On the occurrence of multiscroll and multistable dynamics in a star network of four nonlinearly coupled self-driven Duffing–Rayleigh oscillators

Abstract

The study of oscillator networks is currently the subject of intensive efforts for researchers working in the field of non-linear science. In this article, we are interested in the collective behavior of a star network formed by four mutually coupled Rayleigh–Duffing type oscillators (RDO here after) although each isolated oscillator undergoes a fixed point motion. The coupling considered is non-linear but exploits the intrinsic non-linearity of each of the oscillators so that no non-linear coupling function is necessary as usual. Using analytical techniques, the basic properties of the star system are studied in terms of equilibrium points and their stability, conditions for the appearance of Hopf bifurcations, dissipation and existence of attractors. The direct numerical integration of the mathematical model highlights fascinating phenomena, such as the coexistence of several parallel bifurcation branches, the coexistence of dynamics of the same type or of different types (i.e., regular and chaotic, hidden or self-excited) as well as multi-spiral chaos. These features are uncovered when changing both initial conditions and parameters. The tests carried out in the laboratory using the Arduino module show very good agreement with the results of the theoretical analysis. The study conducted out in this article provides valuable information as a prelude to understanding the behavior of a much more complex network of Rayleigh–Duffing type oscillators and Gunn type microwave oscillators as well.