René Yamapi

Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model

We analyze the bifurcations occurring in the 3D Hindmarsh-Rose neuronal model with and without random signal. When under a sufficient stimulus, the neuron activity takes place; we observe various types of bifurcations that lead to chaotic transitions. Beside the equilibrium solutions and their stability, we also investigate the deterministic bifurcation. It appears that the neuronal activity consists of chaotic transitions between two periodic phases called bursting and spiking solutions. The stochastic bifurcation, defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value, or under certain condition as the collision of a stochastic attractor with a stochastic saddle, occurs when a random Gaussian signal is added. Our study reveals two kinds of stochastic bifurcation: the phenomenological bifurcation (P-bifurcations) and the dynamical bifurcation (D-bifurcations). The asymptotical method is used to analyze phenomenological bifurcation. We find that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.

DYNAMICS AND ACTIVE CONTROL OF MOTION OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR

This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place.

SYNCHRONIZATION OF THE REGULAR AND CHAOTIC STATES OF ELECTROMECHANICAL DEVICES WITH AND WITHOUT DELAY

We consider in this paper the problem of stability and duration of the synchronization process between two electromechanical devices, both in their regular and chaotic states. Stability boundaries are derived through Floquet theory. The influence of the precision on the synchronization time is also analyzed using numerical simulation of the equations of motion.

ADAPTIVE OBSERVER BASED SYNCHRONIZATION OF A CLASS OF UNCERTAIN CHAOTIC SYSTEMS

This study addresses the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework. For a class of uncertain chaotic systems with parameter mismatch and external disturbances, a robust adaptive observer based on the response system is constructed to practically synchronize the uncertain drive chaotic system. Lyapunov stability theory ensures the practical synchronization between the drive and response systems even if Lipschitz constants on function matrices and bounds on uncertainties are unknown. Numerical simulation of two illustrative examples are given to verify the effectiveness of the proposed method.

DYNAMICAL STATES IN A RING OF FOUR MUTUALLY COUPLED SELF-SUSTAINED ELECTRICAL SYSTEMS WITH TIME PERIODIC COUPLING

We investigate in this Letter different dynamical states in the ring of four mutually coupled self-sustained electrical systems with time periodic coupling. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived using the Floquet theory and the Whittaker method. The effects of the amplitude of the periodic parametric perturbations of the coupling parameter on the stability boundaries are analyzed. Numerical simulations are then performed to complement the analytical results.

Adaptive Synchronization of Coupled Self-sustained Electrical Systems

This study addresses the adaptive synchronization of coupled self-sustained electrical systems described by the Rayleigh-Duffing equations. We show that the synchronization of such two coupled systems can be achieved by means of nonlinear feedback coupling. We use the Lyapunov direct method to study the asymptotic stability of the solutions of the synchronization error system. Numerical simulations are given to explain the effectiveness of the proposed control scheme.

Modeling, Stability, Synchronization, and Chaos and Their Applications to Complex Systems

1 Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24 157, Douala, Cameroon 2Dipartimento di Scienze Biologiche ed Ambientali, Universita del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy 3 Laboratoire de Mathematiques Appliquees, Universite du Le Havre, 25 rue Ph. Lebon, BP 540, 76600 Le Havre Cedex, France 4Monell Chemical Senses Center, 3500 Market Street, Philadelphia, PA 19104, USA

Effects of discontinuity of elasticity and damping on the dynamics of an electromechanical transducer

In this paper, we study numerically the effects of discontinuity in elasticity and damping on the dynamics of an electromechanical transducer. Frequency-response curves of oscillatory states are obtained. Bifurcation structures and transitions to nonperiodic or chaotic motion are found.

Dynamics of Disordered Network of Coupled Hindmarsh–Rose Neuronal Models

We investigate the effects of disorder on the synchronized state of a network of Hindmarsh–Rose neuronal models. Disorder, introduced as a perturbation of the neuronal parameters, destroys the network activity by wrecking the synchronized state. The dynamics of the synchronized state is analyzed through the Kuramoto order parameter, adapted to the neuronal Hindmarsh–Rose model. We find that the coupling deeply alters the dynamics of the single units, thus demonstrating that coupling not only affects the relative motion of the units, but also the dynamical behavior of each neuron; Thus, synchronization results in a structural change of the dynamics. The Kuramoto order parameter allows to clarify the nature of the transition from perfect phase synchronization to the disordered states, supporting the notion of an abrupt, second order-like, dynamical phase transition. We find that the system is resilient up to a certain disorder threshold, after that the network abruptly collapses to a desynchronized state. T...

SYNCHRONIZATION OF TWO ELECTROMECHANICAL DEVICES WITH PARAMETRIC COUPLING

The study of synchronization of two electromechanical devices with parametric coupling, in their regular and chaotic states was investigated. It was observed that an analytical study based on the Floquet theory makes it possible to determine the coefficients of coupling, ensuring a complete synchronization. Emphasis was placed on the analysis of amplitude effects on coupling and stability boundaries of the synchronization process. Numerical investigations are then used to support the accuracy of the analytical approach.