between the Duffing oscillator and the external force, and noise also modifies the shape of the hysteresis curve of the monostable Duffing oscillator. There are many ingenious methods of realizing a Duffing equation in practice. However, stochastic resonance is still not used in many real- world applications because it is difficult to tune stochastic resonance in continuous systems.

[0091] In contrast to the Duffing oscillator that has static stiffness terms, adaptive oscillators can learn and store information in dynamic plastic states. This process modifiesDocket: 511607-2030 their resonance. Several examples of adaptive oscillators include the Hopf adaptive frequency oscillator, the 4-state adaptive oscillator, and pendulum adaptive oscillator. Related to the adaptive oscillator’s computational ability, the Hopf and van der Pol oscillator have also been utilized as a physical reservoir computer.

[0092] While stochastic resonance could be a powerful phenomenon in the realms of computing and signal processing, it often has limited applications due to its narrow window of efficacy. Here, the Duffing adaptive oscillator is explored in order to quantify the range of its stochastic resonance effect.

[0093] Equations of Motion. The forced monostable Duffing oscillator can be written as: (11) Here, ^^ is theterm, and ^^^^^^ For^^ଷ ^ 0) is shown in this disclosure, but similar results are expected for both the softeningand bistable cases as well. Now, writing Eq. (1) in state space as:

[0094] constant is removed to prevent ^^-dependent damping in the ^^^ equation. The monostableDuffing adaptive oscillator is now written as: To investigatea sinusoid added with white Gaussian noise, such that: (14)

[0095] In Eq.theamplitude of the noise, and ^^^ is the derivative of a Wiener process. Since the Wienerprocess is not differentiable because it is nowhere smooth, Eq. (13) can be rewritten in an incremental form as follows:Docket: 511607-2030Maruyama method, which is an extension of Euler’s method to systems that have noise. The noise is chosen such that it has a mean equal to zero and a standard deviation equal to√^^^^.

[0096] Simulation Results. In this section, the Euler-Maruyama method is used to numerically integrate the Duffing oscillator and the Duffing adaptive oscillator to find the relationship between the forcing frequency and noise amplitude on the stochastic resonance. The signal-to-noise ratio (SNR) is used to measure the stochastic resonance effect. A log scale is used for the following SNR figures.

[0097] For comparison, the non-adaptive Duffing oscillator in Eq. (1) was first simulated using the Euler-Maruyama method. The results are shown in FIG.14. For the non-adaptive Duffing oscillator, stochastic resonance is observed over a relatively small frequency range.Here, ^^ ൌ 0.05, ^^^ ൌ 1, ^^ଷ ൌ 1, and ^^ ൌ 0.1. The stochastic resonance effect can be seen fora relatively narrow band of frequencies. Notably, higher frequencies require a larger amount of noise for stochastic resonance to occur, but this effect is diminished for larger amounts of noise. A portion of the stochastic resonance effect is attributable to the hysteresis curve. For higher frequencies, a noise-hardening effect shifts the resonance past the hysteresis curve with no noise.

[0098] The Duffing adaptive oscillator in Eq. (3) was also simulated using the Euler- Maruyama method, and the results are shown in FIG.15. For the Duffing adaptive oscillator, stochastic resonance is observed over a wide range of frequencies. For large values of ^^,the noise prevents the adaptive oscillator from correctly learning. Here, ^^ ൌ 0.05, ^^ଷ ൌ 1,^^ఠ ൌ 0.1, and ^^ ൌ 0.1. For the Duffing adaptive oscillator, the adaptive state learns aresonance condition, which aids in boosting the external deterministic signal. The stochastic resonance region is extended because of the ^^ state. Again, for very large noise values, the signal-to-noise ratio is diminished.

[0099] Experimental Results. The Duffing oscillator and the Duffing adaptive oscillator are implemented as field-programmable analog array (FPAA) circuits. FPAAs are reconfigurable analog circuits, which use switched-capacitor technology. Because of their reconfigurability, FPAAs have been used to construct other nonlinear oscillators, such as the Lorenz system, the van der Pol oscillator, the 4-state Hopf adaptive oscillator, and the chaotic adaptive pendulum.Docket: 511607-2030

[0100] A circuit schematic for the Duffing adaptive oscillator is shown in FIG.16. This circuit uses several configurable analog modules (CAMs), including inverse gain (െG),multiplication (ൈ), integration (^ ), summation (Σ), and sample-and-hold (^^ି^). The Duffingadaptive oscillator is implemented using two of the four FPAA AN231E04 chips on an Anadigm QuadApex board (Anadigm, Paso Robles, CA). The circuit schematic for the non-adaptive Duffing oscillator replaces the ^^^ portion of the schematic shown in FIG. 16 with aDC voltage to create a constant resonance frequency. The forcing function, which includes both a sinusoid and noise, is constructed in MATLAB and sent to the FPAA using a National Instruments 9263 module. A National Instruments 9201 module was used to collect the states’ voltages back to MATLAB for processing.

[0101] A frequency-amplitude response was performed for both the non-adaptive Duffing oscillator and the Duffing adaptive oscillator, and the results are shown in FIG.17. Both the upsweeps and downsweeps were implemented to exhibit the hysteresis present in the Duffing system, which is absent in the Duffing adaptive oscillator. For both oscillators,^^ ൌ 0.4, ^^ଷ ൌ 1.25, and ^^ ൌ 0.11. For the Duffing oscillator, ^^^ ^ 506 Hz, and for the Duffingadaptive oscillator, ^^ఠ ൌ 0.25. As the monostable Duffing oscillator has a hardeningstiffness, there is a hysteresis when comparing the frequency sweep in the upward and downward directions. The Duffing adaptive oscillator, on the other hand, does not have any hysteresis when comparing the frequency sweep in the two directions. It should be noted that the amplitude of the Duffing adaptive oscillator nearly coincides with the non-adaptive Duffing oscillator’s maximal amplitude.

[0102] For comparison, the non-adaptive Duffing oscillator circuit was first explored. The results are shown in FIG.18. For the non-adaptive Duffing oscillator circuit, stochasticresonance is observed over a relatively small frequency range. Here, ^^ ൌ 0.4, ^^^ ^ 506 Hz,^^ଷ ൌ 1.25, and ^^ ൌ 0.11. Similar to the simulations, the stochastic resonance effect can beseen for a relatively narrow band of frequencies. The Duffing adaptive oscillator circuit represented in FIG.16 was also experimentally tested. The results are shown in FIG.19. For the Duffing adaptive oscillator circuit, adaptive stochastic resonance is observed over a broad range of frequencies. In comparing the amplitude of the SNR in FIG.18, the amplitude of the SNR is for the Duffing adaptive oscillator is higher over a large range of bothfrequencies and noise amplitudes. Here, ^^ ൌ 0.4, ^^ఠ ൌ 0.25, ^^ଷ ൌ 1.25, and ^^ ൌ 0.11. Theadaptive stochastic resonance effect can be seen over a broad range of frequencies and noise amplitude values.

[0103] In FIG.18, stochastic resonance can be observed in a relatively narrow band, and higher frequencies are not significantly boosted. However, in FIG.19, the adaptive stochastic resonance provides a boosted signal for a wide band.Docket: 511607-2030

[0104] Stochastic resonance is a phenomenon in which a weak signal is boosted by noise. This phenomenon is utilized by many biological systems, but its use in technology is limited due to its frequency-dependence. This frequency-dependence may be observed in FIGS.14 and 18. Stochastic resonance is due to the nonlinearity of the oscillator. By concatenating the Duffing oscillator with an adaptive state, an adaptive form of stochastic resonance is achieved that overcomes this limitation. Adaptive stochastic resonance allows a stochastic resonance effect to occur over a wide range of frequencies.

[0105] As noise is ubiquitous, stochastic resonance is an important nonlinear effect. This adaptive stochastic resonance effect has direct applications to signal processing and high quality factor actuators & sensors. Moreover, the Duffing adaptive oscillator is a powerful physical reservoir computer architecture. This adaptive stochastic resonance allows it to be more robust in noisy environments. Duffing Adaptive Oscillator Physical Reservoir Computer

[0106] The Duffing adaptive oscillator is constructed as a mechanical system. Of particular interest, this system is capable of both physical reservoir computing and analog signal processing. Here, a mechanical realization of the Duffing adaptive oscillator is presented, and its potential as a physical reservoir computer is demonstrated using several tasks. To gain a better understanding of this oscillator, continuation methods are used to compare its dynamics with the non-adaptive Duffing oscillator.

[0107] Introduction. Adaptive oscillators (AOs), as the name implies, are a type of nonlinear oscillator with adaptive states. AOs can both learn and store information from an external stimuli in these plastic states. These oscillators have typically been utilized for central pattern generators and analog frequency analyzers. Adaptive oscillators are composed of a base oscillator and one or more adaptive states. The most common adaptive state is frequency, but two other popular adaptive states are amplitude and learning rate for the adaptive oscillator itself. The most common base oscillators are the Hopf and van der Pol oscillators since they can adjust their phase to synchronize with an external sinusoidal force, but the pendulum adaptive oscillator has been implemented as well. For the Hopf adaptive oscillator, analysis using the full Fokker-Planck equation was compared with experiments to explore the effects of noise on learning.

[0108] In this disclosure, the Duffing oscillator is implemented for the first time as an adaptive oscillator. The Duffing oscillator appears in a wide range of physical systems. The Duffing oscillator is a very important nonlinear oscillator, as it exhibits hysteresis, stochastic resonance, and chaotic motion. Further, arrays of Duffing oscillators can exhibit intrinsic localized modes; these are persistent vibratory modes that are spatially localized. Intrinsic localized modes have been proposed as a method of sending packets of energy through an array.Docket: 511607-2030

[0109] Physical reservoir computing is a type of neuromorphic computing that repurposes the nonlinear dynamics of a physical system as a computational resource. Several physical systems have been used as physical reservoir computers, such as a Duffing array, the van der Pol oscillator, a harmonic array, a Mackey-Glass oscillator, soft robots, and quantum systems.

[0110] Adaptive oscillators possess a form of dynamic intelligence that is notably different to other types of vibratory control or phase locked loops. Adaptive oscillators’ dynamic learning, which is typically used for analog frequency analysis, can be repurposed as a type of self-learning physical reservoir computing for reconfigurable tasks. Additionally, the adaptive oscillator can also be used as a multiplex-free, morphable logic gate. Taking these two different tasks together, this points to the potential of the AO to be used as a powerful processor for both AI inference and traditional computing tasks simultaneously. With this in mind, a mechanical realization of a Duffing adaptive oscillator is proposed that has nearly separable linear and nonlinear stiffness terms. By scaling this system to the nano or micro range, very fast mechanical computing can be unlocked.

[0111] Equations. The Duffing oscillator can either be monostable hardening, monostable softening, or bistable. (It should be noted that there is a fourth unstable case.) The monostable Duffing oscillator can be written as: (16) Here, theand the external force is ^^^^^^. Here, the mass has been divided throughout the equation. Eq. (16) can be written in state space as: Eq. (17) canIn Eq. (18), thedynamic state, ^^. The ^^ఠterm is a constant, which affects the learning rate of the frequency state.

[0112] Experimental Setup. An experimental Duffing adaptive oscillator is designed, constructed, and tested. This system comprises a cantilevered beam, which has a movable roller to modify the beam’s length. A set of magnets (e.g., one on the free end of the beamDocket: 511607-2030 that moves with the beam and a static magnet that is attached above it) provide a Duffingstiffness term. A hardening monostable Duffing term (^^^ ^ 0 and ^^ଷ ^ 0) is demonstratedhere, but modifications to this experiment would allow both a softening monostable and bistable realizations.

[0113] The effective mass of the beam is given by: (19) The beam’s mass,magnet’s mass, the range of deflections used in this experiment. The linear stiffness term is given by: (20) Here, ^^ is the Young’sof the beam. For the rectangular cross section beam used in the experiment, ^^ ൌ^ଶ where ^^ and ^^ are the width and thickness of the beam, respectively. Plugging this into Eq. (20): Assuming that theof the cantilever, ^^, then Coulomb’s law for magnetic charges can be used to obtain the force between the magnets as:

[0114] permeability, andis the gap distance between the magnet at the tip of the beam (at zero beam deflection) and the magnet on the frame. Thus, the magnetic force, ^^^^^, provides a linear and Duffing force. As the beam’s length can be modified to counteract the linear term from the magnetic force, this setup allows the linear and Duffing terms to be modified separately. By choosing attracting magnets, a hardening Duffing term is achieved, while, by choosing repelling magnets, a softening Duffing term is achieved. Further, these magnets can be either permanent or electromagnetic. For the experimental setup demonstrated here, the magnets are attracting, permanent magnets; the magnet attached to the frame can beDocket: 511607-2030 manually adjusted with a screw to set the gap distance, ^^^. By combining the linear and Duffing stiffness from Eqs. (21) and (22), the total linear and Duffing stiffness terms can be written as: Thus, the naturalThus, by using Eq.setup by modifying setup FIG.20. Duffing adaptive oscillator is constructed as a mechanical experimental prototype. A cantilevered beam is secured to a fixture, while a linear bearing with two rollers is adjusted with a motor with a screw to modify the beam’s length. An adjustable magnet above the beam can create a hardening monostable, softening monostable, or bistable Duffing term.

[0115] The non-adaptive Duffing oscillator has a hysteresis in its’ frequency response, and it only has a high amplitude at a smaller band of frequencies. This frequency-amplitude responses for various non-adaptive Duffing oscillators are shown in FIG.21. Frequency sweeps were performed on the mechanical prototype, with the beam's length set to multiple static values. For these frequency sweeps, the forcing frequency was slowly varied. The beam's amplitude of oscillations is plotted against the forcing frequency. The sampled amplitudes for each frequency are plotted for the upsweep (circles) and downsweep (Xs), and a curve is plotted to guide the eye.

[0116] The Duffing adaptive oscillator has a large amplitude of oscillation for a broad range of forcing frequencies, which is shown in FIG.22. This is because the Duffing adaptive oscillator can learn the resonance condition to increase its amplitude of oscillation. For a static forcing frequency, the amplitude of the Duffing adaptive oscillator is shown in FIG.23 as it learns the resonance condition. The vertical dashed line represents the point at which it undergoes a phase shift with respect to the external forcing signal.

[0117] Duffing AO Physical Reservoir Computer The Duffing adaptive oscillator given by Eq. (3) can be used as an analog frequency analyzer to obtain the primaryDocket: 511607-2030 frequency component of the external forcing function, ^^^^^^. This quantity is explicitly represented as the cantilevered beam’s length. However, it is much more interesting to consider the Duffing adaptive oscillator as a self-learning physical reservoir computer. To demonstrate this in the experiment, a complicated sinusoidal signal, ℎ^^^^, is sent to the oscillator, and the mechanical Duffing adaptive oscillator’s displacement, ^^^^^^, is recorded. In FIG.24, ℎ^^^^ and ^^^^^^ are shown for the experiment. Nonlinear autoregressive moving average (NARMA) tasks of various orders can then be predicted using the oscillator’s response, ^^^^^^. The predictions for several orders of NARMA tasks are shown in FIG.25.

[0118] To demonstrate this through simulations, the external force is defined as multiplication of a desired signal, such as the Lorenz chaotic time series, multiplied by a sinusoid. Specifically, the Lorenz chaotic time series, ^^ௗ^^^^, is a discretized signal; this is used to define a continuous signal, ^^^^^^^, such that each discretized value is held for ^^^seconds, where ^^^is the pseudo-period or clock cycle: (25)computer, ^^^can be set equal to the inverse of the sampling rate, ^^^, such that ^^^ൌிೞ. This pseudo-period can be used to also define the pseudo-frequency, such that ^^ ଶగ ^ ൌ . During ^்the time ^^^, ℎ^^^^ holds the value of 1 ^ ^^^^^^^. During each pseudo-period, virtual nodes arecollected from the physical reservoir computer. In contrast, real nodes are used for other physical reservoir computers. As examples of this, each oscillator in an oscillator array or voltage sensors along a wire can be used as physical nodes. However, time multiplexing can be used to create many virtual nodes. A ridge regression can used to train the physical reservoir computer for different tasks. The continuous time series data is first multiplied by a sinusoid, such that: (26) An example of theadaptive oscillator is sent the Lorenz chaotic time series, which is encoded in Eq. (26). On the top, the ^^ state is plotted with the ^^^^^^^ function for reference. On the bottom, the ^^ state is plotted with the ^^^value for reference. The physical reservoir computer self-learns the correct resonance to work effectively as a computer.

[0120] Using this method, a one-step ahead prediction task was performed for the Lorenz time series. This is shown in FIG.27. The Duffing adaptive oscillator physical reservoir computer was trained to complete a one-step ahead prediction task. The root mean square error for the prediction was 0.046.Docket: 511607-2030

[0121] Discussion. In this paper, a mechanical Duffing adaptive oscillator is demonstrated as a physical experiment, and it was implemented as a physical reservoir computer. This system can be scaled to the nano- or micro-scale to increase the frequencies involved for fast, mechanical computing. Several other Duffing adaptive oscillator realizations were also considered, which provide a practical basis for creating other Duffing adaptive oscillators for analog frequency analyzers or physical reservoir computers.

[0122] It should also be noted that the magnetic force in this Duffing adaptive oscillator could be set equal to zero by removing the magnets. This creates a linear adaptive oscillator. Although not the focus of this paper, the linear adaptive oscillator can learn and store information in a similar method as the Duffing adaptive oscillator (e.g., the frequency content can be learned and stored in the ^^ state).

[0123] This Duffing adaptive oscillator could be further extended to include other plastic states, such as amplitude, phase, learning rate, or Duffing stiffness. Although adaptive oscillators have received relatively little attention, their unique learning abilities suggest that they could be a compact and powerful computational resource for both generalized computing and AI inference tasks. Adaptive Stochastic Resonance in the Duffing Adaptive Oscillator Physical Reservoir Computer

[0124] The Duffing adaptive oscillator can be used as a physical reservoir computer, in which the oscillator’s dynamics are used for reconfigurable computing tasks. Since the Duffing oscillator is capable of stochastic resonance, the Duffing adaptive oscillator possesses intrinsic noise-enhanced signal processing due to its nonlinear dynamics. In this paper, the Duffing adaptive oscillator physical reservoir computer is presented, and its robustness to noise is demonstrated. By tapping into the different states of this oscillator, reconfigurable AI inference tasks can be completed, while other states provide morphable logic gates without the need of time-multiplexing. This combination of computing techniques (i.e., powerful & reconfigurable AI inference and fast & morphable logic gates) makes the Duffing adaptive oscillator an ideal physical reservoir computer, which also has superior performance in the presence of noise.

[0125] Introduction. In physical reservoir computing, the nonlinear dynamics of a physical system are repurposed for computation. Physical reservoir computers (PRCs) can be broken into three components: the input (e.g., an external force on the nonlinear system), the reservoir (e.g., the nonlinear system), and the readout function (e.g., the trained output). More than a novelty, physical reservoir computers are an untapped potential for a shift to non-von Neumann architectures, as the dynamics both store memory and performDocket: 511607-2030 computation. By leveraging this non-von Neumann architecture, powerful computing with a small footprint can be realized.

[0126] More impressively, physical reservoir computers can be utilized for both AI inference tasks (for instance, wake word recognition, digit recognition, time series prediction) and for traditional logic gates. Previously, the latter task was done by time-multiplexing a response. However, this time-multiplexing response is cumbersome. Recently, a more streamlined approach was proposed that removes the need for time-multiplexing. Moreover, this approach creates a morphable logic gate, which can act as all of the basic logic gates without changing the base system.

[0127] This new multiplex-free method uses adaptive oscillators as the nonlinear reservoir. Adaptive oscillators are a subset of nonlinear oscillators, which can natively learn and store information in their dynamic states. Notably, this process does not use traditional machine learning, but it is instead a dynamic process. Adaptive oscillators can be constructed from other oscillators by concatenating their dynamic states. They are a likely an ideal candidate for PRCs because of their intrinsic learning abilities.

[0128] Previously, adaptive oscillators have been used for analog frequency analysis, central pattern generators, robotic locomotion control, medical gait analysis, and vibratory energy harvesters. Although relatively little work has been pursued on adaptive oscillators (especially experimental work), several adaptive oscillator experiments have been previously constructed. These include a four-state Hopf adaptive oscillator, three-state Hopf adaptive oscillator, mechanical pendulum adaptive oscillator, electrical pendulum adaptive oscillator, and adaptive Duffing oscillator, which is considered here.

[0129] Stochastic resonance is a phenomenon in which a weak signal is boosted by noise by exploiting nonlinearity. Notably, stochastic resonance is an important phenomenon in biological sensors (for instance, mechanoreceptors in crayfish and tactile sensation in humans) and biological neural networks (e.g., the cricket cercal sensory system and the human brain). Stochastic resonance has been demonstrated in a wide range of physical systems, including an optomechanical system, and bistable nanomechanical oscillators, quantum systems, climate modeling, chemical systems, and even ant foraging models. While stochastic resonance is prolific in nature, it is not properly utilized in technology because the phenomenon is only observed for a limited range of frequencies.

[0130] While reservoirs provide robustness to noise, noise enhancement has not been previously demonstrated. Taking inspiration from the Duffing oscillator’s noise enhancement capabilities, a Duffing adaptive oscillator is constructed as a physical reservoir computer. Adaptive stochastic resonance is observed, in which noise boosts the signal over a wide range of frequencies. Adaptive stochastic resonance is more applicable to technologies related to physical computing hardware, as it is efficacious over a wide range of frequenciesDocket: 511607-2030 and enhances the computational abilities in the presence of noise. Moreover, as the weak signal is amplified, adaptive stochastic resonance provides a technique to reduce the power requirements for computing.

[0131] The Duffing physical reservoir computer will be described. The results of simulations and the experiment are provided. The exclusive or (XOR) task is used to demonstrate the adaptive stochastic resonance in the Duffing physical reservoir computer. The theoretical and technological ramifications of this are discussed. The normalized information rate, which is used as the metric for the physical reservoir computer’s XOR calculation, is outlined in detail as well.

[0132] Results. The nondimensionalized Duffing equation can be written as follows: (27) Here, ^^ is theand ^^^^^^ is the space:When ^^ଷ ^ 0,monostable Duffing oscillators have multiple dynamic solutions and hysteresis, and classical stochastic resonance can be observed. The case considered in this paper is the hardening monostableDuffing oscillator, in which ^^ଷ ^ 0.

[0133] The monostable Duffing adaptive oscillator can be obtained by concatenating anadaptive frequency state, ^^^ , as follows:In Eq. (29), thegoverned by ^^^ . A constant, ^^ఠ, affects the learning rate of the ^^ state.

[0134] Now, the external force is set equal to an encoded signal with additive white Gaussian noise: (30) ^^^^^^ encodes the bits,corresponds to sin^Ωி^^^. The white Gaussian noise has an amplitude of ^^, where ^^^ is thederivative of a Wiener process. The Wiener process is nowhere smooth, and so it is not differentiable. Thus, Eq. (29) can be rewritten in an incremental form:Docket: 511607-2030method is then used to numerically integrate Eq. (31); this method is an extension of Euler’s method to stochastic ordinary differential equations. The mean of the noise is equal to zero, and the standard deviation of the noise is equal to√^^^^, where ^^^^ is the step size of the integration.

[0136] To highlight adaptive stochastic resonance, an exclusive OR (XOR) logical task was chosen. “False” was encoded as a frequency of Ω^^^^^ൌ 2 Hz, and “true” was encodedas a frequency of Ω௧^௨^ ൌ 2.02 Hz. This signal, ^^^^^^ ൌ sinΩ^^^, was sent to the Duffingadaptive oscillator. The ^^ state was modified to produce ^^^^ௗ, such that:

[0137] In theInstead, a single physical real node is used within each clock cycle (e.g., pseudo-period). The beginning of each clock cycle is depicted by a dashed, vertical line. FIG.28 shows the response of the Duffing adaptive oscillator is shown here when sending the encoded information to system as ^^^^^^. In the top plot, the ^^ state is shown. In the middle plot, the ^^ state is shown. The Ω௧^௨^and Ω^^^^^values are depicted by dashed, horizontal lines. In the bottom plot, ^^^^ௗis plotted, where the Xs denote the physical node that is used in the ridge regression. Using this real node and a bias in the ridge regression, ^^^^ௗcan be used to emulate an XOR task. Using the nodes plotted in FIG.28 and a bias, a ridge regression was used to train the physical reservoir computer to output an exclusive OR task shown in FIG. 29.80% of the time history was used for training, and 20% of the time history was used for testing. Using this method, the physical reservoir computer performed the XOR task with perfect accuracy.

[0138] This XOR task was then used to explore whether the Duffing adaptive oscillator can exhibit adaptive stochastic resonance. In FIG.30, the forcing amplitude, ^^, is plotted against the training data set size, ^^. In each case, the information rate, ^^^^, is used as the metric for an exclusive OR (XOR) task. The performance of the PRC is compared without(^^ ൌ 0) and with (^^ ൌ 0.05) noise. With a training data set size of less than approximatelyDocket: 511607-2030 200, the PRC does not work. While the noise degrades performance of larger values of the external force, it also enhances the performance for some values of ^^. For instance, the 4th plotted column from the right has a higher ^^^^ for the noise case as compared to the no noisecase above approximately ^^ ൌ 600. This adaptive stochastic resonance phenomenon isshown at approximately ^^ ^ 0.55. For larger values of ^^, the ^^ ൌ 0.05 case has larger valuesof the ^^^^ as compared to the ^^ ൌ 0.00 case.

[0139] Thus, for some combinations of parameters, the Duffing adaptive oscillator can exhibit adaptive stochastic resonance. This phenomenon is similar to the stochastic resonance effect found in the non-adaptive Duffing oscillator, but this adaptive stochastic resonance effect has a much broader range of applications.

[0140] Discussion. For the Hopf physical reservoir computer, it was found that its computational abilities are enhanced by resonance criteria. Arnold tongues are synchronization regions that appear in the nonlinear oscillator, which reappear in the Hopf physical reservoir computer as regions of high computation. Numbers from the Farey sequence also appear when studying the synchronization of a nonlinear oscillator with an external force, and this Farey sequence is also mirrored in the computational ability of the Hopf physical reservoir computer. This previous work provides insights into creating robust physical reservoir computers by taking advantage of resonance criteria. However, it also poses a significant challenge, as these resonance conditions require rather precise frequencies. For Arnold tongues, robustness is gained at the cost of an amplified signal, which increases the energy requirements.

[0141] There are multiple types of plasticity in neuromorphic computing. At the neuronal level, spike-frequency adaptation is governed by the relationship between a neuron’s spiking rate and a steady state stimulus. This is quite different than an adaptive oscillator’s dynamics, as the neuron itself is not learning information but only exhibiting different responses based on external stimuli. The adaptive oscillator, on the other hand, is learning and storing information in a plastic state. When an array of spiking neurons are coupled with synapses, they form a spiking neural networks. Spiking neural networks have another type of plasticity, in which the synapses can be modified to store information.

[0142] In light of the recent discovery of adaptive stochastic resonance, this paper shows that this phenomenon can also be used to enhance the computational ability of a physical reservoir computer. A Duffing adaptive oscillator is used as the reservoir, and a XOR task was used to quantify this adaptive stochastic resonance phenomenon in the context of physical reservoir computing. Other Duffing Adaptive Oscillator Designs

[0143] The Duffing adaptive oscillator can be implemented in other configurations, and these other configurations can be utilized as physical reservoir computers.Docket: 511607-2030

[0144] Josephson Junction Duffing AO. Josephson junctions consist of two superconductors that are weakly coupled by a region that may be either non- superconducting or a weaker superconductor. Josephson junctions exhibit macroscopic quantum phenomena. Further, Josephson junctions are often modeled as Duffing oscillators. They convert a DC voltage into high frequency electromagnetic oscillations, such that: (33) where ^^ is the voltage per Josephson junctions can beproportional to a DC voltage, a Josephson junction-based Duffing adaptive oscillator can be constructed using the same methodology as described for the cantilever beam. As Josephson junctions can exhibit frequencies into the terahertz, the Josephson junction- based Duffing adaptive oscillator would be an extremely fast physical reservoir computer.

[0145] Spline Duffing AO. Considering the Duffing beam above, the potential energy curve can be constructed from the springs. On the other hand, a spline can be used to directly construct the potential energy curve instead. In this Duffing AO, a rail that is in the shape of a spline of the approximate potential energy curve; this rail is actuated at locations along the spline to modify the potential energy function’s curvature. Examples of the spline are depicted in FIG.31. The potential energy of the Duffing adaptive oscillator can be approximated by a spline. On the top, the monostable Duffing adaptive oscillator is depicted; while, on the bottom, the bistable Duffing adaptive oscillator is depicted. The arrows correspond to possible locations of actuators that are used to modify the spline rail in accordance with the adaptive oscillator’s equations of motion. The ball corresponds to a mass cart that moves along the rail. This spline Duffing AO could be constructed of a flexible rail with a linear bearing or cart, or it could be constructed of a rolamite.

[0146] Gyroscope Duffing AO. Gyroscopes, which are commonly used for GPS and other aerospace applications, can be modeled as a Duffing oscillator. By changing the electrostatic force of the gyroscope’s comb, the gyroscope can be implemented as a Duffing adaptive oscillator.

[0147] Clamped-Clamped Beam Duffing AO. The clamped-clamped beam can be modeled as a Duffing oscillator. By applying a tension or compression to the beam at the ends, this system can be realized as a Duffing AO.

[0148] Large Deformation Cantilever Duffing AO. In this paper, magnets were used to create a Duffing potential. However, large deformation cantilevers can also be modeled as Duffing oscillators, even without magnets. The linear natural frequency of these beams canDocket: 511607-2030 be modified by changing the length. These large deformation cantilevers could be constructed as Duffing AOs using a similar method.

[0149] String Duffing AO. For a modal approximation, the vibrations of strings (e.g., guitar strings) can be modeled as a Duffing oscillator. By varying the tension, the first linear natural frequency of the string can be tuned. By forcing the string with an electromagnetic force, a Duffing AO can be constructed from the string for physical reservoir computing.

[0150] Helmholtz AO & Helmholtz-Duffing AO. Whereas the Duffing oscillator has a cubic nonlinearity, the Helmholtz oscillator has a quadratic nonlinearity. The ear drum is an example of a Helmholtz oscillator, as it is an asymmetric oscillator. A realization of a Helmholtz adaptive oscillator is shown in FIG.32. In the Helmholtz oscillator here, adaptation can be included in the system by adjusting the vertical spring.

[0151] Crystal Duffing AO. Quartz crystals are commonly used for timing oscillators in electronics. These crystals are modeled as Duffing oscillators. Their linear resonance can be modified by mechanically loading the crystal. This effect can be used to create a crystal Duffing AO.

[0152] Simulations of Duffing AO. The Duffing adaptive oscillator can also be simulated, with several examples of this provided above. These simulations can be used as the reservoir computer for software-based machine learning techniques.

[0153] Other Aspects of Technology Integration. The Duffing adaptive oscillator can also be implemented in tandem with other techniques. For instance, the Duffing adaptive oscillator can be coupled to an optical cavity for optomechanical or optoelectrical physical reservoir computing.

[0154] Moreover, higher harmonics for spatially continuous systems can be modeled as Duffing oscillators (e.g., higher harmonics for crystal oscillators). These higher harmonics can be converted to adaptive oscillators as well, which effectively creates an array of Duffing oscillators from a single physical oscillator. Each of these higher harmonics can be utilized as a separate Duffing adaptive oscillator physical reservoir computer simultaneously. Unique Aspects of the Duffing Adaptive Oscillator

[0155] The Duffing adaptive oscillator can be constructed from a Duffing oscillator. The Duffing oscillator has three configurations that are not divergent: monostable hardening(^^^ ^ 0, ^^ଷ ^ 0), monostable softening (^^^ ^ 0, ^^ଷ ^ 0), and bistable (^^^ ^ 0, ^^ଷ ^ 0). Allthree of these configurations can be implemented as a Duffing oscillator. The bistable case uses a moving average (or other signal processing or analytical technique) to approximate the center of the oscillations.

[0156] The Duffing adaptive oscillator exhibits adaptive stochastic resonance. While nonadaptive stochastic resonance is dependent on the resonance frequency of the DuffingDocket: 511607-2030 oscillator, adaptive stochastic resonance boosts a weak signal using noise over a broad range of frequencies.

[0157] Adaptive stochastic resonance can be used for weak signal detection in a wide variety of sensors, including sensors for mechanical, electrical, optical, and quantum systems. For instance, adaptive stochastic resonance can be used with cell phones to use electromagnetic noise to boost a weak signal.

[0158] Noise energy harvesting can be utilized through adaptive stochastic resonance.

[0159] The Duffing adaptive oscillator can be implemented from many systems, including a mechanical beam, electrical circuits (e.g., FPAA, PCB, VLSI), Josephson junctions (or arrays of Josephson junctions), actuated splines, gyroscopes, clamped- clamped beams, large deformation cantilevered beams, large deformation strings, Helmholtz-Duffing oscillators (with additional quadratic spring stiffness), and crystals (e.g., timing crystals already used in electronics). The Helmholtz adaptive oscillator could be implemented using the same methodology. Unique Aspects of the Duffing AO Physical Reservoir Computer

[0160] Each of the Duffing adaptive oscillators can be used as a physical reservoir computer. The adaptive states can be used for self-learning, in which they are used to reprogram the adaptive oscillator physical reservoir computer.

[0161] The Duffing adaptive oscillators can be used as a physical reservoir computer with time multiplexing.

[0162] The Duffing adaptive oscillators can be used as a physical reservoir computer without time multiplexing.

[0163] The Duffing adaptive oscillator physical reservoir computer can perform generalized calculations, including time-series tasks and morphable logic tasks. These logic gates are utilized without modifying the Duffing adaptive oscillator, but, instead, only a different set of weights (found with a ridge regression or other method) are needed. Thus, multiple logic gates can be computed in parallel, using a single Duffing adaptive oscillator.

[0164] The Duffing adaptive oscillator physical reservoir computer exhibits adaptive stochastic resonance, which can be utilized for computational tasks.

[0165] The Duffing adaptive oscillator physical reservoir computer can be implemented with crystals, commonly used for timing oscillators in electronics, as a morphable artificial intelligence processor (computing such as artificial intelligence inference tasks) & conventional processor (computing such as logic gates).

[0166] The Duffing adaptive oscillator physical reservoir computer can be implemented as Josephson junction(s) as a morphable artificial intelligence processor & conventional processor. This can be implemented using either regular superconductors or, if room temperature superconductors are achieved, room temperature superconductors.Docket: 511607-2030 Other Unique Aspects

[0167] The Duffing adaptive oscillator can also be implemented in tandem with other techniques. For instance, the Duffing adaptive oscillator can be coupled to an optical cavity for optomechanical or optoelectrical physical reservoir computing.

[0168] Spatially-distributed systems have an infinite number of higher order modes. These nonlinear modes can often individually be modeled as Duffing oscillators. Each of these modes can be realized as a Duffing adaptive oscillator for simultaneous physical reservoir computing. Oscillator-based Computer as Physical Unclonable Function

[0169] The adaptive oscillator itself is capable of a type innate neuromorphic processing, as it can dynamically learn and store information about an external signal. When used as a physical reservoir computer, it has been demonstrated that the adaptive oscillator possesses both self-learning and the ability to be used as a multiplex-free morphable logic gate. In addition to these useful traits, the Duffing adaptive oscillator is also a physical unclonable function. Here, a bifurcation diagram is proposed as a novel challenge-response pair. Notably, since the bifurcation diagram is asymptotically approached by the oscillator's dynamics, it is easily reproducible. Further, small changes to the oscillator's parameters produce large changes to the bifurcation diagram, which makes it act as a “fingerprint” of the oscillator. From an applied perspective, this means that the Duffing adaptive oscillator could be implemented as a trusted artificial intelligence hardware device.

[0170] Introduction. Adaptive oscillators are a subset of nonlinear oscillators, which have the ability to both learn and store information in dynamic plastic states. Adaptive oscillators have a wide variety of potential applications, such as robotic locomotion, central pattern generators, and analog signal analyzers. Adaptive oscillators have been implemented as both electrical and mechanical systems.

[0171] On the other hand, physical reservoir computers are a subset of neuromorphic computers, which utilize the dynamics of a physical system for computation. Physical reservoir computing typically requires lower training costs and is robust to overfitting. Physical reservoir computers have been used for many tasks, including digit recognition, time series prediction, wake word recognition, and image recognition. The real power of physical reservoir computing is in its ability to tap into the physics of different physical systems, allowing these systems to act as the physical embodiment of a neural network. Physical reservoir computers have been created from soft robots, optoelectronics, quantum systems, oscillator arrays, and limit cycle oscillators. Notably, by using the adaptive oscillator that possesses innate computing abilities, a physical reservoir computer can be created that both exhibits self-learning and can be used as a multiplex-free morphable logic gate.Docket: 511607-2030

[0172] Physical unclonable functions (PUFs) can be used as cryptographic keys for the authentication of hardware. PUFs may be thought of as a physical object's “fingerprint”. Since physical devices have fabrication variations, these hardware variations can be used to authenticate the hardware itself, which is potentially a practical solution for secret key generation. Many different physical systems have been shown to be PUFs. Quantum readouts, optical systems using shaped wavefronts and light scattering, micromagnet arrays, optical waveguides, spin-orbit-torque-magnetic random access memory, high-capacity crossbar memory, and analog circuits. Some PUFs directly utilize chaos. For instance, transient chaos in a field programmable gate array has been used as a PUF, and a chaos- based ring oscillator PUF was implemented on a field programmable analog array.

[0173] The adaptive oscillator is capable of functioning as a physical reservoir computer, and it is demonstrated here that it is also a PUF. Taken together, this means that the oscillator-based physical reservoir computer is a cybersecure artificial intelligence hardware chip. Moreover, the bifurcation diagram of the oscillator is used to create the challenge-response pair. This allows the challenge-response pair to be arbitrarily difficult to emulate for an attacker.

[0174] Bistable Duffing Adaptive Oscillator Physical Reservoir Computer Equations. The bistable Duffing adaptive oscillator has been shown to function as a physical reservoir computer. The bistable Duffing adaptive oscillator is a Duffing oscillator with an adaptive state that learns other information, such as the frequency. The equation of motion for the non-adaptive bistable Duffing oscillator is given by: (34) where f(t) is theoscillator can be described by equations: ζ is the dampingof oscillations. ω is the resonance frequency of the oscillator, which changesaccording to the ω^ equation, rather than a static term. Since the ^^ state oscillates about anon-zero quantity when the oscillator is trapped in a well, the xୡ^୰is introduced as a correction term. The orbit’s center, xୡ^୰, is approximated by taking a moving average during one period of the oscillator’s ^^ state oscillations.

[0175] To function as a physical reservoir computer, information must be encoded and sent to the oscillator as an external force, ^^^^^^. As the focus of this paper is on the oscillator-Docket: 511607-2030 based computer acting as a PUF, only a single example of the physical reservoir computing ability of the bistable Duffing adaptive oscillator will be shown. A Nonlinear AutoRegressive Moving Average (NARMA) time series prediction task will be implemented. The NARMA equation for an arbitrary function, ^^^^^^, is defined as: (36)(α,series. Here, the arbitrary function ^^ is defined as: (37)(α, β, γ, δ) are arbitrary constants, which are chosen here to be (0.3,0.05,1.5,0.1),respectively, ^^ is the order of NARMA task, and ^^ is the number of steps in the NARMA series. Here, the arbitrary function ^^ is defined as:

[0176] To send the information to the oscillator, ℎ^^^^ is used as an intermediary function to encode the amplitudes. The pseudo-period, ^^^, is defined, which can also be used todefine the pseudo-frequency, such that ω^ ൌ 2π⁄ ^^^ . ℎ^^^^ is defined such that it holds thevalue of ^^^^^^^ for a pseudo-period: (38) The(39)

[0177] Virtualridge regression is used to train the physical reservoir computer for different tasks, with 80% of the data used for training and the remaining 20% used for testing.

[0178] In FIG.33, the predictions from the bistable Duffing adaptive oscillator physical reservoir computer for different orders of NARMA tasks are shown here. The normalizedmean square error (NMSE) for the 5th, 10th, 15th, and 20th order tasks are 1.98 ൈ 10ି^, 2.33ൈ 10ିହ, 2.52ൈ 10ିସ, 1.04 ൈ 10ିଷ, respectively.

[0179] Physical Unclonable Function. To demonstrate that the oscillator-based computer is a PUF, clones of the oscillator will be made. The functionality of the clones will be shown. Next, the original oscillator-based computer will be shown to have robustness to varied initial conditions, while the clones produce drastically different responses to the challenge-response pair.Docket: 511607-2030

[0180] The bistable Duffing adaptive oscillator has three unique system parameters:ζ, ^^ଷ, ^^ன. To create the clones, the system parameters are drawn from the continuousuniform distribution such that the interval is (-1.01^^,+1.01^^), where ^^ is one of the three parameters. Thus, the clone's system parameters are chosen such that they are േ1% of the original oscillator's parameters.

[0181] The bifurcation diagram of each oscillator acts as the “fingerprint”. The challenge-response pair for the oscillator-based computer uses the chaotic response of the oscillator as the response. Thus, the “challenge” is a frequency sweep, and the “response” is a bifurcation diagram. This bifurcation diagram “fingerprint” is quite reproducible for a given oscillator. However, even with small changes in the oscillator's parameters, the bifurcation diagram is quantitatively different for the clones. It should be noted that the bistable Duffing adaptive oscillator can be chaotic, which is verified by the presence of a period-3 orbit. The bifurcation diagram of the original bistable Duffing adaptive oscillator is shown in FIG.34. It should be noted that the bistable Duffing adaptive oscillator can be chaotic, which is verified by the presence of a period-3 orbit. An example of a period-3 orbit is shown in FIG.35 for the bistable Duffing adaptive oscillator to verify that the bistable Duffing adaptive oscillator is chaotic. The trajectory is shown on the top in the x-y-ω space and the oscillations of the ω state are shown on the bottom, with X’s representing the Poincare sections. The horizontal dashed lines are plotted to guide the eye.

[0182] Functionality of Clones. To verify the functionality of the clones, 100 clones were created following the procedure described above. The original oscillator was trained to predict the 5th order NARMA task (see FIG.36), and the weights from the ridge regression were saved. Next, the 100 clones were each forced with the same ^^^^^^ that was used for the original oscillator.

[0183] Without retraining the cloned oscillators, each of the clones were used to predict the NARMA task. Even without retraining, the oscillator-based computer was still able to predict the task. By retraining the weights for each clone, a similar NMSE value could be achieved. These results are shown in the top of FIG.36. By using the weights that were calculated from the original oscillator's prediction, the clones were still able to perform the prediction task. By retraining the weights for each clone, the bottom of FIG.36 shows that the clones' NMSE values was decreased further. In each plot, the vertical dashed line represents the NMSE of the original oscillator. Thus, the clones can function as a physical reservoir computer with or without retraining them, but their functionality is improved if they are retrained.

[0184] Robustness. The robustness of the oscillator is verified using a bifurcation diagram. The “challenge” (frequency sweep) is used to excite the oscillator, and its “response” (bifurcation diagram) is used as the fingerprint. Each initial condition of theDocket: 511607-2030 original simulation was modified by adding an independently drawn random number that was sampled from the continuous uniform distribution for the interval (-0.1, +0.1). Even though the bistable Duffing adaptive oscillator is chaotic, its bifurcation diagram is robust to random initial conditions for the frequency sweep.

[0185] A 2D histogram of the bifurcation diagrams of the 100 simulations of the original oscillator is shown in the top of FIG.37 with random initial conditions, where the shade of each pixel is dictated by the number of bifurcation diagrams that had a Poincaré section in that location. The “crispness” of the 2D histogram pictorially shows that the bifurcation diagram is robust to changes in the initial conditions.

[0186] Comparing the image of the bifurcation diagram of the original oscillator and that of another simulation with different initial conditions, the number of different pixels is calculated for each set of simulations. The bottom of FIG.37 shows the number of different pixels between the image of the bifurcation diagram of the original oscillator and that of the same oscillator with different initial conditions was calculated and plotted as a histogram. The low number of differences verifies that the oscillator is robust to random initial conditions.

[0187] Unclonability. The unclonability of the oscillator is verified using a bifurcation diagram. As in the last subsection, the “challenge” (frequency sweep) is used to excite the clone, and its “response” (bifurcation diagram) is used as the fingerprint. Since the bistable Duffing adaptive oscillator is chaotic, small changes to the oscillator's parameters produce quantitative changes to the bifurcation diagram.

[0188] A 2D histogram of the bifurcation diagrams of the 100 clones is shown in the top of FIG.38, where the shade of each pixel is dictated by the number of bifurcation diagrams that had a Poincaré section in that location. The bifurcation diagram of the original oscillator is overlaid on top of the 2D histogram for comparison. From comparing the two, it can be noted that many bifurcation points are shifted. In addition, for a given Ω value, ωusually has a different number of Poincaré sections.

[0189] Comparing the image of the bifurcation diagram of the original oscillator and that of a clone, the number of different pixels is calculated for each clone. The number of different pixels between the image of the bifurcation diagram of the original oscillator and that of the clones was calculated and plotted as a histogram shown in the bottom of FIG.38. The high number of differences verifies that the oscillator is unclonable.

[0190] Discussion. Here, an oscillator-based physical reservoir computer was shown to function as a physical unclonable function. In general, the monostable and bistable Duffing oscillator (as well as other adaptive oscillators) could be utilized as PUFs, using the framework here or other methods. Other challenges could include a single frequency input, a multiple frequency input, impulses, steps, pulses, and other waveforms. The response couldDocket: 511607-2030 include the bifurcation diagram, the FFT, the frequency amplitude relationship, and other signal processes used for chaotic oscillators. Here, a bistable Duffing adaptive oscillator was used as the physical reservoir, and a NARMA time series prediction task was used as a benchmark. Since the oscillator can exhibit chaotic behavior, its bifurcation diagram was proposed as its “fingerprint.”

[0191] Further, PUFs can be qualified as either weak or strong. A strong PUF is needed to be able to withstand a large number of challenge-response pairs within a fixed amount of time; an adversary, given a polynomial-sized sample of challenge-response pairs, should not be able to predict the response to a new random challenge; further, it should not be feasible to fabricate two PUFS with the exact same response.

[0192] With the strength of the PUF in mind, the bifurcation diagram has extremely fine structures, which can be seen by zooming in on a portion of it. By leveraging the chaotic response of the oscillator, the strength of the PUF could be made arbitrarily high by modifying the frequency sweep (for instance, sweeping over a larger range or taking a fine scan over a smaller set of frequencies). Since the oscillator can function as both a physical reservoir computer and a physical unclonable function, this implies that it can be constructed as a cybersecure artificial intelligence chip.

[0193] In addition to adaptive stochastic resonance, noise also can enhance the learning rate of the Duffing adaptive oscillator. Consider the Duffing adaptive oscillator represented by: Noise canthe Duffing adaptive oscillator:can be used to find the time it takes for the Duffing adaptive oscillator to learn the external forcing frequency. For an optimal noise level, the learning rate is enhanced, which can be seen in FIG.39. The Fokker-Planck equation, whose solution is the time-dependent probability density function of the oscillator, can be written as:Docket: 511607-2030. Theshown in FIG.40. For an optimal amount of noise, the Duffing adaptive oscillator has an enhanced learning rate.

[0195] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.

[0196] The term "substantially" is meant to permit deviations from the descriptive term that don't negatively impact the intended purpose. Descriptive terms are implicitly understood to be modified by the word substantially, even if the term is not explicitly modified by the word substantially.Docket: 511607-2030

[0197] It should be noted that ratios, concentrations, amounts, and other numerical data may be expressed herein in a range format. It is to be understood that such a range format is used for convenience and brevity, and thus, should be interpreted in a flexible manner to include not only the numerical values explicitly recited as the limits of the range, but also to include all the individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly recited. To illustrate, a concentration range of “about 0.1% to about 5%” should be interpreted to include not only the explicitly recited concentration of about 0.1 wt% to about 5 wt%, but also include individual concentrations (e.g., 1%, 2%, 3%, and 4%) and the sub-ranges (e.g., 0.5%, 1.1%, 2.2%, 3.3%, and 4.4%) within the indicated range. The term “about” can include traditional rounding according to significant figures of numerical values. In addition, the phrase “about ‘x’ to ‘y’” includes “about ‘x’ to about ‘y’”.

Claims

Docket: 511607-2030 CLAIMS Therefore, at least the following is claimed:

1. A physical reservoir computer (PRC), comprising: a Duffing adaptive oscillator (AO) configured for adaptive stochastic resonance, the PRC configured to utilize the adaptive stochastic resonance to perform reconfigurable tasks.

2. The physical reservoir computer of claim 1, wherein the Duffing AO is not divergent.

3. The physical reservoir computer of claim 2, wherein the Duffing AO is monostable hardening (^^^ ^ 0, ^^ଷ ^ 0).

4. The physical reservoir computer of claim 2, wherein the Duffing AO is monostable softening (^^^ ^ 0, ^^ଷ ^ 0).

5. The physical reservoir computer of claim 2, wherein the Duffing AO is bistable (^^^^ 0, ^^ଷ ^ 0).

6. The physical reservoir computer of claim 1, wherein the Duffing AO is implemented as an electrical circuit.

7. The physical reservoir computer of claim 6, wherein the electrical circuit is implemented using a FPAA, a FPGA, a PCB, or a VLSI.

8. The physical reservoir computer of claim 1, wherein the Duffing AO is implemented as a mechanical system.

9. The physical reservoir computer of claim 8, wherein the mechanical system comprises a mechanical beam, clamped-clamped beams, large deformation cantilevered beams, or string in tension.

10. The physical reservoir computer of claim 8, wherein the mechanical system comprises a quartz crystal.

11. The physical reservoir computer of claim 1, wherein the Duffing AO is implemented as an electromechanical system.Docket: 511607-2030 12. The physical reservoir computer of claim 11, wherein the electromechanical system comprises a comb drive tuning fork (commonly used for MEMS gyroscopes).

13. The physical reservoir computer of claim 1, wherein the Duffing AO is implemented as a flexible rail with a linear bearing, cart, or rolamite.

14. The physical reservoir computer of claim 1, wherein the Duffing AO is implemented as a quantum system.

15. The physical reservoir computer of claim 14, wherein the quantum system comprises a Josephson junction.

16. The physical reservoir computer of claim 14, wherein the quantum system comprises a tunnel diode oscillator.

17. The physical reservoir computer of claim 1, wherein the PRC utilizes time multiplexing.

18. The physical reservoir computer of claim 1, wherein the PRC does not utilize time multiplexing.

19. The physical reservoir computer of claim 1, wherein the PRC utilizes spatial multiplexing.

20. The physical reservoir computer of claim 1, wherein the PRC does not utilize spatial multiplexing.

21. The physical reservoir computer of claim 1, wherein the Duffing AO is configured as a morphable logic gate.

22. The physical reservoir computer of claim 1, wherein the PRC performs different tasks in parallel with different vibratory modes.

23. The physical reservoir computer of claim 22, wherein various signals are encoded with frequencies that correspond to the different vibratory modes.Docket: 511607-2030 24. A physical reservoir computer (PRC), comprising: a Helmholtz adaptive oscillator (AO).

25. The physical reservoir computer of claim 24, wherein the Helmholtz AO is implemented as a mechanical system.

26. The physical reservoir computer of claim 25, wherein the mechanical system comprises a plate or membrane.

27. The physical reservoir computer of claim 24, wherein the Helmholtz AO is implemented as an electrical system.

28. The physical reservoir computer of claim 27, wherein the electrical circuit is implemented using a FPAA, a FPGA, a PCB, or a VLSI.

29. A vibratory energy harvester, comprising: a Duffing adaptive oscillator comprising an adjustable rail that is shaped like a potential energy curve, comb drive tuning fork, clamped-clamped beam, large deformation cantilever beam, string, plate, or Josephson junction; a controller configured to adjust the potential energy curve, electrostatic force, tension, length, or voltage; and circuitry to convert and store energy in a battery or capacitor.

30. A signal booster for sensors, comprising: a Duffing adaptive oscillator comprising an adjustable rail that is shaped like a potential energy curve, comb drive tuning fork, clamped-clamped beam, large deformation cantilever beam, string, plate, or Josephson junction; and a controller configured to adjust the potential energy curve, electrostatic force, tension, length, or voltage.

31. The Duffing AO of claim 30, wherein adaptive stochastic resonance uses noise to boost the signal-to-noise ratio of a weak signal.

32. The Duffing AO of claim 31, wherein adaptive stochastic resonance boosts a weak electromagnetic or optical signal.Docket: 511607-2030 33. The Duffing AO of claim 31, wherein adaptive stochastic resonance boosts a weak acoustic signal.

34. The Duffing AO of claim 31, wherein adaptive stochastic resonance boosts a weak quantum signal.

35. The Duffing AO of claim 31, wherein adaptive stochastic resonance boosts a weak mechanical signal.

36. The Duffing AO of claim 31, wherein adaptive stochastic resonance boosts a weak electrical signal.

37. The Duffing AO of claim 30, wherein noise enhances the learning rate.

38. A spline adaptive oscillator, comprising: a flexible rail, actuators placed at one or more locations to modify the shape of the rail, and a controller configured to adjust the actuators to create the desired frequency of a movable cart or linear bearing.

39. A cybersecure authentication key, comprising: a Duffing adaptive oscillator (AO) and a challenge-response pair.

40. The cybersecure authentication key of claim 39, wherein the challenge-response pair is a bifurcation diagram.

Images