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.