Nonlinear Electronic Oscillator Research Articles

Observation of a phase transition from a continuous to a discrete time crystal

Discrete (DTCs) and continuous time crystals (CTCs) are novel dynamical many-body states, that are characterized by robust self-sustained oscillations, emerging via spontaneous breaking of discrete or continuous time translation symmetry. DTCs are periodically driven systems that oscillate with a subharmonic of the external drive, while CTCs are continuously driven and oscillate with a frequency intrinsic to the system. Here, we explore a phase transition from a continuous time crystal to a discrete time crystal. A CTC with a characteristic oscillation frequency ωCTC is prepared in a continuously pumped atom-cavity system. Modulating the pump intensity of the CTC with a frequency ωdr close to 2ωCTC leads to robust locking of ωCTC to ωdr/2 , and hence a DTC arises. This phase transition in a quantum many-body system is related to subharmonic injection locking of non-linear mechanical and electronic oscillators or lasers.

Real-time tracking of coherent oscillations of electrons in a nanodevice by photo-assisted tunnelling

Coherent collective oscillations of electrons excited in metallic nanostructures (localized surface plasmons) can confine incident light to atomic scales and enable strong light-matter interactions, which depend nonlinearly on the local field. Direct sampling of such collective electron oscillations in real-time is crucial to performing petahertz scale optical modulation, control, and readout in a quantum nanodevice. Here, we demonstrate real-time tracking of collective electron oscillations in an Au bowtie nanoantenna, by recording photo-assisted tunnelling currents generated by such oscillations in this quantum nanodevice. The collective electron oscillations show a noninstantaneous response to the driving laser fields with a T2 decay time of nearly 8 femtoseconds. The contributions of linear and nonlinear electron oscillations in the generated tunnelling currents were precisely determined. A phase control of electron oscillations in the nanodevice is illustrated. Functioning in ambient conditions, the excitation, phase control, and read-out of coherent electron oscillations pave the way toward on-chip light-wave electronics in quantum nanodevices.

An Ising machine based on networks of subharmonic electrical resonators

Combinatorial optimization problems are difficult to solve with conventional algorithms. Here we explore networks of nonlinear electronic oscillators evolving dynamically towards the solution to such problems. We show that when driven into subharmonic response, such oscillator networks can minimize the Ising Hamiltonian on non-trivial antiferromagnetically-coupled 3-regular graphs. In this context, the spin-up and spin-down states of the Ising machine are represented by the oscillators’ response at the even or odd driving cycles. Our experimental setting of driven nonlinear oscillators coupled via a programmable switch matrix leads to a unique energy minimizer when one exists, and probes frustration where appropriate. Theoretical modeling of the electronic oscillators and their couplings allows us to accurately reproduce the qualitative features of the experimental results and extends the results to larger graphs. This suggests the promise of this setup as a prototypical one for exploring the capabilities of such an unconventional computing platform.

Identifiability of structural networks of nonlinear electronic oscillators

The interplay between structure and function is critical in the understanding of complex systems, their dynamics and their behavior. We investigated the interplay between structural and functional networks by means of the differential identifiability framework, which here quantifies the ability of identifying a particular network structure based on (1) the observation of its functional network and (2) the comparison with a prior observation under different initial conditions. We carried out an experiment consisting of the construction of M=20 different structural networks composed of N=28 nonlinear electronic circuits and studied the regions where network structures are identifiable. Specifically, we analyzed how differential identifiability is related to the coupling strength between dynamical units (modifying the level of synchronization) and what are the consequences of increasing the amount of noise existing in the functional networks. We observed that differential identifiability reaches its highest value for low to intermediate coupling strengths. Furthermore, it is possible to increase the identifiability parameter by including a principal component analysis in the comparison of functional networks, being especially beneficial for scenarios where noise reaches intermediate levels. Finally, we showed that the regime of the parameter space where differential identifiability is the highest is highly overlapped with the region where structural and functional networks correlate the most.

Analysis of harmonic generation by a hydrogen-like atom using a quasi-classical non-linear oscillator model with realistic electron potential

A theoretical model of the nonlinear electron oscillations in an atom exposed to electro-magnetic wave is described. This model considers electron motion as a nonharmonic periodic function, unlike the current approach based on a perturbation solution of the linear harmonic oscillator. Our work demonstrates the use of realistic electron potential and correct radiation damping in the modeling of optical electron response. One of the main benefits of our approach is that it provides a way of computing the efficiencies of the up- and down-conversion of laser frequency.

Apparent remote synchronization of amplitudes: A demodulation and interference effect.

A form of "remote synchronization" was recently described, wherein amplitude fluctuations across a ring of non-identical, non-linear electronic oscillators become entrained into spatially-structured patterns. According to linear models and mutual information, synchronization and causality dip at a certain distance, then recover before eventually fading. Here, the underlying mechanism is finally elucidated through novel experiments and simulations. The system non-linearity is found to have a dual role: it supports chaotic dynamics, and it enables the energy exchange between the lower and higher sidebands of a predominant frequency. This frequency acts as carrier signal in an arrangement resembling standard amplitude modulation, wherein the lower sideband and the demodulated baseband signals spectrally overlap. Due to a spatially-dependent phase relationship, at a certain distance near-complete destructive interference occurs between them, causing the observed dip. Methods suitable for detecting non-trivial entrainment, such as transfer entropy and the auxiliary system approach, nevertheless, reveal that synchronization and causality actually decrease with distance monotonically. Remoteness is, therefore, arguably only apparent, as also reflected in the propagation of external perturbations. These results demonstrate a complex mechanism of dynamical interdependence, and exemplify how it can lead to incorrectly inferring synchronization and causality.

Уединённая правополяризованная электромагнитная волнав релятивистской плазме

In this paper of the nonlinear laser wave propagation in the hot plasma along the strong exter-nal magnetic field under the electron cyclotron resonance conditions is investigated. The strongnonlinearity of such a process is caused by the relativistic electron movement and resonancewave ponderomotive force acting on the electrons. The system of equations for the enveloperight hand polarized laser pulse is derived using the hydrodynamics and Maxwell’s equations.The numerical integration of this system for the cold plasma case discovered the soliton solu-tions. This kind of solutions take a form of the envelope solitons containing inside them the plasma oscillations. The analytical expression for the energy density integral in a cold plasma isderived. It follows from the numerical results that for a hot plasma under cyclotron resonanceconditions the soliton solution becomes unstable. In this case the energy density conservationbreaks down, but electron momentum density conserves. It is concluded that the nonlinear sat-uration of the field amplitude is due to the plasma charge separation under electromagneticradiation pressure. In this case the discrete set of the envelope soliton carrier frequency is de-termined by the ratio of the frequency of the nonlinear longitudinal electron oscillations to theLangmuir frequency of plasma. For the low density plasma the discrete frequency spectrumobtained by the numerical integration transforms to the continuous one.

A Circuit for Automatic Measurement of Bifurcation Diagram in Nonlinear Electronic Oscillators

Experimental bifurcation diagram is very helpful to describe the global behaviour of nonlinear electronic oscillators. The diagram allows insight gain of the oscillators behaviour for a parameter variation. However, when the bifurcation parameter is a resistance, experimental bifurcation diagrams are hard to achieve. In this case, if the potentiometer resistance has been used to change the gain, it is possible to replace the potentiometer by an alternative circuit using operational transconductance amplifier. This kind of circuit is presented in this work and experimental results are provided to piecewise affine Rossler electronic oscillator. The results obtained show a very good matching with the theoretical diagram. Then, if the bifurcation parameter is a resistance, voltage or current, we have implemented a circuit to extract the experimental bifurcation diagram of all nonlinear electronic oscillators.

Critical phenomena at a first-order phase transition in a lattice of glow lamps: Experimental findings and analogy to neural activity.

Networks of non-linear electronic oscillators have shown potential as physical models of neural dynamics. However, two properties of brain activity, namely, criticality and metastability, remain under-investigated with this approach. Here, we present a simple circuit that exhibits both phenomena. The apparatus consists of a two-dimensional square lattice of capacitively coupled glow (neon) lamps. The dynamics of lamp breakdown (flash) events are controlled by a DC voltage globally connected to all nodes via fixed resistors. Depending on this parameter, two phases having distinct event rate and degree of spatiotemporal order are observed. The transition between them is hysteretic, thus a first-order one, and it is possible to enter a metastability region, wherein, approaching a spinodal point, critical phenomena emerge. Avalanches of events occur according to power-law distributions having exponents ≈3/2 for size and ≈2 for duration, and fractal structure is evident as power-law scaling of the Fano factor. These critical exponents overlap observations in biological neural networks; hence, this circuit may have value as building block to realize corresponding physical models.

Experimental implementation of maximally synchronizable networks

Maximally synchronizable networks (MSNs) are acyclic directed networks that maximize synchronizability. In this paper, we investigate the feasibility of transforming networks of coupled oscillators into their corresponding MSNs. By tuning the weights of any given network so as to reach the lowest possible eigenratio λN/λ2, the synchronized state is guaranteed to be maintained across the longest possible range of coupling strengths. We check the robustness of the resulting MSNs with an experimental implementation of a network of nonlinear electronic oscillators and study the propagation of the synchronization errors through the network. Importantly, a method to study the effects of topological uncertainties on the synchronizability is proposed and explored both theoretically and experimentally.

Synchronization of Heterogeneous Oscillators by Noninvasive Time-Delayed Cross Coupling.

We demonstrate that nonidentical systems, in particular, nonlinear oscillators with different time scales, can be synchronized if a mutual coupling via time-delayed control signals is implemented. Each oscillator settles on an unstable state, say a fixed point or an unstable periodic orbit, with a coupling force which vanishes in the long time limit. We present the underlying theoretical considerations and numerical simulations, and, moreover, demonstrate the concept experimentally in nonlinear electronic oscillators.

Wave breaking of nonlinear electron oscillations in a warm magnetized plasma

Wave breaking phenomena of nonlinear electron oscillations around a homogeneous background of massive ions have been studied in a warm magnetized plasma by using Lagrangian variables. An inhomogeneity in the background magnetic field is shown to induce phase mixing and thus breaking of the oscillations. A nonlinear analysis in Lagrangian variables predicts that wave breaking may disappear above a critical value of the electron temperature. An estimate for the critical temperature has been provided.

Nonlinear electron oscillations in a warm plasma

A class of nonstationary solutions for the nonlinear electron oscillations of a warm plasma are presented using a Lagrangian fluid description. The solution illustrates the nonlinear steepening of an initial Gaussian electron density disturbance and also shows collapse behavior in time. The obtained solution may indicate a class of nonlinear transient structures in an unmagnetized warm plasma.

On Noise Analysis of Oscillators Based on Statistical Mechanics

On Noise Analysis of Oscillators Based on Statistical MechanicsIn this paper a new approach of thermal noise analysis of electronic oscillators is presented. Although nonlinear electronic oscillators are one of the most essential subcircuits in electronic systems typical design concepts for these oscillators are based on ideas of linear circuits. Because the functionality of oscillators depends on nonlinearities, advanced design methods are developed where nonlinearities are an integral part. Since low voltage oscillator concepts have to be developed in modern IC technologies there is a need to include at least thermal noise aspects into the design flow. For this reason we developed new physical descriptions of thermal noise in electronic oscillators where we use ideas from nonequilibrium statistical mechanics as well as the Langevin approach. We illustrate our concepts by some examples.

Nonlinear electron oscillations in a viscous and resistive plasma

Nonlinear, spatially periodic, long-wavelength electrostatic modes of an electron fluid oscillating against a motionless ion fluid (Langmuir waves) are given, with viscous and resistive effects included. The cold plasma approximation is adopted, which requires the wavelength to be sufficiently large. The pertinent requirement valid for large amplitude waves is determined. The general nonlinear solution of the continuity and momentum transfer equations for the electron fluid along with Poisson's equation is obtained in simple parametric form. It is shown that in all typical hydrogen plasmas, the influence of plasma resistivity on the modes in question is negligible. Within the limitations of the solution found, the nonlinear time evolution of any (periodic) initial electron number density profile ne(x,t=0) can be determined (examples). For the modes in question, an idealized model of a strictly cold and collisionless plasma is shown to be applicable to any real plasma, provided that the wavelength λ>>λmin(n(0),Te) , where n(0)=const and Te are the equilibrium values of the electron number density and electron temperature. Within this idealized model, the minimum of the initial electron density n(e)(xmin,t=0) must be larger than half its equilibrium value, n(0)/2 . Otherwise, the corresponding maximum n(e)(xmax,t=τ(p)/2) , obtained after half a period of the plasma oscillation blows up. Relaxation of this restriction on n(e)(x,t=0) as one decreases λ , due to the increase of the electron viscosity effects, is examined in detail. Strong plasma viscosity is shown to change considerably the density profile during the time evolution, e.g., by splitting the largest maximum in two.

Crisis and stochastic resonance in Shinriki’s circuit

In nonlinear electronic oscillators of the Chua type the merging of two precritical attractors leads to crisis-induced chaos–chaos intermittency. This behavior can be induced either by addition of noise to the precritical system or by variation of a control parameter. Close to such a crisis small periodic signals can be enhanced by the addition of noise via the mechanism of stochastic resonance. We present an experimental observation of this effect as well as a thorough description of the critical behavior of the system.

Inducing Chaos in Electronic Circuits by Resonant Perturbations

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We propose a scheme to induce chaotic attractors in electronic circuits. The applications that we are interested in stipulate the following three constraints: 1) the circuit operates in a stable periodic regime far away from chaotic behavior; 2) no parameters or state variables of the circuit are directly accessible to adjustment and 3) the circuit equations are unknown and the only available information is a time series (or a signal) measured from the circuit. Under these conditions, a viable approach to chaos induction is to use external excitations such as a microwave signal, assuming that a proper coupling mechanism exists which allows the circuit to be perturbed by the excitation. The question we address in this paper is how to choose the waveform of the excitation to ensure that sustained chaos (chaotic attractor) can be generated in the circuit. We show that weak resonant perturbations with time-varying frequency and phase are generally able to drive the circuit into a hierarchy of nonlinear resonant states and eventually into chaos. We develop a theory to explain this phenomenon, provide numerical support, and demonstrate the feasibility of the method by laboratory experiments. In particular, our experimental system consists of a Duffing-type of nonlinear electronic oscillator driven by a phase-locked loop (PLL) circuit. The PLL can track the frequency and phase evolution of the target Duffing circuit and deliver resonant perturbations to generate robust chaotic attractors. </para>

Bistable phase control via rocking in a nonlinear electronic oscillator

We experimentally demonstrate the effective rocking [de Valcarcel and Staliunas, Phys. Rev. E 67, 026604 (2003)] of a nonlinear electronic circuit operating in a periodic regime. Namely, we show that by driving a Chua circuit with a periodic signal, whose phase alternates (also periodically) in time, we lock the oscillation frequency of the circuit to that of the driving signal, and its phase to one of two possible values shifted by pi, and lying between the alternating phases of the input signal. In this way, we show that a rocked nonlinear oscillator displays phase bistability. We interpret the experimental results via a theoretical analysis of rocking on a simple oscillator model, based on a normal form description (complex Landau equation) of the rocked Hopf bifurcation.

Internal crisis in a second-order non-linear non-autonomous electronic oscillator

The internal crisis of a second-order non-linear non-autonomous chaotic electronic circuit is studied. The phase portraits consist of two interacting sub-attractors, a chaotic and a periodic one. Maximal Lyapunov exponents were calculated, for both the periodic and the chaotic waveforms, in order to confirm their nature. Transitions between the chaotic and the periodic sub-attractors become more frequent by increasing the circuit driving frequency. The frequency distribution of the corresponding laminar lengths and their average values indicate that an internal crisis takes place in this circuit, manifested in the intermittent behaviour of the corresponding orbits.

Strong Coupling of Nonlinear Electronic and Biological Oscillators: Reaching the “Amplitude Death” Regime

Interaction between an electronic and a biological circuit has been investigated for a pair of electrically connected nonlinear oscillators, with a spontaneously oscillating olivary neuron as the single-cell biological element. By varying the coupling strength between the oscillators, we observe a range of behaviors predicted by model calculations, including a reversible low-energy dissipation "amplitude death" where the oscillations in the coupled system cease entirely.