Heterogeneous multistability and antimonotonicity for a new 3D system with a triple well nonlinearity: theoretical study, control and microcontroller implementation

Abstract

Abstract We propose a new 3D autonomous multistable jerk-like system with a nonlinear term consisting of a six-order triple well function. The presence of six equilibrium points with symmetrical locations along the x-axis represents one of the main distinguishing properties of the new system. Strikingly, the stability analysis of equilibria reveals a cascade of Hopf bifurcations at three specific values of a single control parameter, which results in several forms of complexity. Accordingly, various forms of coexisting attractors such as stable fixed points, limit cycles of diverse periodicities, and chaotic attractors are depicted for some special parameter values. Moreover, It is found that the new jerk-like system with six order triple well polynomial function exhibit extremely complex nonlinear behaviors such as anti-monotone bifurcations, hysteresis and parallel bifurcation branches. These latter aspects explain the presence of multiple (i.e. up to four) coexisting asymmetric attractors for some special rank of parameters. In the presence of multiple competing dynamics, we resort to basins of attraction in order to highlight the how the state space is magnetized. The combination of dynamic features discussed in the new jerk-like system with triple well polynomials nonlinearity introduced in this article is unique and rarely reported. An electronic version of the new system with triple well polynomial nonlinearity is implemented in PSpice. Moreover, a hardware digital implementation of the system is also carried out using an Arduino microcontroller. A very good agreement is captured between PSpice simulation results, the laboratory measurements and the theoretical predictions.