From coexisting attractors to multi-spiral chaos in a ring of three coupled excitation-free Duffing oscillators

Abstract

The dynamics of a ring of three unidirectional coupled excitation-free double well Duffing oscillators is considered. The coupling technique followed in this work consists in the amplitude modulation of an oscillator with a proportion of the amplitude of its neighboring counterpart. The rate equation of this model is a sixth-order autonomous nonlinear odd symmetric system endowed with twenty seven equilibrium points eight of which undergo Hopf bifurcation. A detailed analysis of the model based on classic nonlinear analysis tools (e.g. bifurcation diagrams, phase portraits, basins of attraction) discloses attractive and interesting dynamical features such as eight parallel bifurcation threes, Hopf bifurcations, and various types of coexisting modes of oscillations (e.g. eight coexisting limit cycles, four competing double spiral chaotic attractors, two coexisting four-spiral chaotic attractors), and eight-spiral chaotic attractor, just to cite a few. The electronic circuit implementation of the three coupled Duffing oscillators system was executed based on simple available electronic components. The simulation study of the proposed analog circuit in PSpice matches well with the findings the theoretical study. Our results yield the conclusion that coupling excitation-free oscillators of Duffing type in a ring topology can be seen as a useful approach to obtain multi-spiral chaotic attractors.