System and method for synchronizing nonlinear resonators in chaotic regime

Abstract

Title: System and Method for Synchronizing Nonlinear Resonators in Chaotic Regime. The invention relates to a system comprising a first and a second generator (E1, E2) delivering excitation signals to a first and a second resonator (R1, R2) generating chaotic signals. The resonators (R1, R2) are structurally different and coupled to each other. Advantageously, this system allows the synchronization of chaotic signals and comprises: a dissipation correction module acting on the modulation frequencies fm1, fm2 of the excitation signals; a nonlinearity correction module acting on the amplitudes of the excitation signals; a resonance frequency correction module acting on the carrier frequencies fc1, fc2 of the excitation signals; and an amplitude correction module acting on the amplitudes of the chaotic signals. The invention also relates to a synchronization method based on this system.Figure for the abridged version: Fig.9.

Description

Title of the invention: System and method for synchronizing nonlinear resonators in chaotic regimes. Technical field

[0001] The field of the invention is that of the control of devices and methods for generating chaotic signals. The invention relates more particularly to the coupling and synchronization of nonlinear resonators in chaotic regimes, when these resonators are not structurally identical. STATE OF THE ART

[0002] Modern cryptography is based in particular on the generation and processing of signals from intrinsically random physical processes. Such signals can be observed in resonant components, or resonators, of different kinds (electronic, optical, mechanical...).

[0003] Micro / Nano-Electro-Mechanical Systems (M / NEMS) are an example of resonators that inherently exhibit sources of randomness. Due to their inherent nonlinearity, it is possible to put these resonators into a chaotic regime [YC Wang et al. "Chaos in MEMS, parameter estimation and its potential application," in IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 45, no. 10, pp. 1013-1020, Oct. 1998]. In particular, for a given excitation frequency and force, the resonator dynamics can become complex, non-periodic, and chaotic. This chaotic regime is characterized, in particular, by its non-reproducibility and the impossibility of predicting the system's state in the medium and long term.

[0004] This chaotic regime can be advantageously exploited to encrypt information. To decrypt this information, it has been shown that it is possible to synchronize two chaotic systems. This counterintuitive property establishes that two identical chaotic systems, sometimes called "twins," can have the same temporal behavior while exhibiting dynamics similar to those of noise. This dynamic is called "identical synchronization." However, the requirement for two identical chaotic systems hinders the development of this chaotic cryptography technology.

[0005] To overcome this drawback, a so-called "generalized" synchronization method has been developed. Although this method facilitates synchronization between two structurally different chaotic systems, the simplifications introduced by this method are such that they limit or even negate its usefulness for chaotic cryptography.

[0006] An object of the present invention is therefore to propose a system and a method for synchronizing structurally different chaotic resonators.

[0007] The other objects, features and advantages of the present invention will become apparent from an examination of the following description and accompanying drawings. It is understood that other advantages may be incorporated. SUMMARY

[0008] To achieve this objective, according to one embodiment, a system is provided comprising: - a first resonator characterized by • a first transduction coefficient Tl', • a first non-linearity coefficient al, • a first resonant frequency fol, • a first dissipation Afl, - a second resonator characterized by • a second transduction coefficient T2', • a second non-linearity coefficient a2, • a second resonance frequency f02, • a second dissipation Af2.

[0009] The first and second resonators are structurally different so that at least one condition is verified among: Tl VT2', al^a2, fol^fo2, Afl^Af2.

[0010] The system also includes: - a first generator configured to provide, to the first resonator, a first excitation signal U1 oscillating at a first carrier frequency fcl and modulated at a first modulation frequency fml, said first resonator being configured to generate, from the first excitation signal Ul, a first chaotic signal Ul' in chaotic regime, via a variable XI, - a second generator configured to provide, to the second resonator, a second excitation signal U2 oscillating at a second carrier frequency fc2 and modulated at a second modulation frequency fm2, said second resonator being configured to generate, from the second excitation signal U2, a second chaotic signal U2' in chaotic regime, via a variable X2, - a first demodulator D1 configured to provide a first signal U1” corresponding to the demodulation of the first chaotic signal U1' at the first carrier frequency fcl, - a second demodulator D2 configured to provide a second signal U2” corresponding to the demodulation of the second chaotic signal U2' at the second carrier frequency fc2, - a coupling loop configured to inject into the second resonator a US synchronization signal oscillating at the second carrier frequency fc2 and modulated by the difference of the first and second signals Ul”-U2”.

[0011] Advantageously, the system comprises at least one of the following: - a dissipation correction module configured to modify the first modulation frequency fml of the first excitation signal Ul and / or the second modulation frequency fm2 of the second excitation signal U2 so that fml / Af1 = fm2 / Af2, and to speed up or slow down the first chaotic signal Ul' and / or the second chaotic signal U2' so that the first and second chaotic signals Ul', U2' evolve at the same characteristic frequency, - a non-linearity correction module configured to modify an amplitude Al of the first excitation signal Ul and / or an amplitude A2 of the second excitation signal U2, such that al.XI2 / Afl = a2.X22 / Af2, - a resonance frequency correction module configured to modify the first carrier frequency fcl of the first excitation signal Ul and / or the second carrier frequency fc2 of the second excitation signal U2 such that (fol - fcl) / Afl = (f02 - fc2) / Af2, - an amplitude correction module configured to modify an amplitude Al' or Al” of the first chaotic signal Ul' or the first signal Ul” respectively, and / or to modify an amplitude A2' or A2'' of the second chaotic signal U2' or the second signal U2” respectively, so that Al” = A2”.

[0012] These modules advantageously allow for the simulation of identical characteristics for structurally different resonators. It is thus possible to synchronize the chaotic signals from these two different resonators. This opens up numerous applications in chaotic cryptography. For example, one can imagine a chaotic MEMS accelerometer synchronizing with a chaotic MEMS microphone in order to exchange information securely using chaotic cryptography. The system's range of applications is significantly increased.

[0013] Another advantage of this system is that the resonators themselves are not modified. Only the excitation and chaotic signals are corrected by adding the different modules. This makes it possible to maintain Structurally distinct resonators, while maintaining their integrity, enable the synchronization of chaotic signals emanating from them. This simplifies system implementation.

[0014] Another aspect of the invention relates to a method for synchronizing the first and second resonators of the system described above. This method comprises the following steps: - Correct the dissipation of the first and / or second resonator by modifying the first modulation frequency fml of the first excitation signal U1 and / or the second modulation frequency fm2 of the second excitation signal U2 so that fml / Afl = fm2 / Af2, - Accelerate or decelerate the first signal Ul” and / or the second signal U2” so that the first and second signals Ul”, U2” evolve at the same characteristic frequency, - Correct the non-linearity of the first and / or second resonator by modifying an amplitude Al of the first excitation signal Ul and / or an amplitude A2 of the second excitation signal U2, so that al.XI2 / Afl = a2.X22 / Af2, - Correct the resonance frequency of the first and / or second resonator by modifying the first carrier frequency fcl of the first excitation signal Ul and / or the second carrier frequency fc2 of the second excitation signal U2 so that (fol - fcl) / Afl = (f02 - fc2) / Af2, - Correct the amplitude Al' or A1 ” of the first chaotic signal U1 ' or of the first signal Ul” respectively, and / or correct the amplitude A2' or A2” of the second chaotic signal U2' or of the second signal U2' ' respectively, so that Al” = A2''. BRIEF DESCRIPTION OF THE FIGURES

[0015] The aims, objects, features and advantages of the invention will become clearer from the detailed description of an embodiment thereof, which is illustrated by the following accompanying drawings in which:

[0016] [Fig. 1] The [Fig. 1] illustrates the operation of a MEMS type resonator in linear and non-linear regime.

[0017] [Fig.2] Fig.2 illustrates a part of the system comprising a non- linear and its generator, as well as various intrinsic and extrinsic parameters relating to this part, according to an embodiment of the present invention.

[0018] [Fig.3] The [Fig.3] illustrates an example of a chaotic signal emitted by a non-linear resonator after demodulation when excited in chaotic regime, according to an embodiment of the present invention.

[0019] [Fig.4] The [Fig.4] illustrates a synchronization system by coupling two substantially identical non-linear resonators, according to the prior art.

[0020] [Fig.5] The [Fig.5] illustrates a synchronization system by coupling two structurally different non-linear resonators, according to an embodiment of the present invention.

[0021] [Fig.6] The [Fig.6] illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0022] [Fig.7] The [Fig.7] illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0023] [Fig.8] The [Fig.8] illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0024] [Fig.9] The [Fig.9] illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0025] The drawings are given by way of example and are not limiting of the invention. They constitute schematic representations of principle intended to facilitate understanding of the invention and are not necessarily to scale with practical applications. DETAILED DESCRIPTION

[0026] Before proceeding to a detailed review of embodiments of the invention, optional features that may be used in combination or alternatively are listed below:

[0027] According to one example, the system comprises both the dissipation correction module and the non-linearity correction module and the resonance frequency correction module and the amplitude correction module. This makes it possible to correct all the intrinsic parameters of one and / or the other of the resonators.

[0028] In one example, the dissipation correction module includes a voltage-controlled oscillator VCOlm delivering the first modulation frequency fml at the first generator and / or a voltage-controlled oscillator VC02m delivering the second modulation frequency fm2 at the second generator. The difference in dissipation of the resonators is compensated by adjusting the modulation frequency(ies) of the generators. In one example, the dissipation correction module includes a buffer replicating one of the first and second signals Ul”, U2”, by speeding them up or slowing them down. This allows adjusting the time scales on which the chaotic dynamics from the resonators develop, so as to obtain the same temporality for the demodulated chaotic signals Ul”, U2”.

[0029] According to one example, the nonlinearity correction module includes a gain G1 located at the output of the first generator and / or a gain G2 located at the output of the second generator. The nonlinearity of the resonators is corrected by adjusting the amplitudes of the excitation signals.

[0030] According to one example, the resonance frequency correction module comprises a voltage-controlled oscillator VCOlc delivering the first carrier frequency fcl at the first generator and / or a voltage-controlled oscillator VCO2c delivering the second carrier frequency fc2 at the second generator. The difference in the resonance frequencies of the resonators is compensated by adjusting the carrier frequency(ies) of the generators.

[0031] According to one example, the amplitude correction module includes a gain Gl' located at the input or output of the first demodulator D1, and / or a gain G2' located at the input or output of the second demodulator D2.

[0032] In one example, the first and second resonators are micrometric or nanometric electromechanical devices. In another example, the variables XI, X2 correspond respectively to displacement amplitudes of a moving element of the first and second resonators.

[0033] According to one example, the coupling loop comprises: - a signal subtraction element comprising • a first input receiving the first signal U1”, • a second input receiving the second signal U2' ', • an output providing a signal Ul” - U2’’ to be modulated, - a modulator M comprising: • an input receiving the signal to be modulated Ul” - U2’’, • an output providing the modulated US synchronization signal, - a signal addition element comprising: • an input receiving the second excitation signal U2, • an input receiving the US synchronization signal, • an output connected to the second resonator and providing the second resonator with the sum of the second excitation signal U2 and the synchronization signal US.

[0034] According to one example, the demodulator Dl can be placed at the input or output of the gain Gl'.

[0035] According to one example, the demodulator D2 can be placed at the input or output of the gain G2'.

[0036] According to one example, the system includes a gain G interposed between the output of the subtraction element and the input of the addition element receiving the synchronization signal.

[0037] According to one example, the gain G replaces one of the GU gain and the G2' gain. This makes it possible to reduce the cost and / or size of the system.

[0038] Micro / Nanometric M / NEMS devices are micro / nanometric devices that convert a mechanical process into an electrical one, and vice versa. Resonant M / NEMS devices are characterized by a vibrating mass that transfers mechanical energy into electrical energy, or vice versa. This allows, for example, the design of an energy harvester or a force sensor, depending on the chosen geometry.

[0039] One of the geometries usable within the framework of the present invention consists, for example, of a doubly fixed micro- or nano-beam. This beam typically forms a nonlinear resonator. The application of a periodic excitation signal makes this beam vibrate according to different regimes. An excitation signal of moderate amplitude and whose frequency is close to the resonance of the structure will generate a predictable vibration of the beam, characterized by its resonance frequency f0. The resonator operates in the linear regime. When the excitation signal has a sufficiently high amplitude, the dynamic behavior of the beam becomes bistable, that is to say, two vibration amplitudes are accessible for the same set of parameters. The resonator then operates in the nonlinear regime.When modulation is applied to the periodic excitation signal, and the modulation frequency and amplitude are cleverly chosen, the beam will erratically alternate between its two bistability states; the beam's behavior becomes unpredictable and chaotic. The resonator then operates in a chaotic regime.

[0040] The chaotic dynamics reside in the vibration amplitude of the nonlinear resonator. The resonator vibrates periodically at the frequency fcl, and it is the amplitude of this vibration that is chaotic. In other words, the periodic vibration is modulated by the chaotic dynamics. Since the chaos resides not in the carrier but in the modulation, only this modulating signal matters for chaotic synchronization. In a conventional chaotic synchronization application, the entire signal (carrier and modulating chaotic signal) is transferred into the coupling loop. In the case of the present invention, only the chaotic signal needs to be transferred. To extract the modulation from a signal comprising a carrier and a modulating signal, it is necessary to use a demodulator, such as for example synchronous detection.

[0041] The chaotic signals Ul' and U2' each possess a characteristic frequency, even though they are chaotic (that is, these signals Ul' and U2' resemble noise—due to the chaos—but one frequency is emphasized more than the others). This characteristic frequency is none other than the modulation frequency. Thus, Ul has a modulation frequency fml, and the first signals Ul', Ul' have a characteristic frequency fml. U2 has a modulation frequency fm2, and the second signals U2', U2'' have a characteristic frequency fm2.

[0042] This is typically the element of the dissipation correction module, configured to slow down or speed up the dynamics, which allows the characteristic frequency of the first signal Ul” to be modified so that it has the characteristic frequency fm2. This allows U2” to be subtracted from Ul”.

[0043] In the present invention, for the sake of brevity, the first signal U1 corresponds to the first demodulated chaotic signal U1. The second signal U2 corresponds to the second demodulated chaotic signal U2.

[0044] Resonant microdisks are another example of resonators exhibiting significant nonlinearity. Other geometries are perfectly conceivable for the nonlinear resonators of the present invention.

[0045] A nonlinear resonator (regardless of its nature: electronic, mechanical, optical, etc.) is primarily characterized by four parameters: its transduction coefficient (T), its nonlinearity coefficient (α), its resonance frequency (f0), and its dissipation (Af). These parameters are intrinsic: they depend on the physical properties of the component and can therefore be considered fixed. These parameters are determined and characterized in both the linear and nonlinear regimes.

[0046] The terms "resonant component" and "resonator" are used synonymously. The terms "demodulator" and "demodulation element or module" are used synonymously. The terms "modulator" and "modulation element or module" are used synonymously.

[0047] Dimensional values ​​are understood to be within manufacturing and measurement tolerances.

[0048] The terms "approximately", "about", and "on the order of" mean, when referring to a value, "within 5%" of that value, or, when referring to an angular orientation, "within 5°" of that orientation. Thus, a direction substantially normal to a plane means a direction having an angle of 90+5° with respect to the plane.

[0049] In the following, the resonators described and illustrated by way of non-limiting examples are electromechanical devices that can be excited with a voltage generated by a generator. This voltage corresponds to the excitation signal. This voltage induces, in a known manner, a mechanical displacement of a structural part of the device, for example, a beam or a disk. This displacement can be followed by the variation of another voltage corresponding to the readout or output signal of the resonator.

[0050] Figure 1 illustrates the nonlinear mechanical behavior of a doubly fixed beam resonator. When the beam 100 is set in motion by a periodic excitation signal of moderate amplitude and with an excitation frequency close to the resonant frequency of the resonator, it adopts a predictable resonant behavior 101, characterized by the curve C11 typical of a resonant phenomenon. When the beam 100 is set in motion by a periodic excitation signal of high amplitude, it adopts a nonlinear behavior 102, characterized by the curve C12 exhibiting a hysteresis C121, C122. By applying modulation (in amplitude, frequency, or phase) to the periodic excitation signal, the resonator alternates between the two states present in the hysteresis, leading to chaotic behavior. It is this latter chaotic operating regime of the resonator that is exploited in the present invention.

[0051] Figure 2 illustrates the various parameters that characterize the nonlinear resonator and its generator. The generator delivers an excitation signal characterized by its amplitude or voltage U, its carrier frequency fc, and its modulation frequency fm. The resonator transforms this electrical excitation signal into a mechanical displacement via the electromechanical transduction coefficient T. In particular, in both linear and nonlinear regimes, we observe X = TU. This mechanical displacement generates an electrical signal via the mechano-electrical transduction coefficient T', and regardless of the resonator's regime (linear, nonlinear, or chaotic), we observe the relationship U' = T'.X. The amplitude of the chaotic electrical signal is directly proportional to the amplitude of the mechanical displacement. The transduction is linear here.

[0052] In the non-linear regime of the resonator, the non-linearity mainly observed in mechanics, called Duffing, is written aX2, with a the coefficient of non-linearity.

[0053] The linear resonator has a resonance frequency f0. In chaotic regime, it is the shift between the frequency fc of the excitation signal and the resonance frequency f0 that allows the generated chaos to be characterized.

[0054] The resonator also exhibits a dissipation Af. From a physical point of view, the dissipation corresponds to the losses of the resonator. Changing the dissipation transforms the frequency response of the resonator by creating a broadening or narrowing of the resonance curve.

[0055] The parameters T', a, f0, and Af are intrinsic. They characterize the resonator. The parameter T is also intrinsic to the resonator and should also be corrected, but this correction is already included in the method for correcting a, so T does not need to be considered further. The parameters U, fc, and fm are extrinsic. They originate from the excitation signal. U' is the signal from the resonator excited by the excitation signal, and U” is the demodulation of the signal U'. Figure 3 illustrates such a demodulated chaotic signal U” generated from a resonator in chaotic operation.

[0056] In the context of chaotic synchronization, two resonators can be coupled using a feedback loop. There are no particular constraints on this coupling, which can be optical, electronic, mechanical, magnetic, or otherwise.

[0057] Figure 4 illustrates a typical coupling situation between two identical or substantially identical resonators RA and RB. Resonator RA is excited by generator EA, which delivers an excitation signal UA with a carrier frequency fcA and a modulation frequency fmA. Resonator RA, characterized by its transduction coefficient TA', its nonlinearity coefficient aA, its resonance frequency f0A, and its dissipation AfA, generates a chaotic signal UA' in response to the excitation signal UA. Similarly, resonator RB is excited by generator EB, which delivers an excitation signal UB with a carrier frequency fcB and a modulation frequency fmB. Resonator RB, characterized by its transduction coefficient TB', its nonlinearity coefficient aB, its resonance frequency f0B, and its dissipation AfB, generates a chaotic signal UB' in response to the excitation signal UB.The RA and RB resonators are coupled by the coupling loop 30. This coupling loop 30 allows a "synchronization" signal, corresponding to the difference between the chaotic and UB' signals, to be fed back into the RB resonator. The coupling loop 30 typically comprises a signal subtraction element 31, configured to produce the synchronization signal UA' - UB', and a signal addition element 32, configured to add the synchronization signal to the excitation signal UB. A gain can be inserted between the subtraction element 31 and the addition element 32, as is known, to improve the amplitude of the synchronization signal.

[0058] This coupling loop 30 allows, in a known manner, the synchronization of the chaotic signals UA' and UB' when the resonators RA and RB are identical, that is, when TA' = TB', aA = aB, f0A = f0B and AfA = AfB. The resonators RA and RB are exponentially sensitive to variations. A minute change in pressure The influence of temperature and / or humidity will induce a different dynamic for resonator RA than for resonator RB, even if the parameters of resonators RA and RB are identical. The chaotic signals UA' and UB' remain similar (they exhibit the same statistics and figures of merit), but their temporal evolution differs. However, thanks to coupling, the reinjection of the synchronizing signal into resonator RB will force it to align with the dynamics of resonator RA. This is the synchronization mechanism, which will result in UA'(t) = UB'(t). When the chaotic signals UA' and UB' are identical (and therefore synchronized), the coupling becomes zero. But as soon as there is a deviation, the coupling reappears and allows resynchronization of resonators RA and RB.This synchronization is only possible if the parameters of the two resonators RA, RB are identical, and if the excitation signals UA(t), UB(t) are identical.

[0059] In the context of the present invention, a synchronization system has been advantageously developed for situations where the resonators are not structurally or intrinsically identical. In this system according to the invention, the extrinsic parameters are modified so as to virtually adjust the intrinsic parameters of the resonators. It is thus possible to obtain, by modifying only the extrinsic parameters, similar chaotic signal amplitudes, similar non-linearity effects, similar operating points, and similar dynamic ranges for two different resonators. The chaotic signals generated by these two different resonators will be similar and can therefore be synchronized.

[0060] In the following, resonators RI and R2 are structurally different. Resonator RI is excited by generator E1, which delivers an excitation signal U1 with a carrier frequency fcl and a modulation frequency fml. Resonator RI, characterized by its transduction coefficient Tl', its nonlinearity coefficient al, its resonance frequency fol, and its dissipation Afl, generates a chaotic signal Ul' in response to the excitation signal UL. This chaotic signal Ul' is then demodulated from the carrier fcl by means of demodulator DI, which generates the demodulated chaotic signal Ul'. Similarly, resonator R2 is excited by generator E2, which delivers an excitation signal U2 with a carrier frequency fc2 and a modulation frequency fm2.The resonator R2, characterized by its transduction coefficient T2', its non-linearity coefficient a2, its resonance frequency f02 and its dissipation Af2, generates a chaotic signal U2' in response to the excitation signal U2. This chaotic signal U2' is then demodulated from the carrier fc2 by the demodulator D2 which generates the demodulated chaotic signal U2'. At least one of the following conditions is satisfied: T1 VT2' and / or al^a2 and / or f0l^f02 and / or Afl^Af2.

[0061] Figure 5 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, the demodulation modules DI and D2, the modulation module M added within the coupling loop 30, and a dissipation correction module. In this case, Afl^Af2.

[0062] The correction of the dissipation of the resonators RI, R2 is the first correction to be made to the system, since the effects of dissipation impact in particular the corrections on the non-linearity and the resonance frequency of the resonators.

[0063] To correct the dissipation of resonators RI, R2, the modulation frequencies fml and / or fm2 are modified so that:

[0064] [Math.l] fif 2 A.fl “A / 2

[0065] This frequency modulation correction can be obtained via voltage-controlled oscillators VCOlm (illustrated by reference 11 in [Fig.5]), VC02m (illustrated by reference 21 in [Fig.5]) at the generators El, E2. This frequency modulation correction can also be carried out by other methods.

[0066] To take into account the effects of dissipation on the non-linearity and resonance frequency of the resonators, care shall be taken to ensure that:

[0067] [Math.2] «IXF _ «2A22 A / l - A / 2 And

[0068] [Math.3] f02-f2 A / l - A / 2

[0069] The non-linearity between RI and R2 is not the same, but the non-linearity relative to dissipation is. Similarly, the frequency shift is not the same, but this shift relative to dissipation is. For example, if resonator RI has a resonant frequency of 10 kHz and a dissipation of 100 Hz, and is excited at 9 kHz, the ratio above has a value of 10: (10000-9000) / 100 = 10. If resonator R2 has a resonant frequency of 20 kHz with a dissipation of 10 Hz, it will then be necessary to excite it at 19.9 kHz to maintain this ratio of 10: (20000-19900) / 10 = 10. The equivalence between the resonators RI and R2 is relative.

[0070] This relative equivalence implies that the dynamics of the demodulated chaotic signals U1”, U2”, are similar, but on different timescales. The figures of merit of the two chaotic resonators are identical, but one of the two chaotic signals is “faster” than the other. The dissipation correction module therefore includes an element 40 to slow down or speed up the dynamics chaotic of the resonator RI so that its speed is perceived as identical to that of the resonator R2. This element 40 can include a buffer memory and a signal replicator with a modifiable internal clock, for example. Other components are obviously conceivable to fulfill the function of element 40. In the example illustrated in [Fig. 5], it is the first signal Ul” that is accelerated or slowed down, so as to present the same time scale as the second signal U2”.

[0071] Figure 6 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, the demodulation modules DI and D2, the modulation module M added within the coupling loop 30, and a non-linearity correction module. In this case, al^a2.

[0072] The nonlinearity of the resonators RI and R2, which can be written as a1.X12 and a2.X22 respectively, can be advantageously corrected by modifying the amplitudes of U1 and U2, respectively. Since XI varies linearly with Ul, a modification of Ul results in a modification of the overall nonlinearity a1.X12 of the resonator RL. Similarly, since X2 varies linearly with U2, a modification of U2 results in a modification of the overall nonlinearity a2.X22 of the resonator R2. The modification of the amplitudes of Ul and / or U2 can be carried out via gains G1 and / or G2 placed at the output of the generators El and / or E2, respectively, as illustrated in [Fig. 6]. Alternatively, this modification of the amplitudes can be carried out directly at the generators El and / or E2. The modification of the amplitudes is carried out in such a way as to satisfy the relation [Math2] given above.

[0073] Figure 7 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, the demodulation modules DI and D2, the modulation module M added within the coupling loop 30, and a resonance frequency correction module. In this case, f0 Ufo2.

[0074] The resonance frequencies fol, f02 are characteristic of the RI, R2 resonators. With regard to the chaotic dynamics of each of these RI, R2 resonators, it is the shift between the frequency fcl, fc2 of the excitation signal Ul, U2 and the resonance frequency fol, f02 that characterizes or modifies the chaotic dynamics. This is an intrinsic property of the way chaos is generated via these resonators, which is notably observed in the paper "Bichromatic synchronized chaos in driven coupled electro-optomechanical nanoresonators, G. Madiot et al., Phys. Rev. A 104, 023525 (2021)". In practice, these frequency shifts can be controlled by modifying the frequencies fcl, fc2 of the excitation signals Ul, U2. This resonance frequency correction can be achieved via voltage-controlled oscillators VCOlc (illustrated by reference 12 on the [Fig.7]), VC02c (illustrated by reference 22 on [Fig.7]) at the level of the generators El, E2. The modification of the frequencies fcl, fc2 is carried out in such a way as to satisfy the relation [Math3] provided above.

[0075] Figure 8 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, the demodulation modules DI and D2, the modulation module M added within the coupling loop 30, and an amplitude correction module. In this case, Tl VT2'.

[0076] When the transduction coefficients of resonators RI and R2 are different, the displacements XI and X2 associated with the same excitation signal are different. To overcome this difference without structurally modifying the resonators, one solution is to adjust U1 and / or U2 to obtain Ul' = U2'. Adjusting U1 and / or U2 is nevertheless preferred for correcting the nonlinearity of resonators RI and R2, as described above. Another solution is to directly modify U1' and / or U2' to achieve the same result (UL = U2'). Modifying the amplitudes of Ul' and / or U2' can be done via gains Gl' and / or G2' placed at the output of resonators RI and / or R2, respectively, as illustrated in [Fig. 8]. According to one possibility, one of the gains Gl' or G2' can correspond to the gain inserted between the subtraction element 31 and the addition element 32 of the coupling loop 30.

[0077] Figure 9 illustrates one embodiment of the synchronization system according to The invention comprises a conventional coupling loop 30, the demodulation modules DI and D2, the modulation module M added within the coupling loop 30, and all the correction modules described above (dissipation correction, nonlinearity correction, resonance frequency correction, amplitude correction). In this case, Tl VT2' and al^a2 and fol^fo2 and Afl^Af2. The various elements 11, 12, 21, 22, Dl, D2, M, Gl, Gl', G2, G2', 40 of the coupling loop and of each of the modules are shown in [Fig. 9]. Care will be taken to perform the dissipation correction first, which notably impacts the nonlinearity correction and the resonance frequency correction, as indicated above.

[0078] The implementation in the synchronization system according to the invention of the set of elements 11, 12, 21, 22, Dl, D2, M, Gl, Gl', G2, G2', 40 advantageously allows the synchronization of two arbitrarily different chaotic resonators RI, R2.

[0079] This synchronization system has, for example, been implemented for RI, R2 resonators in the form of submillimeter diameter and 100m thickness disks, placed under vacuum (clmbar), whose transduction is carried out piezoelectrically. The resonance frequencies of these two disks are f01 = 70 kHz and f02 = 160 kHz. The non-linearity coefficients of these two disks are a1 = 49 kHz / V2 and a2 = 23 kHz / V2. The transduction coefficients of these two disks are such that T1*TT = 0.22 V / V and T2*T2' = 0.47 V / V. The dissipations of these two disks are equivalent: Afl = Af2 = 130 Hz. The two resonators RI, R2 are therefore quite different and exhibit very different intrinsic characteristics. When these two resonators RI, R2 are excited in chaotic operation, by applying the frequency offsets and gains to the different signals as indicated above, the two resonators RI, R2 synchronize with a deviation of less than 2% on average.

[0080] It is clear that the present invention advantageously allows the synchronization of two intrinsically different chaotic resonators by modifying only extrinsic parameters and without structurally altering these resonators. The invention thus makes it possible to obtain chaotic twins from structurally different resonators.

[0081] In particular, MEMS resonators already exhibit similarities: they are characterized, notably, by a resonant frequency, dissipation, non-linearity, excitation force, and displacement, for a wide range of envisaged MEMS applications (accelerometer, microphone, gyroscope, energy harvester, gas sensor, etc.). The invention makes it possible, in practice, to normalize the dynamics of two MEMS resonators by adjusting the forces and frequencies involved. This ultimately yields two identical dynamics, even in the chaotic regime. Thus, after normalization, two different MEMS resonators will exhibit chaos with similar figures of merit. These two different MEMS resonators will then be akin to "twins," which will allow them to be synchronized.

[0082] The invention is not limited to the embodiments described above. The invention extends to all elements or components equivalent to elements 11, 12, 21, 22, Dl, D2, M, Gl, Gl', G2, G2', 40, that is to say to all elements or components which perform the same function as elements 11, 12, 21, 22, Dl, D2, M, Gl, Gl', G2, G2', 40 of the synchronization system according to the invention.

Claims

Demands

1. System comprising: • a first resonator (RI) characterized by - a first transduction coefficient Tl', - a first non-linearity coefficient al, - a first resonant frequency fol, - a first dissipation Afl, • a second resonator (R2) characterized by - a second transduction coefficient T2', - a second non-linearity coefficient a2, - a second resonance frequency f02, - a second dissipation Af2, the first and second resonators (RI, R2) being structurally different such that at least one condition is satisfied among: Tl VT2', al^a2, f0l^f02, AfUAf2, a first generator (E1) configured to provide, to the first resonator (RI), a first excitation signal U1 oscillating at a first carrier frequency fcl and modulated at a first modulation frequency fml, said first resonator being configured to generate, from the first excitation signal Ul, a first chaotic signal Ul' in chaotic regime, via a variable XI, a second generator (E2) configured to provide, to the second resonator (R2), a second excitation signal U2 oscillating at a second carrier frequency fc2 and modulated at a second modulation frequency fm2, said second resonator being configured to generate, from the second excitation signal U2, a second chaotic signal U2' in chaotic regime, via a variable X2, a first demodulator (Dl) configured to provide a first signal U1” corresponding to the demodulation of the first chaotic signal U1’ at the first carrier frequency fcl, a second demodulator (D2) configured to provide a second signal U2” corresponding to the demodulation of the

2. second chaotic signal U2' at the second carrier frequency fc2, • a coupling loop (30) configured to inject into the second resonator (R2) a synchronizing signal US oscillating at the second carrier frequency fc2 and modulated by the difference between the first and second signals Ul”-U2”, the system being characterized in that it comprises at least one of the following: • a dissipation correction module configured to modify the first modulation frequency fml of the first excitation signal Ul and / or the second modulation frequency fm2 of the second excitation signal U2 so that fml / Afl = fm2 / Af2, and to speed up or slow down the first signal Ul” and / or the second signal U2” so that the first and second signals Ul”, U2” evolve at the same characteristic frequency, • a non-linearity correction module configured to modify an amplitude Al of the first excitation signal Ul and / or an amplitude A2 of the second excitation signal U2, such that al.XI2 / Afl = a2.X22 / Af2, • a resonance frequency correction module configured to modify the first carrier frequency fcl of the first excitation signal Ul and / or the second carrier frequency fc2 of the second excitation signal U2 such that (fol - fcl) / Afl = (f02 - fc2) / Af2, • an amplitude correction module configured to modify an amplitude Al' or Al” of the first chaotic signal U1' or of the signal U1” respectively, and / or to modify an amplitude A2' or A2'' of the second chaotic signal U2' or of the signal U2” respectively, so that Al” = A2”. System according to the preceding claim comprising the dissipation correction module and the non-linearity correction module and the resonance frequency correction module and the amplitude correction module.

3. System according to any one of the preceding claims, wherein the dissipation correction module comprises a voltage-controlled oscillator VCOlm (11) delivering the first modulation frequency fml at the first generator (El) and / or a voltage-controlled oscillator VC02m (21) delivering the second modulation frequency fm2 at the second generator (E2), and a buffer memory (40) replicating one of the first and second signals Ul”, U2”, speeding it up or slowing it down.

4. System according to any one of the preceding claims, wherein the non-linearity correction module comprises a gain G1 located at the output of the first generator (E1) and / or a gain G2 located at the output of the second generator (E2).

5. System according to any one of the preceding claims, wherein the resonance frequency correction module comprises a voltage-controlled oscillator VCOlc (12) delivering the first carrier frequency fcl at the first generator (El) and / or a voltage-controlled oscillator VCO2c (22) delivering the second carrier frequency fc2 at the second generator (E2).

6. System according to any one of the preceding claims, wherein the amplitude correction module comprises a gain Gl' located at the input or output of the first demodulator D1, and / or a gain G2' located at the input or output of the second demodulator D2.

7. System according to any one of the preceding claims, wherein the first and second resonators (RI, R2) are micrometric or nanometric electromechanical devices and wherein the variables XI, X2 correspond respectively to displacement amplitudes of a moving element (100) of the first and second resonators (RI, R2).

8. A system according to any one of the preceding claims, wherein the coupling loop (30) comprises: • a signal subtraction element (31) comprising - a first input receiving the first signal U1”, - a second input receiving the second signal U2”,

9.

10.

11. - an output providing a signal Ul'' - U2'' to be modulated, • a modulator (M) comprising: - an input receiving the signal to be modulated U1' ' -U2”, - an output providing the modulated US synchronization signal, • a signal addition element (32) comprising: - an input receiving the second excitation signal U2, - an input receiving the modulated US synchronization signal, - an output connected to the second resonator (R2) and providing the second resonator (R2) with the sum of the second excitation signal U2 and the synchronization signal US. A system according to the preceding claim comprising a gain G interposed between the output of the subtraction element (31) and the input of the addition element (32) receiving the synchronization signal. A system according to the preceding claim and claim 6 in combination, wherein the gain G replaces one of the gain Gl' and the gain G2'. A method for synchronizing the first and second resonators (RI, R2) of a system according to any one of the preceding claims, said method comprising the following steps: • Correct the dissipation of the first and / or second resonator (RI, R2) by modifying the first modulation frequency fml of the first excitation signal Ul and / or the second modulation frequency fm2 of the second excitation signal U2 so that fml / Afl = fm2 / Af2, • Accelerate or decelerate the first signal U1” and / or the second signal U2” so that the first and second signals U1”, U2” evolve at the same characteristic frequency, • Correct the non-linearity of the first and / or second resonator (RI, R2) by modifying an amplitude Al of the first excitation signal Ul and / or an amplitude A2 of the

12.

13. second excitation signal U2, such that al.Xl2 / Afl = a2.X22 / Af2, • Correct the resonance frequency of the first and / or second resonator (RI, R2) by modifying the first carrier frequency fcl of the first excitation signal U1 and / or the second carrier frequency fc2 of the second excitation signal U2 so that (fOl - fcl) / Afl = (fO2 -fc2) / Af2, • Correct the amplitude Al' or A1 ” of the first chaotic signal U1' or the first signal U1 ” respectively, and / or correct the amplitude A2' or A2” of the second chaotic signal U2' or the second signal U2' ' respectively, so that Al” = A2''. Method according to the preceding claim, wherein the correction of the dissipation of the first and / or second resonator (RI, R2) is carried out before the other steps. A method according to any one of the two preceding claims, wherein the correction of the amplitude Al' or Al” and / or of the amplitude A2' or A2” is carried out after the other steps.

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