Multi-scroll in bidirectional coupling schemes of two second-order non-oscillatory systems: theoretical analysis and PSpice implementation

In this study, we provide a way to generate multi-scroll dynamics between two second-order non-oscillatory systems just through a bidirectional coupling between the subsystems. Indeed, we consider two systems able to describe the trajectory of a fixed point when they are independent. Both the systems have three fixed points with the considered nonlinearity and are not able to generate a multi-scroll dynamics in their uncoupled state. We subsequently prove that the proposed coupling scheme induces a multitude of equilibrium points (up to nine) in the coupled system, which leads to the birth of multi-scroll dynamics and other highly nonlinear and complex dynamics, including the Hopf bifurcation phenomenon, crisis phenomena, a period-doubling route, multistability (up to six solutions), double-scroll and four-scroll dynamics, coexistence of parallel bifurcation branches to name a few. We would like to stress that the approach used here to generate the multi-scroll behaviors is quite different from those used in the literature. To obtain more information about the dynamics of the proposed new coupled system, a detailed analysis of the listed complex behaviors is performed by means of Routh stability criterion, mono- and bi-parametric Lyapunov diagrams, bifurcation maps, phase spaces trajectory plots and basins of attraction. A supplementary parameter k introduced in the system allows us to show the dilatation aspect of equilibria and the four-scroll dynamics in the same line. Experimental implementations in the PSpice environment validate different dynamics researched during the theoretical and numerical study. At this level, we want to highlight the fact that the coupling approach thus proposed is an efficient and effective method to induce higher order multi-scroll attractors between non-oscillatory systems.