Magnetic coupling based control of a chaotic circuit: Case of the van der Pol oscillator coupled to a linear circuit

Abstract

This paper investigates the control of the dynamics of the van der Pol oscillator coupled to Linear circuit (VDPCL oscillator). The method consists of carrying out a magnetic coupling (wireless interaction) of a VDPCL oscillator to a magnetic core coil supplied by a sinusoidal voltage. One of the most important outlets of the proposed control scheme is the ability to control the dynamic of the oscillator by varying three external parameters as well as to generate very rich and complicated behaviors including attractors not yet reported in the non-controlled model. In this impetus, the frequency and the amplitude of the excitation source of the RL circuit as well as the magnetic coupling coefficient are thus used to adjust the VDPCL oscillator dynamics in a desired state. The basic properties of the model are discussed, such as fixed point stability, symmetry and dissipation. Using the control parameters mentioned above, the dynamic behaviors of the model are studied using the classical tools of nonlinear dynamics such as the two-parameters and one parameter Lyapunov exponents, the one-parameter bifurcation diagrams, phase portraits and time series. Numerical simulation reveals a rich repertoire of dynamic behaviors in the system, including periodicity, period doubling, quasi-periodicity, 2-torus, 3-torus and chaos. In particular, complex and striking phenomena such as antimonotonicity, intermittency, coexistence of attractors and amplitude modulation are reported in the model when monitoring the frequency of the sinusoidal excitation. Sample coexisting bifurcation diagrams are drawn and the coexistence of attractors is illustrated using the phase portraits and the cross section of the basin of attraction. Pspice based simulations and laboratory experimental measurements are included to confirm the theoretical results and the feasibility of the proposed control scheme.