Anaclet Fomethe

Dynamical Properties and Finite-Time Hybrid Projective Synchronization Using Fractional Nonsingular Sliding Mode Surface in Fractional-Order Two-Stage Colpitts Oscillators

The dynamics and robust finite-time hybrid projective synchronization of a fractional-order four-dimensional nonlinear system based on a two-stage Colpitts oscillator is investigated. The study of the fractional order stability of the equilibrium states of the system is carried out. The bifurcation diagram confirms the occurrence of Hopf bifurcation in the proposed system when the fractional-order passes a sequence of critical values; the Lyapunov exponent shows the different chaotic sequences of the system. Further, a fractional nonsingular terminal sliding surface and an appropriate robust fractional sliding mode control law are proposed for the finite-time hybrid projective synchronization of a fractional-order chaotic two-stage Colpitts oscillator by taking into account the effects of model uncertainties and the external disturbances. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. Finally, some numerical simulations are presented to demonstrate the effectiveness and applicability of the proposed technique.

Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators

We propose the use of a simple, cheap, and easy technique for the study of dynamic and synchronization of the coupled systems: effects of the magnetic coupling on the dynamics and of synchronization of two Colpitts oscillators (wireless interaction). We derive a smooth mathematical model to describe the dynamic system. The stability of the equilibrium states is investigated. The coupled system exhibits spectral characteristics such as chaos and hyperchaos in some parameter ranges of the coupling. The numerical exploration of the dynamics system reveals various bifurcations scenarios including period-doubling and interior crisis transitions to chaos. Moreover, various interesting dynamical phenomena such as transient chaos, coexistence of solution, and multistability (hysteresis) are observed when the magnetic coupling factor varies. Theoretical reasons for such phenomena are provided and experimentally confirmed with practical measurements in a wireless transfer.

Structural static stability and dynamic chaos of active electromagnetic bearing systems: Analytical investigations and numerical simulations:

The complete nonlinear dynamics of an active magnetic bearing (AMB) with proportional and derivative controller is revisited analytically. Through the stability analysis of the second-order nonlinear differential equation governing the dynamics of the system, a state diagram is obtained in the proportional gain and precontrol current space. This diagram is fundamental and may help in making important design decisions namely in the identification of a controller for the complete range of the rotor mass, in order to ensure structural stability. In particular, we find that there exists a threshold for the proportional gain whose expression depends both on the inner radius of the bearing and on the bias current, and below which no stable dynamics of the system can occur. In the operating range, we show that the system exhibits a rich dynamics characterized by the possibility for the existence of many kinds of nonlinear localized excitations in the transient state, which may lead to an irregular or a chaotic behavior of the shaft. In this regime, the expressions of the critical running speed obtained by means of the Melnikov theory indicate its dependence on the AMB characteristic parameters.

Nonlinear excitations in a continuous bi-inductance electrical line

The dynamics of nonlinear excitations in an electrical bi-inductance transmission line are examined by means of the multiple scales method. In the continuum approximation using an appropriate decoupling ansatz for the voltage of the two different cells, we consider modulated waves and show that their propagation in the network is governed by a nonlinear Schrodinger equation instead of a Korteweg–de Vries equation. We have also established the existence of two frequency modes and study separately the wave dynamics in each mode. It appears from our investigations that the continuous bi-inductance electrical line supports a dark soliton in the low frequency mode and only a bright soliton in the high frequency mode. The study of the network properties has revealed that a nonlinear bi-inductance electrical transmission line can be considered as a superposition of two independent nonlinear mono-inductance transmission lines with different inductances.