Field-Programmable Gate Array-Based Chaos Oscillator Implementation for Analog-Discrete and Discrete-Analog Chaotic Synchronization Applications

This work focuses on evaluating the behavior of analog chaos oscillators in field-programmable gate arrays (FPGAs). This work is motivated by a new approach to designing chaos-based communication systems using chaos oscillator circuits implemented in hardware in the transmitter and the mathematical models of the oscillator implemented on an FPGA in the receiver. Such a hybrid approach opens new possibilities for chaos-based modulation schemes for wireless sensor network (WSN) applications. This work brings a hybrid chaos-based communication system closer to realization by implementing the chaos oscillators on an FPGA and achieving analog-discrete and discrete-analog chaotic synchronization. First, this paper derives a model that simulates the dynamics of Vilnius and RC chaos oscillators using Euler-Cromer numerical integration in fixed-point arithmetic. The derived MATLAB model precisely describes the digital design and is thus directly transferred to VHDL. The synthesized digital design is compiled onto an FPGA chip and is then used to achieve analog-discrete and discrete-analog Pecora-Carroll chaotic synchronization.

Analytic integrability of generalized 3-dimensional chaotic systems

Numerous recently introduced chaotic systems exhibit straightforward algebraic representations. In this study, we explore the potential for identifying a global analytic first integral in a generalized 3-dimensional chaotic system (2). Our work involves detailing the model of a new 3-D chaotic system characterized by three Lyapunov exponents-positive, zero, and negative. We depict the phase trajectories, illustrate bifurcation patterns, and visualize Lyapunov exponent graphs. The investigation encompasses both local and global analytic first integrals for the system, providing results on the existence and non-existence of these integrals for different parameter values. Our findings reveal that the system lacks a global first integral, and the presence or absence of analytic first integrals is contingent upon specific parameter values. Additionally, we present a formal series for the system, demonstrating 3D and 2D projections of the system (2) for a given set of initial conditions achieved by selecting alternative values for parameters a, b, c, d, r and l.

Global μ-synchronization for coupling delayed complex dynamical networks via event-triggered delayed impulsive control

This paper mainly focuses on solving the global μ-synchronization issue of complex dynamical networks (CDNs) by a novel event-triggered impulsive control (ETIC) method with time delays. This method combines the advantages of impulsive control and event-triggered control and gets rid of the limitation that the Lyapunov function decreases strictly monotonically with the sequence of event triggers. An event-triggered mechanism is specifically designed to realize μ-synchronization for CDNs in this paper, which means that event-triggered control has been applied to μ-synchronization field for the first time. Compared with periodic impulsive control, ETIC only produces control effects when the event-triggered mechanism are met, which is more in line with the actual situation. By Lyapunov-Razumikhin, recursion, etc, some valid global μ-synchronization criteria of CDNs are obtained and also Zeno behavior is avoided. Additionally, coupled delays and uncertainties are considered in CDNs. Finally, two numerical examples are shown to demonstrate the correctness of the designed ETIC strategy.

Investigation on the monolithically integrated chaotic optical transmitting chip based on parallel EAMs

Chaotic optical communication ensures information security at the physical layer. However, the monolithic integration of lasers and lithium niobate Mach-Zehnder modulators remains a challenge, limiting the progress of integrated chaotic optical communication systems based on an electro-optic feedback. Here, we propose the monolithically integrated chaotic optical transmitting chip based on the parallel EAMs and validate its performance from the perspectives of phase portraits, fast Fourier transform (FFT), probability density function (PDF), largest Lyapunov exponents, and bifurcation. The results demonstrate the feasibility of the chip, which is beneficial for the miniaturization and integration of the system.

Singular perturbation analysis in a coupled Chua's circuit with diffusion

This paper is concerned with the traveling wave solutions of a singularly perturbed system, which arises from the coupled arrays of Chua's circuit. By the geometric singular perturbation theory and invariant manifold theory, we prove that there exists a heteroclinic cycle consisting of the traveling front and back waves with the same wave speed. In particular, the expression of corresponding wave speed is also obtained. Furthermore, we show that the chaotic behavior induced by this heteroclinic cycle is hyperchaos.

Transmission and Decryption of the Audio Signal Masked with ECG by FDM Method

Today, the use of these methods as hybrids has provided the motivation to be a solution to important problems, since the existing methods are insufficient at some points in ensuring the security of personal data. In data security, the inability to decrypt and decrypt the signal to be encrypted retrospectively has always been the subject of research in terms of privacy. At this point, it was preferred to use the electrocardiography (ECG) signal, which is a signal that shows the vital signs of the human body and is also difficult to copy. In the study, firstly, the emulator circuit was obtained by using the mathematical model of the ECG signal. With this obtained signal, the audio signals are masked. The audio signal masked on the transmitter side and the signals providing synchronization were transmitted to the receiver side over a single channel using the frequency division multiplexing (FDM) method. Then, the sliding mode control (SMC) method was chosen for the synchronization of the ECG emulator circuits on the receiver and transmitter side. Histogram, spectral, mean square error (MSE), peak signal to noise ratio (PSNR), key space and key sensitivity, NSCR (number of sample change rate), UACI (unified average changing intensity) and PESQ (perceptual evaluation of speech quality) analyses were used to check the accuracy of the system. These analyses showed that the ECG encoding method has faster unit change, reduces synchronization time, minimizes losses and improves the security of the masked signal compared to other methods sent from two channels. Finally, use of an arrhythmia ECG signal for the synchronization signal on both the transmitter and receiver sides, the synchronization of this signal with the SMC method and the testing of a live audio recording in addition to the conversation, distinguishes the study from other existing studies and reveals its originality.

Experimental Study on FM-CSK Communication System for WSN

The current paper presents the experimental study of the frequency-modulated chaos shift keying (FM-CSK) communication system. The proposed system has the potential to enhance the security aspects of the physical layer to meet the needs for safe data transmission in wireless sensor networks (WSN). Compared to common digital FM-DCSK, the studied analog FM-CSK communication system provides a more straightforward design. As chaos oscillators are the core elements of the FM-CSK communication system, the paper investigates the selected oscillators’ properties, including the implementation aspects—the impact of reactive element value deviations on dynamics and synchronization stability. Chaotic dynamics are evaluated with the mean-square-displacement-based 0–1 test, while correlation analysis is used to evaluate synchronization. The impact of the different chaos oscillators’ employment on FM-CSK communication system performance is examined, and the bit error ratio is used for noise immunity evaluation.

Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems

In this article, we propose the utilization of chaos-based frequency modulated (CBFM) waveforms for joint monostatic and bistatic radar-communication systems. Short-duration pulses generated via chaotic oscillators are used for wideband radar imaging, while information is embedded in the pulses using chaos shift keying (CSK). A self-synchronization technique for chaotic systems decodes the information at the communication receiver and reconstructs the transmitted waveform at the bistatic radar receiver. Using a nonlinear detection scheme, we show that the CBFM waveforms closely follow the theoretical bit-error rate (BER) associated with bipolar phase-shift keying (BPSK). We utilize the same nonlinear detection scheme to optimize the target detection at the bistatic radar receiver. The ambiguity function for both the monostatic and bistatic cases resembles a thumbtack ambiguity function with a pseudo-random sidelobe distribution. Furthermore, we characterize the high-resolution imaging capability of the CBFM waveforms in the presence of noise and considering a complex target.

Didactic model of a simple driven microwave resonant T-L circuit for chaos, multistability and antimonotonicity

A simple driven bipolar junction transistor (BJT) based two-component circuit is presented, to be used as didactic tool by Lecturers, seeking to introduce some elements of complex dynamics to undergraduate and graduate students, using familiar electronic components to avoid the traditional black-box consideration of active elements. Although the effect of the base-emitter (BE) junction is practically suppressed in the model, chaotic phenomena are detected in the circuit at high frequencies (HF), due to both the reactant behavior of the second component, a coil, and to the birth of parasitic capacitances as well as to the effect of the weak nonlinearity from the base-collector (BC) junction of the BJT, which is otherwise always neglected to the favor of the predominant but now suppressed base-emitter one. The behavior of the circuit is analyzed in terms of stability, phase space, time series and bifurcation diagrams, Lyapunov exponents, as well as frequency spectra and Poincaré map section. We find that a limit cycle attractor widens to chaotic attractors through the splitting and the inverse splitting of periods known as antimonotonicity. Coexisting bifurcations confirm the existence of multi-stability behaviors, marked by the simultaneous apparition of different attractors (periodic and chaotic ones) for the same values of system parameters and different initial conditions. This contribution provides an enriching complement in the dynamics of simple chaotic circuits functioning at high frequencies. Experimental lab results are completed with PSpice simulations and theoretical ones.

A Novel Chaotic System with Two Circles of Equilibrium Points: Multistability, Electronic Circuit and FPGA Realization

This paper introduces a new chaotic system with two circles of equilibrium points. The dynamical properties of the proposed dynamical system are investigated through evaluating Lyapunov exponents, bifurcation diagram and multistability. The qualitative study shows that the new system exhibits coexisting periodic and chaotic attractors for different values of parameters. The new chaotic system is implemented in both analog and digital electronics. In the former case, we introduce the analog circuit of the proposed chaotic system with two circles of equilibrium points using amplifiers, which is simulated in MultiSIM software, version 13.0 and the results of which are in good agreement with the numerical simulations using MATLAB. In addition, we perform the digital implementation of the new chaotic system using field-programmable gate arrays (FPGA), the experimental observations of the attractors of which confirm its suitability to generate chaotic behavior.

Switch Elements with S-Shaped Current-Voltage Characteristic in Models of Neural Oscillators

In this paper, we present circuit solutions based on a switch element with the S-type I–V characteristic implemented using the classic FitzHugh–Nagumo and FitzHugh–Rinzel models. Using the proposed simplified electrical circuits allows the modeling of the integrate-and-fire neuron and burst oscillation modes with the emulation of the mammalian cold receptor patterns. The circuits were studied using the experimental I–V characteristic of an NbO2 switch with a stable section of negative differential resistance (NDR) and a VO2 switch with an unstable NDR, considering the temperature dependences of the threshold characteristics. The results are relevant for modern neuroelectronics and have practical significance for the introduction of the neurodynamic models in circuit design and the brain–machine interface. The proposed systems of differential equations with the piecewise linear approximation of the S-type I–V characteristic may be of scientific interest for further analytical and numerical research and development of neural networks with artificial intelligence.

Shilnikov problem in Filippov dynamical systems

In this paper, we introduce the concept of sliding Shilnikov orbits for 3D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely, a pseudo-saddle-focus. A version of Shilnikov's theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first to exhibit such a sliding phenomenon.

Intelligent Ball Bearing Fault Diagnosis Using Fractional Lorenz Chaos Extension Detection

In this study we used a non-autonomous Chua's circuit, and the fractional Lorenz chaos system. This was combined with the Extension theory detection method to analyze the voltage signals. The bearing vibration signals, measured using an acceleration sensor, were introduced into the master and slave systems through a Chua's circuit. In a chaotic system, minor differences can cause significant changes that generate dynamic errors. The matter-element model extension can be used to determine the bearing condition. Extension theory can be used to establish classical and sectional domains using the dynamic errors of the fault conditions. The results obtained were compared with those from discrete Fourier transform analysis, wavelet analysis and an integer order chaos system. The diagnostic rate of the fractional-order master and slave chaotic system could reach 100% if the fractional-order parameter adjustment was used. This study presents a very efficient and inexpensive method for monitoring the state of ball bearings.

Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media

The classical connection between symmetry breaking and the onset of chaos in dynamical systems harks back to the seminal theory of Noether [Transp. Theory Statist. Phys. 1, 186 (1918)10.1080/00411457108231446]. We study the Lagrangian kinematics of steady 3D Stokes flow through simple cubic and body-centered cubic (bcc) crystalline lattices of close-packed spheres, and uncover an important exception. While breaking of point-group symmetries is a necessary condition for chaotic mixing in both lattices, a further space-group (glide) symmetry of the bcc lattice generates a transition from globally regular to globally chaotic dynamics. This finding provides new insights into chaotic mixing in porous media and has significant implications for understanding the impact of symmetries upon generic dynamical systems.

Complete synchronization of the global coupled dynamical network induced by Poisson noises

The different Poisson noise-induced complete synchronization of the global coupled dynamical network is investigated. Based on the stability theory of stochastic differential equations driven by Poisson process, we can prove that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1. Furthermore, numerical examples are provided to show the agreement between theoretical and numerical analysis.

Noise activated bistable sensor based on chaotic system with output defined by temporal coding and firing rate

Traditional bistable sensors use external bias signal to drive its response between states and their detection strategy is based on the output power spectral density or the residence time difference (RTD) in two sensor states. Recently, the noise activated nonlinear dynamic sensors driven only by noise based on RTD technique have been proposed. Here, we present experimental results of dc voltage measurements by noise-driven bistable sensor based on electronic Chua's circuit operating in a chaotic regime where two single scroll attractors coexist. The output of the sensor is quantified by the proportion of the time the sensor stays in one state to the total observation time and by the spike-count rate with spikes defined by crossings between attractors. The relationship between the stimuli and particular observable for different noise intensities is obtained, the usefulness of each coding scheme is discussed, and the optimal noise intensity for detection is indicated. It is shown that the obtained relationship is the same for any observation time when population coding is used. The optimal time window for both detection and the number of units in population coding is found. Our results may be useful for analyses and understanding of the neural activity and in designing bistable storage elements at length scales where thermal fluctuations drastically increase and the effect of noise must be taken into consideration.

Outer synchronization between two complex dynamical networks with discontinuous coupling

In this paper, we study the outer synchronization between two complex networks with discontinuous coupling. Sufficient conditions for complete outer synchronization and generalized outer synchronization are obtained based on the stability theory of differential equations. The theoretical results show that two networks can achieve outer synchronization even if two networks are switched off sometimes and the speed of synchronization is proportional to the on-off rate. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results.

Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks

We study the effect of noise on the outer synchronization between two unidirectionally coupled complex networks and find analytically that outer synchronization could be achieved via white-noise-based coupling. It is also demonstrated that, if two networks have both conventional linear coupling and white-noise-based coupling, the critical deterministic coupling strength between two complex networks for synchronization transition decreases with an increase in the intensity of noise. We provide numerical results to illustrate the feasibility and effectiveness of the theoretical results.

Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies

In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically demonstrated.

Ghost resonance in the chaotic Chua's circuit

We experimentally investigate the ghost resonance phenomenon in the electronic circuit of Chua operating in the chaotic regime. The circuit can be stimulated to jump between two single-scroll attractors by an external periodic signal with an amplitude above an intrinsic threshold. For subthreshold signals, jumps between the chaotic attractors can be promoted by a superposed white noise. We show that the circuit output can exhibit a well-defined ghost resonance signature, i.e., a resonance on a frequency that is absent in a multicomponent input signal, when the amplitudes of the input components are properly related. Further, we show that ghost resonance can be induced by the Chua's circuit's own chaotic dynamics when it is driven by a suprathreshold multicomponent signal without the need of an external noise source.

Theoretical analysis of multiplicative-noise-induced complete synchronization in global coupled dynamical network

In this paper, based on the theory of stochastic differential equation, we study the effect of noise on the synchronization of global coupled dynamical network, when noise presents in coupling term. The theoretical result shows that noise can really induce synchronization. To verify the theoretical result, Cellular Neural Network neural model and Rössler-like system are performed as numerical examples.

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.

Signal design using nonlinear oscillators and evolutionary algorithms: application to phase-locked loop disruption

This work describes an approach for efficiently shaping the response characteristics of a fixed dynamical system by forcing with a designed input. We obtain improved inputs by using an evolutionary algorithm to search a space of possible waveforms generated by a set of nonlinear, ordinary differential equations (ODEs). Good solutions are those that result in a desired system response subject to some input efficiency constraint, such as signal power. In particular, we seek to find inputs that best disrupt a phase-locked loop (PLL). Three sets of nonlinear ODEs are investigated and found to have different disruption capabilities against a model PLL. These differences are explored and implications for their use as input signal models are discussed. The PLL was chosen here as an archetypal example but the approach has broad applicability to any input∕output system for which a desired input cannot be obtained analytically.

Mixed outer synchronization of coupled complex networks with time-varying coupling delay

In this paper, the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay is investigated. In particular, we introduce a new type of outer synchronization behavior, i.e., mixed outer synchronization (MOS), in which different state variables of the corresponding nodes can evolve into complete synchronization, antisynchronization, and even amplitude death simultaneously for an appropriate choice of the scaling matrix. A novel nonfragile linear state feedback controller is designed to realize the MOS between two networks and proved analytically by using Lyapunov-Krasovskii stability theory. Finally, numerical simulations are provided to demonstrate the feasibility and efficacy of our proposed control approach.

Digital controller design for absolute value function constrained nonlinear systems via scalar sign function approach

In this paper, a scalar sign function-based digital design methodology is developed for modeling and control of a class of analog nonlinear systems that are restricted by the absolute value function constraints. As is found to be not uncommon, many real systems are subject to the constraints which are described by the non-smooth functions such as absolute value function. The non-smooth and nonlinear nature poses significant challenges to the modeling and control work. To overcome these difficulties, a novel idea proposed in this work is to use a scalar sign function approach to effectively transform the original nonlinear and non-smooth model into a smooth nonlinear rational function model. Upon the resulting smooth model, a systematic digital controller design procedure is established, in which an optimal linearization method, LQR design and digital implementation through an advanced digital redesign technique are sequentially applied. The example of tracking control of a piezoelectric actuator system is utilized throughout the paper for illustrating the proposed methodology.

The effect of noise on the complete synchronization of two bidirectionally coupled piecewise linear chaotic systems

In this paper, we study the synchronization of two bidirectionally coupled piecewise linear chaotic systems when the coupling strength is disturbed by the common or different noise. Based on stochastic differential equation theory, we verify that the noise can really induce the occurrence of synchronization, and the sufficient conditions of synchronization with probability 1 are established. We also find that with the common noise it is easier to induce the synchronization than with different noise. Moreover, two examples are provided and some numerical simulations are performed to verify the theoretical results.

Experimental synchronization of single-transistor-based chaotic circuits

This work deals with nonautonomous chaotic circuits and, in particular, with the experimental characterization of the synchronization properties of two simple nonautonomous circuits. Two single-transistor chaotic circuits, which are among the simplest chaotic oscillators, are investigated. We studied synchronization of these circuits and found that the most appropriate technique to synchronize two single-transistor chaotic circuits is that based on the design of an inverse circuit.

A simple time-delay feedback anticontrol method made rigorous

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis.

Anticontrol of chaos in continuous-time systems via time-delay feedback

In this paper, a systematic design approach based on time-delay feedback is developed for anticontrol of chaos in a continuous-time system. This anticontrol method can drive a finite-dimensional, continuous-time, autonomous system from nonchaotic to chaotic, and can also enhance the existing chaos of an originally chaotic system. Asymptotic analysis is used to establish an approximate relationship between a time-delay differential equation and a discrete map. Anticontrol of chaos is then accomplished based on this relationship and the differential-geometry control theory. Several examples are given to verify the effectiveness of the methodology and to illustrate the systematic design procedure. (c) 2000 American Institute of Physics.

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which self-similarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N-1 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features.

A universal circuit for studying chaotic phenomena

In this paper, an overview of the results on an autonomous chaotic electronic circuit, called Chua’s oscillator, is given. Along with brief descriptions of numerical and analytical investigations on Chua’s oscillator, we present some of its potential applications. The significance of the oscillator for the study of general dynamical systems is discussed.