Nonlinear Electronic Circuits Research Articles

Dynamical complexity as a proxy for the network degree distribution

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological roles, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons, and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually coupled simple nonlinear electronic circuits

A novel generalized analytical solution for the normalized state equations of a class of mutually coupled chaotic systems is presented. To the best of our knowledge, for the first time synchronization dynamics of mutually coupled chaotic systems is studied analytically. The coupled dynamics obtained through the analytical solutions has been validated by numerical simulation results. Furthermore, we provide a suitable condition for the occurrence of synchronization in mutually coupled, second-order, non-autonomous chaotic systems through the analysis of the difference system on the stability of fixed points. The bifurcation of the eigenvalues of the difference system as a function of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states, is presented. The stability of synchronized states in each coupled system discussed in this article is studied using the Master Stability Function. Finally, the electronic circuit experimental results confirming the analytical and numerical results are presented.

Effect of fractional-order, time-delay and noisy parameter on slow-passage phenomenon in a nonlinear oscillator

In this paper, we report the effect of fractional order, time delay and noisy parameter on slow passage phenomenon in a nonlinear oscillator. We consider a second order LCR based nonlinear electronic circuit with a time varying resistor and use sinusoidal modulation on the resistor to change the resistance value. The time dependent parameter of a dynamical system causes slow passage effect which leads to bifurcation delay in the system dynamics and leaving the actual bifurcation point unpredictable. We find that the fractional order of the system significantly changes the magnitude of bifurcation delay and brings the system to oscillatory state. While the time delay in dynamical systems destroys the stable steady state leading it to oscillatory state. We study both these fractional order and time delay and their combined effect on the slow passage effect. We have also included the noise with the sinusoidal periodic modulation on the resistor to understand the effect of noise on the slow passage effect and found that the noise enhances the oscillatory behaviour of the system.

Robust impulsive reset observers of a class of switched nonlinear systems with unknown inputs

This paper is concerned with observer designs for a class of switched nonlinear systems with unknown inputs under average dwell time (ADT) switching. Firstly, for the unknown input (UI) elimination purpose, an output derivative-based method is considered. Then, in order to avoid using differentiator signals and meanwhile maintain a satisfactory estimation performance, a full-order impulsive reset switching observer is developed. The existence condition of such an observer is given via a Riccati equation. Besides, we also devote to the construction of the reduced-order observer. And furthermore, it is found that the condition under which a full-order observer exists also guarantees a reduced-order reset switching observer. Finally, a switched nonlinear electronic circuit system is used as the simulation example to verify the effectiveness of the obtained methods.

Convergence of Volterra series in simulation of nonlinear processes in electronic circuits

Physical and mathematical reasons for the divergence of functional Volterra series in calculation of nonlinear electronic circuits described by algebraic differential equations are considered. It is shown that the main reason is the character of the feedbacks involving nonlinear elements and leading to a high level of repeated combinational interactions. On model examples, the disruption of the iterative process due to the loss of continuity and differentiability of nonlinear operators on the real axis or on the complex plane is demonstrated.

Optimal phase synchronization in networks of phase-coherent chaotic oscillators.

We investigate the existence of an optimal interplay between the natural frequencies of a group of chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function J(ω,L) linking the natural frequencies ωi of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix L, the latter accounting for the specific organization of the network of interactions between oscillators. We use the classical Rössler system to show that the synchrony alignment function obtained for phase oscillators can be extended to phase-coherent chaotic systems. Finally, we carry out a series of experiments with nonlinear electronic circuits to show the robustness of the theoretical predictions despite the intrinsic noise and parameter mismatch of the electronic components.

Synchronization of strange non-chaotic attractors via unidirectional coupling of quasiperiodically-forced systems

In this paper, we present a numerical investigation on the robust synchronization phenomenon observed in a unidirectionally-coupled quasiperiodically-forced simple nonlinear electronic circuit system exhibiting strange non-chaotic attractors (SNAs) in its dynamics. The SNA obtained in the simple quasiperiodic system is characterized for its SNA behavior. Then, we studied the nature of the synchronized state in unidirectionally coupled SNAs by using the Master-Slave approach. The stability of the synchronized state is studied through the master stability functions (MSF) obtained for coupling different state variables of the drive and response system. The property of robust synchronization is analyzed for one type of coupling of the state variables through phase portraits, conditional lyapunov exponents and the Kaplan-Yorke dimension. The phenomenon of complete synchronization of SNAs via a unidirectional coupling scheme is reported for the first time.

Finding hidden oscillations in the operation of nonlinear electronic circuits

Several dynamic behaviours can simultaneously coexist in the operation of a nonlinear electronic circuit. However, some of these dynamics are difficult to be observed due to their sensibility to initial conditions. The use of describing functions to determine the coexistence of multiple dynamics in the operation of an electronic circuit with nonlinear feedback is proposed.

Inter-layer synchronization in multiplex networks of identical layers.

Inter-layer synchronization is a distinctive process of multiplex networks whereby each node in a given layer evolves synchronously with all its replicas in other layers, irrespective of whether or not it is synchronized with the other units of the same layer. We analytically derive the necessary conditions for the existence and stability of such a state, and verify numerically the analytical predictions in several cases where such a state emerges. We further inspect its robustness against a progressive de-multiplexing of the network, and provide experimental evidence by means of multiplexes of nonlinear electronic circuits affected by intrinsic noise and parameter mismatch.

Analytical treatment for synchronizing chaos through unidirectional coupling and implementation of logic gates

The idea of synchronization can be explicitly demonstrated by both numerical and analytical means on a nonlinear electronic circuit. Also, we introduce a scheme to obtain various logic gate structures, using synchronization of chaotic systems. By a small change in the response parameter of unidirectionally coupled nonlinear systems, one is able to construct various logic behaviours by both numerical and analytical methods.

Saddle-Node Bifurcations in Classical and Memristive Circuits

This paper addresses a systematic characterization of saddle-node bifurcations in nonlinear electrical and electronic circuits. Our approach is a circuit-theoretic one, meaning that the bifurcation is analyzed in terms of the devices’ characteristics and the graph-theoretic properties of the digraph underlying the circuit. The analysis is based on a reformulation of independent interest of the saddle-node theorem of Sotomayor for semiexplicit index one differential-algebraic equations (DAEs), which define the natural context to set up nonlinear circuit models. The bifurcation is addressed not only for classical circuits, but also for circuits with memristors. The presence of this device systematically leads to nonisolated equilibria, and in this context the saddle-node bifurcation is shown to yield a bifurcation of manifolds of equilibria; in cases with a single memristor, this phenomenon describes the splitting of a line of equilibria into two, with different stability properties.

Nonideal Behavior of Analog Multipliers for Chaos Generation

In this brief, nonideal behaviors of analog multipliers are explicitly taken into account in the design of nonlinear electronic circuits. The nonidealities of the analog multipliers led to further nonlinear terms that, instead of being considered as parasitic effects, are here explicitly accounted for and exploited to generate new complex dynamics, including chaos. In fact, despite the accuracy of analog devices, nonideal effects are always present, even if often neglected. An analog circuit based on a new nonlinear system is designed and tested, in which the nonideal terms in the multiplier input/output relation are essential for the onset of chaos, thus demonstrating the suitability of the approach.

Master stability functions for a class of coupled simple nonlinear electronic circuits

The stability of the synchronized state of coupled, identical, simple, nonlinear electronic circuits is analyzed using the Master Stabilty Functions of the variational equations. The Master Stability Functions for different types of coupling schemes of the circuit variables are arrived at, and the stability of the synchronized state is observed using the simulation results.

Optimal Parametric Feedback Excitation of Nonlinear Oscillators.

An optimal parametric feedback excitation principle is sought, found, and investigated. The principle is shown to provide an adaptive resonance condition that enables unprecedentedly robust movement generation in a large class of oscillatory dynamical systems. Experimental demonstration of the theory is provided by a nonlinear electronic circuit that realizes self-adaptive parametric excitation without model information, signal processing, and control computation. The observed behavior dramatically differs from the one achievable using classical parametric modulation, which is fundamentally limited by uncertainties in model information and nonlinear effects inevitably present in real world applications.

Enhancing the stability of the synchronization of multivariable coupled oscillators.

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rössler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system.

Delay-based reservoir computing: noise effects in a combined analog and digital implementation.

Reservoir computing is a paradigm in machine learning whose processing capabilities rely on the dynamical behavior of recurrent neural networks. We present a mixed analog and digital implementation of this concept with a nonlinear analog electronic circuit as a main computational unit. In our approach, the reservoir network can be replaced by a single nonlinear element with delay via time-multiplexing. We analyze the influence of noise on the performance of the system for two benchmark tasks: 1) a classification problem and 2) a chaotic time-series prediction task. Special attention is given to the role of quantization noise, which is studied by varying the resolution in the conversion interface between the analog and digital worlds.

Vibrational resonance in an asymmetric bistable system with time-delay feedback

Vibrational resonance is a resonant dynamics induced by a high-frequency periodic force at the low-frequency of the input periodic signal, and the input periodic signal is enhanced by a high-frequency signal. In this paper, a linear time-delayed feedback bistable system with an asymmetric double-well potential driven by both low-frequency and high-frequency periodic forces is constructed. Based on this model, the vibrational resonance phenomenon is investigated. Making use of the method of separating slow motion from fast motion under the conditions of Ω>>ω (Ω is the frequency of the high-frequency signal and ω is the one of the low-frequency signal), equivalent equations to the slow motion and the fast motion are obtained. Neglecting the nonlinear factors, the analytical expression of the response amplitude Q can be obtained, and the effects of the time-delay parameter α and the asymmetric parameter r on the vibrational resonance are discussed in detail. Moreover, the locations at which the vibrational resonance occurs, are obtained by means of solving the condition for a resonance to occur. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter, i.e. in Q-α plot, α can induce the Q which is periodic with the periods of the high-frequency signal and the low-frequency signal. The locations at which the vibrational resonance occurs are not changed by the asymmetric parameter r. However, the resonance amplitude is enhanced with increasing r. Specifically, the resonance amplitude is greatly enhanced when r>0.15. On the other hand, in the symmetric case (r=0), BVR at which the vibrational resonance occurs is periodic with the periods of high-frequency signal and low-frequency signal as α increases, which is shown in BVR-α (B is the amplitude of the high-frequency signal) plot. In Q-Ω plot, Q is presented by multi-resonance at the small values of B and Ω, but Q tends to a fixed value at the small values of B and the large values of Ω. We believe that the above theoretical observations will stimulate the experimental study of vibrational resonance in nonlinear oscillators and electronic circuits with time-delayed feedback.

Non-reciprocal and highly nonlinear active acoustic metamaterials

Unidirectional devices that pass acoustic energy in only one direction have numerous applications and, consequently, have recently received significant attention. However, for most practical applications that require unidirectionality at audio and low frequencies, subwavelength implementations capable of the necessary time-reversal symmetry breaking remain elusive. Here we describe a design approach based on metamaterial techniques that provides highly subwavelength and strongly non-reciprocal devices. We demonstrate this approach by designing and experimentally characterizing a non-reciprocal active acoustic metamaterial unit cell composed of a single piezoelectric membrane augmented by a nonlinear electronic circuit, and sandwiched between Helmholtz cavities tuned to different frequencies. The design is thinner than a tenth of a wavelength, yet it has an isolation factor of >10 dB. The design method generates relatively broadband unidirectional devices and is a good candidate for numerous acoustic applications.

Discontinuous Spirals of Stable Periodic Oscillations

We report the experimental discovery of a remarkable organization of the set of self-generated periodic oscillations in the parameter space of a nonlinear electronic circuit. When control parameters are suitably tuned, the wave pattern complexity of the periodic oscillations is found to increase orderly without bound. Such complex patterns emerge forming self-similar discontinuous phases that combine in an artful way to produce large discontinuous spirals of stability. This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators. Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point. They are useful to optimize locking into desired oscillatory modes and to control complex systems. The organization of oscillations into discontinuous spirals is expected to be generic for a class of nonlinear oscillators.

Dynamics, control and information in delay-coupled systems: an overview

Dynamics, control and information in delay-coupled systems: an overview