Hilaire Fotsin

Coexistence of Multiple Attractors and Crisis Route to Chaos in a Novel Chaotic Jerk Circuit

In this paper, a novel autonomous RC chaotic jerk circuit is introduced and the corresponding dynamics is systematically investigated. The circuit consists of opamps, resistors, capacitors and a pair of semiconductor diodes connected in anti-parallel to synthesize the nonlinear component necessary for chaotic oscillations. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a linear transformation of model MO15 previously introduced in [Sprott, 2010]. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. The bifurcation analysis indicates that chaos arises via the usual paths of period-doubling and symmetry restoring crisis. One of the key contributions of this work is the finding of a region in the parameter space in which the proposed (“elegant”) jerk circuit exhibits the unusual and striking feature of multiple attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). Laboratory experimental results are in good agreement with the theoretical predictions.

DYNAMICS OF TWO NONLINEARLY COUPLED OSCILLATORS

Two nonlinearly coupled oscillators submitted to an external periodic force are studied. The method of multiple scales is used to find solutions in the resonant and nonresonant cases. The response curves are discussed. A set of bifurcations diagrams are obtained showing period-doubling and torus-breakdown routes to chaos.

Synchronizing modified van der Pol–Duffing oscillators with offset terms using observer design: application to secure communications

This study addresses the adaptive synchronization of the modified van der Pol?Duffing (MVDPD) oscillator with offset terms. From our investigations of the system dynamics, we obtain that the system presents a chaotic behaviour at weak values of the offset parameters. Routh?Hurwitz criteria are used to study the asymptotic stability of the steady states. An adaptive observer design method is applied to achieve synchronization of two identical MVDPD oscillators with offset. Numerical simulations are given to validate the proposed synchronization approach. Moreover, as an application, the proposed scheme is applied to secure communication. Also, simulation results verify the proposed schemes success in the communication application.

A strategy for adaptive synchronization of an electrical chaotic circuit based on nonlinear control

This paper deals with the adaptive synchronization of an electrical chaotic oscillator through a nonlinear control law. The nonlinear controller is designed to synchronize two identical chaotic systems. Lyapunov stability theory is applied to prove that under some conditions the drive-response system can achieve practical synchronization. The designed controller contains only continuous nonlinear terms existing in the dynamical systems and is easy to implement in practice. As an illustrative example to verify the validity of the proposed method, a hyperbolic tangent-based electrical and chaotic circuit is chosen and its dynamics is briefly investigated to demonstrate its chaotic behavior.

Implementing a memristive Van der Pol oscillator coupled to a linear oscillator: synchronization and application to secure communication

This paper investigates the dynamics of a memristor-based Van der Pol oscillator coupled to a linear circuit (VDPCL). This chaotic oscillator is a modification of the classical Van der Pol coupled to a linear circuit, and is obtained by replacing the classical cubic nonlinearity by the memristive one. The memristive VDPCL oscillator, in addition to having a very special stability property, exhibits interesting spectral characteristics, which makes it suitable for chaos-based secure communication applications. The memristor is realized by using off-the-shelf components. The basic properties of the circuit are analyzed by means of bifurcation analysis. Chaotic attractors from numerical and experimental analysis are presented, followed by a comparison of results obtained from the modified VDPCL oscillator and those from the classical VDPCL oscillator. An application to synchronization and chaos secure communication is also presented.

Dynamics, Analysis and Implementation of a Multiscroll Memristor-Based Chaotic Circuit

This article introduces a novel four-dimensional autonomous multiscroll chaotic circuit which is derived from the actual simplest memristor-based chaotic circuit. A fourth circuit element — another inductor — is introduced to generate the complex behavior observed. A systematic study of the chaotic behavior is performed with the help of some nonlinear tools such as Lyapunov exponents, phase portraits, and bifurcation diagrams. Multiple scroll attractors are observed in Matlab, Pspice environments and also experimentally. We also observe the phenomenon of antimonotonicity, periodic and chaotic bubbles, multiple periodic-doubling bifurcations, Hopf bifurcations, crises and the phenomenon of intermittency. The chaotic dynamics of this circuit is realized by laboratory experiments, Pspice simulations, numerical and analytical investigations. It is observed that the results from the three environments agree to a great extent. This topology is likely convenient to be used to intentionally generate chaos in memristor-based chaotic circuit applications, given the fact that multiscroll chaotic systems have found important applications as broadband signal generators, pseudorandom number generators for communication engineering and also in biometric authentication.

Noise effects on robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design

In this paper, we report on the synchronization of a pacemaker neuronal ensemble constituted of an AB neuron electrically coupled to two PD neurons. By the virtue of this electrical coupling, they can fire synchronous bursts of action potential. An external master neuron is used to induce to the whole system the desired dynamics, via a nonlinear controller. Such controller is obtained by a combination of sliding mode and feedback control. The proposed controller is able to offset uncertainties in the synchronized systems. We show how noise affects the synchronization of the pacemaker neuronal ensemble, and briefly discuss its potential benefits in our synchronization scheme. An extended Hindmarsh–Rose neuronal model is used to represent a single cell dynamic of the network. Numerical simulations and Pspice implementation of the synchronization scheme are presented. We found that, the proposed controller reduces the stochastic resonance of the network when its gain increases.

An Online Simplified Nonlinear Controller for Transient Stabilization Enhancement of DFIG in Multi-Machine Power Systems

An adaptive nonlinear controller for transient stability and voltage regulation of power systems based DFIG in multimachine configuration is presented using a standard third order dynamical model of the DFIG. Finite time estimators for the unmeasurable time derivative of the quadrature component of the DFIG stator current, mechanical input, unknown direct axis transient open circuit time constant (function of the rotor resistance) are presented. The main feature of the proposed control scheme is its robustness with respect to large perturbations and parameter variations. Numerical results are presented to illustrate the performance of the proposed control scheme and its robustness properties.

A simple and robust gray image encryption scheme using chaotic logistic map and artificial neural network

A robust gray image encryption scheme using chaotic logistic map and artificial neural network (ANN) is introduced. In the proposed method, an external secret key is used to derive the initial conditions for the logistic chaotic maps which are employed to generate weights and biases matrices of the multilayer perceptron (MLP). During the learning process with the backpropagation algorithm, ANN determines the weight matrix of the connections. The plain image is divided into four subimages which are used for the first diffusion stage. The subimages obtained previously are divided into the square subimage blocks. In the next stage, different initial conditions are employed to generate a key stream which will be used for permutation and diffusion of the subimage blocks. Some security analyses such as entropy analysis, statistical analysis, and key sensitivity analysis are given to demonstrate the key space of the proposed algorithm which is large enough to make brute force attacks infeasible. Computing validation using experimental data with several gray images has been carried out with detailed numerical analysis, in order to validate the high security of the proposed encryption scheme.

Adaptive time-delay synchronization of chaotic systems with uncertainties using a nonlinear feedback coupling

This paper treats the adaptive synchronization problem of a class of uncertain chaotic systems with uncertainties, delay and unknown inputs in a drive-response framework. A robust adaptive observer-based response system is designed to synchronize a given delayed chaotic system without the knowledge of upper bounds of uncertainties and unknown inputs. Furthermore, the unknown inputs can be approximately recovered directly by the concept of equivalent control signal. To highlight our method, we improve the robustness of ciphering in a secure communication system. Computer simulation is also given for the purpose of illustration and verification.