Complex dynamic behaviors in a small network of three ring coupled Rayleigh-Duffing oscillators: Theoretical study and circuit simulation

Abstract

This work focuses on the dynamics of a small network of three ring-coupled unidirectional Rayleigh-Duffing oscillators. The equations governing the Rayleigh-Duffing oscillator, containing a cubic term, make this study a more interesting and complex case to analyze. Coupling is achieved by perturbing the amplitude of each oscillator with a signal proportional to the amplitude of the other. The sixth-order self-driven nonlinear system obtained after coupling is analyzed, and presents up to twenty seven equilibrium points. Amongst these equilibrium points, we determined which can present the Hopf bifurcation. Also, the effects of the coupling coefficients and damping coefficients are analyzed. It is shown that varying these different coefficients leads to the appearance of extremely complex dynamic phenomena such as: instability and bifurcations (i.e coexistence of bifurcation branches), coexistence of up to fifteen attractors (heterogeneous multistability) and eight spiral chaotic attractor. The investigation of the coupled system is carried out by using to both analytical and numerical tools such as Hopf bifurcation theorem, the phase portraits, bifurcation diagrams, Lyapunov exponent diagram, frequency spectrum, to name but a few. The Routh-Hurwitz criterion is also used to analyze the stability of equilibrium points. We compute basins of attraction to highlight different zones corresponding to coexisting attractors. The implementation of an analog circuit of coupled Rayleigh-Duffing oscillators has enabled us to confirm the analytical and numerical results.