Victor Kamdoum Tamba

Chaos in a System with an Absolute Nonlinearity and Chaos Synchronization

A system with an absolute nonlinearity is studied in this work. It is noted that the system is chaotic and has an adjustable amplitude variable, which is suitable for practical uses. Circuit design of such a system has been realized without any multiplier and experimental measurements have been reported. In addition, an adaptive control has been applied to get the synchronization of the system.

Analysis of Three-Dimensional Autonomous Van der Pol–Duffing Type Oscillator and Its Synchronization in Bistable Regime

This chapter proposes a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator which is designed from a nonautonomous VdPD two-dimensional chaotic oscillator driven by an external periodic source through replacing the external periodic drive source with a direct positive feedback loop. The dynamical behavior of the proposed autonomous VdPD type oscillator is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, period-doubling bifurcation, symmetry recovering and breaking bifurcation, double scroll chaos, bistable one scroll chaos and coexisting attractors. Basin of attraction of coexisting attractors is computed showing that is associated with an unstable equilibrium. So the proposed autonomous VdPD type oscillator belongs to chaotic systems with self-excited attractors. A suitable electronic circuit of the proposed autonomous VdPD type oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show a good agreement with the numerical simulations. Finally, synchronization of identical coupled proposed autonomous VdPD type oscillators in bistable regime is studied using the unidirectional linear feedback methods. It is found from the numerical simulations that the quality of synchronization depends on the coupling coefficient as well as the selection of coupling variables.

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.

Dynamic Analysis, Electronic Circuit Realization of Mathieu-Duffing Oscillator and Its Synchronization with Unknown Parameters and External Disturbances

This chapter deals with dynamic analysis, electronic circuit realization and adaptive function projective synchronization (AFPS) of two identical coupled Mathieu-Duffing oscillators with unknown parameters and external disturbances. The dynamics of the Mathieu-Duffing oscillator is investigated with the help of some classical nonlinear analysis techniques such as bifurcation diagrams, Lyapunov exponent plots, phase portraits as well as frequency spectrum. It is found that the oscillator experiences very rich and striking behaviors including periodicity, quasi-periodicity and chaos. An appropriate electronic circuit capable to mimic the dynamics of the Mathieu-Duffing oscillator is designed. The correspondences are established between the parameters of the system model and electronic components of the proposed circuit. A good agreement is obtained between the experimental measurements and numerical results. Furthermore, based on Lyapunov stability theory, adaptive controllers and sufficient parameter updating laws are designed to achieve the function projective synchronization between two identical drive-response structures of Mathieu-Duffing oscillators. The external disturbances are taken into account in the drive and response systems in order to verify the robustness of the proposed strategy. Analytical calculations and numerical simulations are performed to show the effectiveness and feasibility of the method.

Dynamic system with no equilibrium and its chaos anti-synchronization

ABSTRACT Recently, systems with chaos and the absence of equilibria have received a great deal of attention. In our work, a simple five-term system and its anti-synchronization are presented. It is special that the system has a hyperbolic sine nonlinearity and no equilibrium. Such a system generates chaotic behaviours, which are verified by phase portraits, positive Lyapunov exponent as well as an electronic circuit. Moreover, the system displays multistable characteristic when changing its initial conditions. By constructing an adaptive control, chaos anti-synchronization of the system with no equilibrium is obtained and illustrated via a numerical example.

Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity: Analysis, FPGA Implementation, and Synchronization

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.