Autonomous Chaotic Oscillator Research Articles

Wien-Bridge Chaotic Oscillator Circuit with Inductive Memristor Bipole

By diode bridging an inductor to implement a memristor bipole, with active Wien-bridge oscillator, a simple and feasible third-order autonomous memristive chaotic oscillator is presented. The dynamical characteristics of the proposed circuit are investigated both theoretically and numerically, from which it can be found that the circuit has one unstable equilibrium point. Through the analysis of the bifurcation diagram, Lyapunov exponent spectrum and the 0–1 test chaos detection, it is shown that this system displays limit cycle orbit with different periodicity, quasi-periodic behavior, chaotic behavior and bursting behavior. The bursting behavior found in this circuit is periodic, quasi-periodic and chaotic bursting. We confirm the feasibility of the proposed theoretical model using Pspice simulations and a physical realization based on an electronic analog implementation of the model.

Electronically Tunable 3D Autonomous Chaotic Oscillator Employing Single CCCFA and Its Extension to 4D

This work presents the realization of a novel electronically tuneable third-order autonomous chaotic oscillator using a current-controlled current conveyor feedback amplifier (CCCFA). The proposed circuit consists of a single CCCFA, two grounded passive capacitors, one inductor, and two diodes. The chaotic oscillator possesses smooth symmetrical sin-hyperbolic nonlinearity through two antiparallelly connected diodes. The electronically tunable intrinsic resistance at the X terminal serves as the bifurcation control parameter in the circuit and is controlled through bias current. Tuning of bias current enables the formation of different periodic and chaotic attractors. The proposed chaotic oscillator exhibits rich nonlinear dynamical behavior such as periodicity, antimonotonicity, and coexistence of attractors. Multistability is investigated through circuit-level simulation. The chaotic oscillator can easily be implemented in an integrated circuit as the configuration is simple, resistorless, and uses the minimum number of components in the count. Further, a higher dimensional (4D) chaotic oscillator is proposed as an extended circuit of the 3D chaotic oscillator and illustrated as an application in chaos encryption. Simulations are done using TSMC 180nm CMOS technology in PSpice. Experimental results are presented to verify the theoretical and simulation analyses of the proposed circuits.

A chaotic circuit based on a physical memristor

Abstract The memristor is a fundamental two-terminal electrical component unique in that it possesses the properties of non-linearity and memory, which are pervasive across natural systems. It has been proven to be in principle a viable substrate for novel dynamical systems showing chaotic behavior, but the recourse to abstract, idealized mathematical non-linearities throughout the existing literature hinders practical realization using physical devices. In this work, we realize a fully autonomous chaotic oscillator circuit based on self-directed channel memristors. Its architecture comprises two feedback loops, a linear one and a non-linear one involving the memristor. Low-dimensional chaotic dynamics are readily obtained experimentally using tungsten-based as well as carbon-based physical devices, despite their non-idealities. A mathematical model of the circuit, revealing further interesting non-linear features such as bifurcations without parameters, is also offered.

Implementación electrónica basada en FPAA del oscilador caótico de Lorenz

Nowadays, chaotic systems are very interesting topics for engineers, physicists and mathematicians because most real physical systems are inherently non-linear in nature. The first electronic implementations of autonomous chaotic oscillators were developed using operational amplifiers and diodes, different references detail implementations of chaotic circuits and systems using analog integrated circuit technology, discrete devices such as FPGA (Field programmable gate arrays), microcontrollers, etc. However, analog implementations suffer the problem of sensitivity of analog component values and digital implementations suffer the problem of degradation due to the reduced number of bits to perform computer arithmetic operations. The systems of differential equations that model the chaotic oscillators require integrators that can be implemented with FPAA's (Field programmable analogue array), in this work electronic implementations are developed that are measured in laboratory conditions to observe experimental chaotic attractors, which will be used in the implementation of random number generators and secure communication systems for image encryption.

Minimal Realizations of Autonomous Chaotic Oscillators Based on Trans-Immittance Filters

This review paper describes a design process toward fully analog realizations of chaotic dynamics that can be considered canonical (minimum number of the circuit elements), robust (exhibit structurally stable strange attractors), and novel. Each autonomous chaotic lumped circuit proposed in this paper can be understood as a looped system, where linear trans-immittance frequency filter interacts with an active nonlinear two-port. The existence of chaos is demonstrated via well-established numerical algorithms that represent the current standard in the field of nonlinear dynamics, i.e., by calculation of the largest Lyapunov exponent and high-resolution 1-D bifurcation diagrams. The achieved numerical results are put into the context of experimental measurement; observed state trajectories prove a one-to-one correspondence between theoretical expectations and practical outputs, i.e., prescribed strange attractors do not represent the chaotic transients. Finally, short term unpredictability of the chaotic flow is demonstrated via calculation of Kaplan–Yorke dimension that is high, i.e., generated waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography.

Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator

This paper presents a novel third-order autonomous memristive chaotic oscillator, which is accomplished by parallelly coupling a simple memristive diode bridge emulator into a Sallen–Key low-pass filter (LPF). With the modeling of this oscillator, stability analyses of the equilibrium point and numerical simulations of the phase plane orbit, time-domain sequence, bifurcation diagram, and finite-time Lyapunov exponent spectrum are performed, from which period, quasi-period, chaos, and quasi-period to chaos route are found. Particularly, two types of dynamical phenomena of quasi-periodic behavior and point-cycle chaotic bursting that are further identified by using 0–1 test are observed in such a third-order autonomous memristive oscillator, which have been rarely reported in the previous literatures. Additionally, hardware experiments are implemented and the quasi-periodic behavior and point-cycle chaotic bursting are well confirmed.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

Chaos in a Single Op-Amp–Based Jerk Circuit: Experiments and Simulations

We report a five-component autonomous chaotic oscillator of jerky type, hitherto the simplest of its kind, using only one operational amplifier. The key component of the circuit is a junction field-effect transistor operating in its triode region, which provides a nonlinear resistor of antisymmetrical current-voltage characteristic, emulating a Colpitts-like chaotic circuit. We describe the experimental results illustrating the dynamical behavior of the circuit. In addition, we report numerical simulations of a model of the circuit which display good agreement with our measurements.

Chaotic Pulse Generation Induced by a Specific Class of Autonomous Oscillator

The nonlinear dynamics of an autonomous chaotic oscillator, using two different stages operational amplifier coupled by mean of diode employed as the nonlinear device, recently introduced by Giannakopoulos and Deliyannis is considered with some particular modifications. These modifications are necessary for generating new type of oscillations, the regular and chaotic pulse oscillations according to the nature of operational amplifiers. Based on the nonlineardiode equation, the transfer voltage function of operational amplifiers in open loop configuration, and an appropriate selection of the state variables, a mathematical model is derived for a better description of the dynamics of the system. The complexness of oscillations is characterized using the bifurcation diagrams and the phase portraits. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the oscillator to generate both the regular and chaotic pulse oscillations, according to the appropriate choice of its components.

A Simple RLCC-Diode-Opamp Chaotic Oscillator

This paper presents a simple autonomous chaotic oscillator. The design method is primarily based on a linear oscillator constructed by a closed loop connection of two building blocks, i.e. an inverting active integrator and a passive second-order LC integrator. A diode is inserted in parallel to the two building blocks for inducing chaos. The mathematical model reveals a set of three-dimensional ordinary differential equations, containing seven terms with four constants and an exponential nonlinearity. The dynamics properties are investigated in terms of an equilibrium point, Jacobian matrix, chaotic attractors, bifurcation, Lyapunov exponents, and chaotic waveforms in time domain. The proposed chaotic oscillator potentially exhibits complex dynamical behaviors through the utilization of only six minimal electronic components.

Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance.

In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals.

Chaos in semiconductor band-trap impact ionization

We introduce a new spatially-extended semiconductor carriers transport equation model, based on generation–recombination process of the band-trap impact ionization under a longitudinal electric field. By means of numerical studies, we demonstrate the existence of chaos. Also, we present many results such as, the lyapunov spectrum, the bifurcation diagram, the phase portrait and the Poincaré surface of section. In addition the basic electric circuit used is found to be helpful in the implementation of a simple and autonomous chaotic oscillator circuit. Furthermore, the obtained results are interesting in the way that they could be useful in avoiding of undesirable chaotic regime in some switching and memory electronic devices.

Autonomous Silva–Young Type Chaotic Oscillator with Flat Power Spectrum

A novel Silva-Young type fourth-order autonomous chaotic oscillator is described. In comparison with the original nonautonomous Silva-Young circuit it lacks the external periodic driving oscillator, but includes a passive internal second-order resonant LRC circuit, inserted in the positive feedback loop. In contrast to many other nonautonomous and autonomous chaotic oscillators, the suggested circuit exhibits fairly flat power spectrum of the output signal. Hardware experimental results and numerical results are presented. Good agreement between numerical and experimental results is observed. Ill. 5, bibl. 22 (in English; abstracts in English and Lithuanian).http://dx.doi.org/10.5755/j01.eee.115.9.761

Phase synchronization and crisis in coupled periodically driven chaotic oscillators

A nontrivial transition from the synchronous to the nonsynchronous state of two coupled nonautonomous oscillators is studied through the theory of two coupled circle maps (torus maps). For the case of nonidentical systems, the transition occurs without the connection to the change in the sign of Lyapunov exponent. We find that the scaling exponent for the intermittent slips near the critical point, which is measured by rotation sets (fluctuations in rotation vectors), depends on the system parameters unlike the case of phase synchronization for two coupled autonomous chaotic oscillators. We demonstrate that through the model of torus maps this nontrivial transition is induced by a boundary crisis on a torus geometry and intermittent slips could be described as the chaotic transient as a result of the crisis.

Phase synchronization and crisis in coupled periodically driven chaotic oscillators

A nontrivial transition from the synchronous to the nonsynchronous state of two coupled nonautonomous oscillators is studied through the theory of two coupled circle maps (torus maps). For the case of nonidentical systems, the transition occurs without the connection to the change in the sign of Lyapunov exponent. We find that the scaling exponent for the intermittent slips near the critical point, which is measured by rotation sets (fluctuations in rotation vectors), depends on the system parameters unlike the case of phase synchronization for two coupled autonomous chaotic oscillators. We demonstrate that through the model of torus maps this nontrivial transition is induced by a boundary crisis on a torus geometry and intermittent slips could be described as the chaotic transient as a result of the crisis.

Self-induced slow-fast dynamics and swept bifurcation diagrams in weakly desynchronized systems.

In systems close to the state of phase synchronization, the fast timescale of oscillations interacts with the slow timescale of the phase drift. As a result, "fast" dynamics is subjected to a slow modulation, due to which an autonomous system under fixed parameter values can imitate repeated bifurcational transitions. We demonstrate the action of this general mechanism for a set of two coupled autonomous chaotic oscillators and for a chaotic system perturbed by a periodic external force. In both cases, the Poincaré sections of phase portraits resemble bifurcation diagram of a logistic mapping with time-dependent parameter.

Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices

Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic second-order RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive first-order or second-order nonlinear composites, chaos is generated and the evolution of the two-dimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chua's circuit. The well-known Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduced.

A semi-systematic procedure for producing chaos from sinusoidal oscillators using diode-inductor and FET-capacitor composites

A design procedure for producing chaos is proposed. The procedure aims to transfer design issues of analog autonomous chaotic oscillators from the nonlinear domain back to the much simpler linear domain by intentionally modifying sinusoidal oscillator circuits in a semisystematic manner. Design rules that simplify this procedure are developed and then two composite devices, namely, a diode-inductor composite and a FET-capacitor composite are suggested for carrying out the modification procedure. Applications to the classical Wien-bridge oscillator are demonstrated. Experimental results, PSpice simulations, and numerical simulations of the derived models are included.

Chua's circuit decomposition: a systematic design approach for chaotic oscillators

The fact that Chua's circuit can be physically decomposed into a sinusoidal oscillator coupled to an active voltage-controlled nonlinear resistor is demonstrated. The sinusoidal oscillator is the beating heart of Chua's circuit and many novel implementations can be obtained by using different sinusoidal oscillator engines. In particular, inductorless realizations can be derived not by replacing the passive inductor in the classical Chua's circuit configuration with an active RC emulation, but rather by replacing the passive LC tank resonator by a sinusoidal oscillator. We provide several circuit-design examples and verify our results experimentally. Finally, we state a conjecture which we believe forms a basis for the design of autonomous analog chaotic oscillators.

On-off intermittency in chaotic synchronization experiment

Universal power laws typical for the on-off intermittency are observed experimentally in an electronic circuit of two bidirectionally coupled autonomous chaotic oscillators. It is shown that an exponential falloff in the distribution of the laminar phases appears due to a small mismatch of the nearly identical oscillators. The dispersion of the local transverse Lyapunov exponents is estimated from the exponential falloff.