System and method for synchronizing non-linear resonators in chaotic regime

Abstract

The invention relates to a system comprising a first and second generator (E1, E2) delivering excitation signals to a first and second resonator (R1, R2) generating chaotic signals. The resonators (R1, R2) are structurally different and coupled to one another. Advantageously, this system makes it possible to synchronize the chaotic signals and comprises: ● a dissipation correction module acting on the modulation frequencies fm1, fm2 of the excitation signals, ● a non-linearity correction module acting on the amplitudes of the excitation signals, ● a resonant frequency correction module acting on the carrier frequencies fc1, fc2 of the excitation signals, ● an amplitude correction module acting on the amplitudes of the chaotic signals. The invention also relates to a synchronization method based on this system.

Description

[0001] SYSTEM AND METHOD FOR SYNCHRONIZING

[0002] "NON-LINEAR RESONATORS IN CHAOTIC REGIME"

[0003] TECHNICAL FIELD OF THE INVENTION

[0004] 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 non-linear resonators in chaotic regimes, when these resonators are not structurally identical.

[0005] STATE OF THE ART

[0006] Modern cryptography relies in particular on the generation and processing of signals from inherently random physical processes. Such signals can be observed in resonant components, or resonators, of various types (electronic, optical, mechanical, etc.).

[0007] 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 induce a chaotic regime in these resonators [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 by its non-reproducibility and the impossibility of predicting the system's state in the medium and long term.

[0008] 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 a dynamic similar to that of noise. This dynamic is called "identical synchronization." However, the requirement for two identical chaotic systems hinders the development of this chaotic cryptography technology.

[0009] To overcome this drawback, a so-called "generalized" synchronization method was developed. Although this method facilitates synchronization between two structurally different chaotic systems, the simplifications it introduces are such that they limit or even negate its usefulness for chaotic cryptography.

[0010] An object of the present invention is therefore to propose a system and a method for synchronizing structurally different chaotic resonators.

[0011] 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.

[0012] SUMMARY OF THE INVENTION

[0013] To achieve this objective, according to one implementation method, a system is planned comprising:

[0014] - a first resonator characterized by

[0015] • a first transduction coefficient TT,

[0016] • a first non-linearity coefficient a1,

[0017] • a first resonant frequency fol,

[0018] • a first dissipation Af1,

[0019] - a second resonator characterized by

[0020] • a second transduction coefficient T2',

[0021] • a second non-linearity coefficient a2,

[0022] • a second resonance frequency fo2,

[0023] • a second dissipation Af2.

[0024] The first and second resonators are structurally different so that at least one condition is met among: T1' T2', a1 a2, fo1 o2, Af1 Af2.

[0025] The system also includes:

[0026] - a first generator configured to provide, to the first resonator, a first excitation signal U1 oscillating at a first carrier frequency fc1 and modulated at a first modulation frequency fm1, said first resonator being configured to generate, from the first excitation signal U1, a first chaotic signal I11' in chaotic regime, via a variable X1,

[0027] - 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,

[0028] - a first demodulator D1 configured to provide a first signal U1” corresponding to the demodulation of the first chaotic signal I11' at the first carrier frequency fc1 ,

[0029] - 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,

[0030] - 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 between the first and second signals U1” - U2”.

[0031] Advantageously, the system includes at least one of the following:

[0032] - a dissipation correction module configured to modify the first modulation frequency fm1 of the first excitation signal U1 and / or the second modulation frequency fm2 of the second excitation signal U2 so that fm1 / Af1 = fm2 / Af2, and to speed up or slow down the first chaotic signal U1' and / or the second chaotic signal U2' so that the first and second chaotic signals U1', U2' evolve at the same characteristic frequency,

[0033] - a non-linearity correction module configured to modify an amplitude A1 of the first excitation signal U1 and / or an amplitude A2 of the second excitation signal U2, such that a1.X1 2 / Af1 = a2.X2 2 / Af2,

[0034] - a resonance frequency correction module configured to modify the first carrier frequency fc1 of the first excitation signal U1 and / or the second carrier frequency fc2 of the second excitation signal U2 such that (f01 - fc1) / Af1 = (f02 - fc2) / Af2,

[0035] - an amplitude correction module configured to modify an amplitude AT or A1” of the first chaotic signal UT or the first signal UT' respectively, and / or to modify an amplitude A2' or A2” of the second chaotic signal U2' or the second signal U2” respectively, so that A1” = A2”.

[0036] These modules advantageously allow for the simulation of identical characteristics for structurally different resonators. This makes it possible to synchronize 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 to exchange information securely using chaotic cryptography. The system's range of applications is significantly expanded.

[0037] Another advantage of this system is that the resonators themselves remain unchanged. Only the excitation and chaotic signals are corrected through the addition of the different modules. This allows structurally different resonators to retain their integrity while enabling the synchronization of the chaotic signals emanating from them. The system's implementation is thus simplified.

[0038] 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:

[0039] - Correct the dissipation of the first and / or second resonator by modifying the first modulation frequency fm1 of the first excitation signal U1 and / or the second modulation frequency fm2 of the second excitation signal U2 so that fm1 / Af1 = fm2 / Af2,

[0040] - 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,

[0041] - Correct the non-linearity of the first and / or second resonator by modifying an amplitude A1 of the first excitation signal U1 and / or an amplitude A2 of the second excitation signal U2, such that a1.X1 2 / Af1 = a2.X2 2 / Af2, - Correct the resonance frequency of the first and / or second resonator by modifying the first carrier frequency fc1 of the first excitation signal U1 and / or the second carrier frequency fc2 of the second excitation signal U2 so that (f01 - fc1 ) / Af1 = (f02 - fc2) / Af2,

[0042] - Correct the amplitude AT or A1” of the first chaotic signal UT 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 A1” = A2”.

[0043] BRIEF DESCRIPTION OF THE FIGURES

[0044] 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:

[0045] Figure 1 illustrates the operation of a MEMS type resonator in linear and non-linear regimes.

[0046] Figure 2 illustrates a part of the system comprising a non-linear resonator and its generator, as well as various intrinsic and extrinsic parameters relating to this part, according to an embodiment of the present invention.

[0047] Figure 3 illustrates an example of a chaotic signal emitted by a nonlinear resonator after demodulation when excited in chaotic regime, according to an embodiment of the present invention.

[0048] Figure 4 illustrates a synchronization system by coupling two substantially identical non-linear resonators, according to the prior art.

[0049] Figure 5 illustrates a synchronization system by coupling two structurally different non-linear resonators, according to an embodiment of the present invention.

[0050] Figure 6 illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0051] Figure 7 illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0052] Figure 8 illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0053] Figure 9 illustrates a synchronization system by coupling two structurally different non-linear resonators, according to another embodiment of the present invention.

[0054] The drawings are given as examples and are not limiting to the invention. They constitute schematic representations of principle intended to facilitate understanding of the invention and are not necessarily to scale with practical applications.

[0055] DETAILED DESCRIPTION OF THE INVENTION

[0056] Before proceeding with a detailed review of embodiments of the invention, optional features that may be used in combination or alternatively are listed below:

[0057] As an example, the system includes a dissipation correction module, a non-linearity correction module, a resonance frequency correction module, and an amplitude correction module. This allows for the correction of all intrinsic parameters of one or both resonators.

[0058] In one example, the dissipation correction module includes a voltage-controlled oscillator VCO1m delivering the first modulation frequency fm1 at the first generator and / or a voltage-controlled oscillator VCO2m delivering the second modulation frequency fm2 at the second generator. The difference in dissipation between the resonators is compensated for by adjusting the modulation frequency(ies) of the generator(s). In another example, the dissipation correction module includes a buffer that replicates one of the first and second signals U1 and U2, speeding it up or slowing it down. This allows the timescales over which the chaotic dynamics from the resonators develop to be adjusted, so that the demodulated chaotic signals U1 and U2 have the same temporal distribution.

[0059] As an 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.

[0060] In one example, the resonance frequency correction module includes a voltage-controlled oscillator VCO1c delivering the first carrier frequency fc1 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 generator(s). In another example, the amplitude correction module includes a gain G1' 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.

[0061] In one example, the first and second resonators are micrometric or nanometric electromechanical devices. In another example, the variables X1 and X2 correspond respectively to the displacement amplitudes of a moving element of the first and second resonators.

[0062] As an example, the coupling loop includes:

[0063] - a signal subtraction element comprising

[0064] • a first input receiving the first signal U1”,

[0065] • a second input receiving the second signal U2”,

[0066] • an output providing a U1” - U2” signal to be modulated,

[0067] - a modulator M comprising:

[0068] • an input receiving the signal to be modulated U1” - U2”,

[0069] • an output providing the modulated US synchronization signal,

[0070] - a signal addition element comprising:

[0071] • an input receiving the second excitation signal U2,

[0072] • an input receiving the US synchronization signal,

[0073] • 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.

[0074] According to one example, the demodulator D1 can be placed at the input or output of the gain G1'.

[0075] According to one example, the demodulator D2 can be placed at the input or output of the gain G2'.

[0076] 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.

[0077] In one example, gain G replaces one of either gain G1' or gain G2'. This reduces the cost and / or size of the system.

[0078] Micro / N-EMS (micro / neometry-oriented magnetic fields) are devices of micro / nanometer size that convert mechanical processes into electrical processes, and vice versa. Resonant micro / neometry-oriented fields are unique in that they incorporate a vibrating mass that transfers mechanical energy into electrical energy, or vice versa. This allows for the design of applications such as energy harvesters or force sensors, depending on the chosen geometry.

[0079] 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. Applying 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 fo. 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, 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 carefully chosen, the beam will erratically alternate between its two bistable states, making its behavior unpredictable and chaotic. The resonator then operates in a chaotic regime.

[0080] The chaotic dynamics reside in the vibration amplitude of the nonlinear resonator. The resonator vibrates periodically at the frequency fc1, 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 is relevant 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, a demodulator, such as synchronous detection, must be used.

[0081] The chaotic signals UT and U2' each possess a characteristic frequency, even though they are chaotic (that is, these signals UT 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, U1 has a modulation frequency of fm1, and the first signals UT, U1', have a characteristic frequency of fm1. U2 has a modulation frequency of fm2, and the second signals U2', U2', have a characteristic frequency of fm2.

[0082] 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 UT' to be modified so that it has the characteristic frequency fm2. This allows U2" to be subtracted from U1".

[0083] In the present invention, for the sake of brevity, the first signal U1'' corresponds to the first demodulated chaotic signal UT. The second signal U2'' corresponds to the second demodulated chaotic signal U2'.

[0084] Resonant microdisks are another example of resonators exhibiting significant nonlinearity. Other geometries are perfectly conceivable for the nonlinear resonators of the present invention.

[0085] 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 resonant 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.

[0086] 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.

[0087] Dimensional values ​​are understood to be within manufacturing and measurement tolerances.

[0088] 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 approximately normal to a plane means a direction at an angle of 90±5° to the plane.

[0089] In the following sections, the resonators described and illustrated as 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 resonator's readout or output signal.

[0090] 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 resonator's resonance frequency, 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.

[0091] Figure 2 illustrates the different parameters that characterize the nonlinear resonator and its generator. The generator delivers an excitation signal characterized by its amplitude or voltage U, and its carrier frequency f c and its modulation frequency f m The resonator transforms this electrical excitation signal into mechanical displacement via the electromechanical transduction coefficient T. In particular, in 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.

[0092] In the nonlinear regime of the resonator, the nonlinearity mainly observed in mechanics, known as Duffing's nonlinearity, is written as aX 2 , with a being the coefficient of non-linearity.

[0093] The linear resonator has a resonant frequency fo. In chaotic conditions, this is the shift between the frequency f c of the excitation signal and the resonance frequency fo which allows to characterize the chaos generated.

[0094] The resonator also exhibits dissipation Af. From a physical perspective, dissipation corresponds to the resonator's losses. Changing the dissipation alters the resonator's frequency response by broadening or narrowing the resonance curve.

[0095] The parameters T', a, fo, 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, f c and f m These 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.

[0096] In 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.

[0097] Figure 4 illustrates a classic coupling situation between two identical or nearly identical resonators RA and RB. Resonator RA is excited by generator EA, which delivers an excitation signal UA with a carrier frequency f c A and a modulation frequency f m A. The resonator RA, characterized by its transduction coefficient TA', its nonlinearity coefficient aA, its resonance frequency foA, and its dissipation AfA, generates a chaotic signal UA in response to the excitation signal UA. Similarly, the resonator RB is excited by the generator EB, which delivers an excitation signal UB with a carrier frequency f c B and a modulation frequency f mB. The resonator RB, characterized by its transduction coefficient TB', its nonlinearity coefficient aB, its resonant frequency foB, and its dissipation AfB, generates a chaotic signal UB' in response to the excitation signal UB. The resonators RA and RB are coupled by the coupling loop 30. This coupling loop 30 allows a "synchronization" signal, corresponding to the difference between the chaotic signals and UB', to be fed back into the resonator RB. 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.

[0098] This coupling loop 30 allows, in a known manner, the synchronization of the chaotic signals UA' and U B' when the resonators RA and RB are identical, that is, when TA = TB', aA = aB, foA = foB, and AfA = AfB. The resonators RA and RB are exponentially sensitive to variations. A minute change in pressure and / or temperature and / or humidity will induce a different dynamic for the resonator RA than for the resonator RB, even if the parameters of the resonators RA and RB are identical. The chaotic signals UA and U B' remain similar (they exhibit the same statistics, the same figures of merit), but their temporal evolution is different. However, thanks to the coupling, the reinjection of the synchronizing signal into the resonator RB will force the latter to align with the dynamics of the 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 disappears. However, as soon as there is a deviation, the coupling reappears and allows the resonators RA and RB to resynchronize. This synchronization is only possible if the parameters of the two resonators RA and RB are identical, and if the excitation signals UA(t) and UB(t) are identical.

[0099] Within the framework 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 nonlinearity 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.

[0100] In the following diagram, resonators R1 and R2 are structurally different. Resonator R1 is excited by generator E1, which delivers an excitation signal U1 with a carrier frequency f c1 and a modulation frequency f m 1. Resonator R1, characterized by its transduction coefficient TT, its nonlinearity coefficient a1, its resonant frequency fol, and its dissipation Af1, generates a chaotic signal UT in response to the excitation signal UT. This chaotic signal UT is then demodulated from the carrier fc1 by demodulator D1, which generates the demodulated chaotic signal U1. Similarly, resonator R2 is excited by generator E2, which delivers an excitation signal U2 with a carrier frequency f c 2 and a modulation frequency f m2. The resonator R2, characterized by its transduction coefficient T2', its nonlinearity coefficient a2, its resonance frequency fo2, 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 met: T1 T2' and / or a1 a2 and / or f0102 and / or Af1 Af2.

[0101] Figure 5 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, demodulation modules D1 and D2, a modulation module M added within the coupling loop 30, and a dissipation correction module. In this case, Af1 Af2.

[0102] The first correction to be made to the system is to correct the dissipation of resonators R1, R2, since the effects of dissipation impact in particular the corrections on the non-linearity and the resonance frequency of the resonators.

[0103] To correct for the dissipation of resonators R1, R2, the modulation frequencies f m 1 and / or f m 2 are modified so that: / ml fm? tofl f2

[0104] This frequency modulation correction can be obtained via voltage-controlled oscillators VCO1m (illustrated by reference 11 in Figure 5), VCO2m (illustrated by reference 21 in Figure 5) at the generators E1, E2. This frequency modulation correction can also be carried out by other methods.

[0105] To account for the effects of dissipation on the non-linearity and resonance frequency of the resonators, care will be taken to ensure that: al. I 2> a2. X2 2

[0106] A / l A / 2 and f o l - f c l _ f o 2 - f c 2

[0107] Hfl Hf2

[0108] The non-linearity between R1 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 R1 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 need to be excited at 19.9 kHz to maintain this ratio of 10: (20000-19900) / 10 = 10. The equivalence between resonators R1 and R2 is relative.

[0109] 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 chaotic signals is “faster” than the other. The dissipation correction module therefore includes an element 40 to slow down or speed up the chaotic dynamics of resonator R1 so that its speed is perceived as identical to that of resonator R2. This element 40 can include a buffer 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 Figure 5, it is the first signal U1”, which is sped up or slowed down, so as to present the same timescale as the second signal U2”.

[0110] Figure 6 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, demodulation modules D1 and D2, a modulation module M added within the coupling loop 30, and a non-linearity correction module. In this case, a1 a2.

[0111] The non-linearity of resonators R1, R2, which is written respectively as a1.X1 2 and o2.X2 2 This can be advantageously corrected by modifying the amplitudes of U1 and U2, respectively. Since X1 evolves linearly with U1, a change in U1 results in a change in the overall nonlinearity a1.X1 2 of resonator R1. Similarly, since X2 evolves linearly with U2, a change in U2 leads to a change in the overall non-linearity a2.X2 2of resonator R2. The amplitudes of U1 and / or U2 can be modified via gains G1 and / or G2 placed at the output of generators E1 and / or E2, respectively, as illustrated in Figure 6. Alternatively, this amplitude modification can be performed directly at generators E1 and / or E2. The amplitude modification is carried out in such a way as to satisfy relation [Math2] given above.

[0112] Figure 7 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, demodulation modules D1 and D2, a modulation module M added within the coupling loop 30, and a resonance frequency correction module. In this case, f o 1 o2.

[0113] The resonance frequencies fol, fo2 are characteristic of resonators R1, R2. Regarding the chaotic dynamics of each of these resonators R1, R2, it is the shift between the frequency f c 1, f c 2 of the excitation signal LU, U2 and the resonance frequency f01, fo2 characterize or modify 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 f c 1, f c2 of the excitation signals U1, U2. This resonance frequency correction can be obtained via voltage-controlled oscillators VCO1c (illustrated by reference 12 in Figure 7), VCO2c (illustrated by reference 22 in Figure 7) at the generators E1, E2. The modification of the frequencies f c 1, f c 2 is performed in such a way as to satisfy relation [Math3] provided above.

[0114] Figure 8 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, demodulation modules D1 and D2, a modulation module M added within the coupling loop 30, and an amplitude correction module. In this case, T1' T2'.

[0115] When the transduction coefficients of resonators R1 and R2 are different, the displacements X1 and X2 associated with the same excitation signal are different. To compensate for this difference without structurally modifying the resonators, one solution is to adjust U1 and / or U2 to obtain UT = U2'. Adjusting U1 and / or U2 is nevertheless preferred for correcting the nonlinearity of resonators R1 and R2, as described above. Another solution is to directly modify UT and / or U2' to achieve the same result (UT = U2'). The modification of the amplitudes of UT and / or U2' can be carried out via gains GT and / or G2' placed at the output of resonators R1 and / or R2, respectively, as illustrated in Figure 8. According to one possibility, one of the gains GT or G2' can correspond to the gain interposed between the subtraction element 31 and the addition element 32 of the coupling loop 30.

[0116] Figure 9 illustrates an embodiment of the synchronization system according to the invention, comprising a conventional coupling loop 30, the demodulation modules D1 and D2, the modulation module M added within the coupling loop 30, and all the correction modules described above (dissipation correction, non-linearity correction, resonance frequency correction, amplitude correction). In this case, T1' T2' and a1 a2 and fo1 fo2 and Af1 Af2. The various elements 11, 12, 21, 22, D1, D2, M, G1, GT, G2, G2', 40 of the coupling loop and of each of the modules are shown in Figure 9. Care should be taken to perform the dissipation correction first, which notably impacts the non-linearity correction and the resonance frequency correction, as indicated above.

[0117] The implementation in the synchronization system according to the invention of the set of elements 11, 12, 21, 22, D1, D2, M, G1, GT, G2, G2', 40 advantageously allows the synchronization of two arbitrarily different chaotic resonators R1, R2.

[0118] This synchronization system has, for example, been implemented for resonators R1 and R2 in the form of submillimeter diameter, 10 µm thick disks placed under vacuum (<1 mbar), whose transduction is piezoelectric. The resonance frequencies of these two disks are fo1 = 70 kHz and fo2 = 160 kHz. The nonlinearity coefficients of these two disks are a1 = 49 kHz / V 2 and a2 = 23kHz / V 2The transduction coefficients of these two disks are such that T1*TT = 0.22V / V and T2*T2' = 0.47V / V. The dissipations of these two disks are equivalent: Af1 = Af2 = 130Hz. The two resonators R1, R2 are therefore quite different and exhibit very different intrinsic characteristics. When these two resonators R1, R2 are excited in chaotic operation, by applying the frequency offsets and gains to the different signals as indicated above, the two resonators R1, R2 synchronize with a deviation of less than 2% on average.

[0119] 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 the resonators. The invention thus makes it possible to obtain chaotic twins from structurally different resonators.

[0120] In particular, MEMS resonators already exhibit similarities: they are characterized by a resonant frequency, dissipation, nonlinearity, excitation force, and displacement, for a wide range of MEMS applications (accelerometer, microphone, gyroscope, energy harvester, gas sensor, etc.). The invention allows, in practice, the normalization of the dynamics of two MEMS resonators by adjusting the forces and frequencies involved. This ultimately yields two identical dynamics, even in chaotic regimes. 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," making it possible to synchronize them.

[0121] 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, D1, D2, M, G1, GT, G2, G2', 40, that is to say to all elements or components which perform the same function as elements 11, 12, 21, 22, D1, D2, M, G1, GT, G2, G2', 40 of the synchronization system according to the invention.

Claims

DEMANDS 1. System comprising: • a first resonator (R1) characterized by a first transduction coefficient TT, a first non-linearity coefficient a1, a first resonance frequency fol, a first dissipation Af1, • a second resonator (R2) characterized by a second transduction coefficient T2', a second non-linearity coefficient a2, a second resonance frequency fo2, a second dissipation Af2, the first and second resonators (R1, R2) being structurally different such that at least one condition is satisfied among: T1' T2', a1 a2, fo1 fo2, Af1 Af2, • a first generator (E1) configured to provide, to the first resonator (R1), a first excitation signal U1 oscillating at a first carrier frequency fc1 and modulated at a first modulation frequency fm1, said first resonator being configured to generate, from the first excitation signal U1, a first chaotic signal UT in chaotic regime, via a variable X1, • 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 (D1) configured to provide a first signal U1” corresponding to the demodulation of the first chaotic signal UT at the first carrier frequency fc1, • 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 (30) configured to inject into the second resonator (R2) a US synchronization signal oscillating at the second carrier frequency fc2 and modulated by the difference between the first and second signals U1” - 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 fm1 of the first excitation signal U1 and / or the second modulation frequency fm2 of the second excitation signal U2 so that fm1 / Af1 = fm2 / Af2, and to speed up or slow down 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, • a non-linearity correction module configured to modify an amplitude A1 of the first excitation signal U1 and / or an amplitude A2 of the second excitation signal U2, such that a1.X1 2 / Af1 = a2.X2 2 / Af2, • a resonance frequency correction module configured to modify the first carrier frequency fc1 of the first excitation signal U1 and / or the second carrier frequency fc2 of the second excitation signal U2 such that (fol - fc1) / Af1 = (fol2 - fc2) / Af2, • an amplitude correction module configured to modify an amplitude A1' or A1” of the first chaotic signal U1' or signal U1” respectively, and / or to modify an amplitude A2' or A2” of the second chaotic signal U2' or signal U2” respectively, so that A1” = A2”.

2. 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 VCO1m (11) delivering the first modulation frequency fm1 at the level of the first generator (E1) and / or a voltage-controlled oscillator VCO2m (21) delivering the second modulation frequency fm2 at the level of the second generator (E2), and a buffer memory (40) replicating one of the first and second signals U1”, U2” by 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 VCO1c (12) delivering the first carrier frequency fc1 at the level of the first generator (E1) and / or a voltage-controlled oscillator VCO2c (22) delivering the second carrier frequency fc2 at the level of the second generator (E2).

6. System according to any one of the preceding claims, wherein the amplitude correction module comprises a gain G1' 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 (R1, R2) are micrometric or nanometric electromechanical devices and wherein the variables X1, X2 correspond respectively to displacement amplitudes of a moving element (100) of the first and second resonators (R1, 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”, an output providing a signal U1” - U2” to be modulated, • a modulator (M) comprising: an input receiving the signal to be modulated U1” - U2”, an output providing the modulated synchronization signal US, • a signal addition element (32) comprising: an input receiving the second excitation signal U2, an input receiving the modulated synchronization signal US, 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.

9. 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.

10. System according to the preceding claim and claim 6 in combination, wherein the gain G replaces one of the gain GT and the gain G2'.

11. A method for synchronizing the first and second resonators (R1, 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 (R1, R2) by modifying the first modulation frequency fm1 of the first excitation signal U1 and / or the second modulation frequency fm2 of the second excitation signal U2 so that fm1 / Af1 = 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 (R1, R2) by modifying an amplitude A1 of the first excitation signal U1 and / or an amplitude A2 of the second excitation signal U2, such that a1.X1 2 / Af1 = O2.X2 2 / Af2, • Correct the resonance frequency of the first and / or second resonator (R1, R2) by modifying the first carrier frequency fc1 of the first excitation signal U1 and / or the second carrier frequency fc2 of the second excitation signal U2 so that (f01 - fc1) / Af1 = (f02 - fc2) / Af2, • Correct the amplitude AT or A1” of the first chaotic signal UT 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 A1” = A2”.

12. Method according to the preceding claim, wherein the correction of the dissipation of the first and / or second resonator (R1, R2) is carried out before the other steps.

13. A method according to any one of the two preceding claims, wherein the correction of the amplitude AT or A1” and / or of the amplitude A2' or A2” is carried out after the other steps.

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