Pierre Kisito Talla

Dynamics of modulated waves in a nonlinear discrete LC transmission line: dissipative effects

We investigate analytically the dynamics of modulated waves in a nonlinear LC transmission line with dissipative elements. The damped nonlinear Schrodinger equation governing slowly modulated wave propagation is derived. Considering modulated plane wave propagating in the line, we establish their criteria of instability and show that, in the case of weak dissipation, there are no significant changes for the bandwidth frequency where the network may exhibit modulational instability (MI). However, dissipative losses constitute a great limiting factor to the experimental observation of the MI phenomenon. We show that the amplitude decreases, the width increases while the velocity remains constant when the soliton propagates along the dissipative network.

Stability of bright solitons in some physical systems

Dynamical systems described by the modified quintic complex Ginzburg Landau equation and its derivative forms are considered and the stability of their bright soliton solution is investigated numerically by means of the split-step Fourier method. Some discussions related to the way of ensuring the stability of this solution are presented.

An Autonomous Helmholtz Like-Jerk Oscillator: Analysis, Electronic Circuit Realization and Synchronization Issues

This chapter introduces an autonomous self-exited three-dimensional Helmholtz like oscillator which is built by converting the well know autonomous Helmholtz two-dimensional oscillator to a jerk oscillator. Basic properties of the proposed Helmholtz like-jerk oscillator such as dissipativity, equilibrium points and stability are examined. The dynamics of the proposed jerk oscillator is investigated by using bifurcation diagrams, Lyapunov exponent plots, phase portraits, frequency spectra and cross-sections of the basin of attraction. It is found that the proposed jerk oscillator exhibits some interesting phenomena including Hopf bifurcation, period-doubling bifurcation, reverse period-doubling bifurcation and hysteretic behaviors (responsible of the phenomenon of coexistence of multiple attractors). Moreover, the physical existence of the chaotic behavior and the coexistence of multiple attractors found in the proposed autonomous Helmholtz like-jerk oscillator are verified by some laboratory experimental measurements. A good qualitative agreement is shown between the numerical simulations and the experimental results. In addition, the synchronization of two identical coupled Helmholtz like-jerk oscillators is carried out using an extended backstepping control method. Based on the considered approach, generalized weighted controllers are designed to achieve synchronization in chaotic Helmholtz like-jerk oscillators. Numerical simulations are performed to verify the feasibility of the synchronization method. The approach followed in this chapter shows that by combining both numerical and experimental techniques, one can gain deep insight about the dynamics of chaotic systems exhibiting hysteretic behavior.

Complex Dynamical Behavior of a Two-Stage Colpitts Oscillator with Magnetically Coupled Inductors

A five-dimensional (5D) controlled two-stage Colpitts oscillator is introduced and analyzed. This new electronic oscillator is constructed by considering the well-known two-stage Colpitts oscillator with two further elements (coupled inductors and variable resistor). In contrast to current approaches based on piecewise linear (PWL) model, we propose a smooth mathematical model (with exponential nonlinearity) to investigate the dynamics of the oscillator. Several issues, such as the basic dynamical behaviour, bifurcation diagrams, Lyapunov exponents, and frequency spectra of the oscillator, are investigated theoretically and numerically by varying a single control resistor. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling and interior crisis routes as the single control resistor is monitored. Furthermore, an experimental study of controlled Colpitts oscillator is carried out. An appropriate electronic circuit is proposed for the investigations of the complex dynamics behaviour of the system. A very good qualitative agreement is obtained between the theoretical/numerical and experimental results.