Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

Abstract

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.