Coupling Induced Dynamics in a Chain-Network of Four Two-Well Duffing Oscillators: Theoretical Analysis and Microcontroller-Based Experiments

This article focused on the collective dynamics of a chain network made up of four Duffing oscillators (DOS). Although each of the isolated oscillators has three fixed points and remains at rest, the coupling results in a relatively high number (i.e. eighty one) of equilibrium points in the network as well as a surprisingly rich and varied dynamics.

Analytical expressions of the equilibrium points as well as their stability were studied using the Routh-Hurwitz criterion. Results indicate that sixteen of the equilibrium points develop a Hopf bifurcation for the same critical value when one of the parameters of the network varies (i.e. coupling factor, dissipation). The resulting effect is the presence of sixteen limit cycles which in turn experience period doubling to give rise to the formation of chaos with 16 scrolls when the control parameter is further varied.

Several forms of multistability occur on the road to 16-scroll chaos due to coexisting bifurcation branches. For example, we revealed the coexistence of various combinations of solutions, some of which are: sixteen limit cycles, twelve limit cycles with four one-scroll chaotic attractors, four chaotic attractors with four scroll, as well as the coexistence of two attractors with eight scrolls which merge to form the 16-scroll attractor.

These striking features were also confirmed by laboratory experimental measurements taken on a microcontroller digital realization of the chain coupled DOS network.