Bifurcation and chaotic behavior in the discrete BVP oscillator

In this work, a new class of discrete Bonhoeffer-van der Pol (BVP) system with an odd function is proposed and investigated. At first, the necessary and sufficient conditions on the existence and stability of the fixed points for this system are given. We then show the system passes through various bifurcations of codimension one, including pitchfork bifurcation, saddle-node bifurcation, flip bifurcation and Neimark-Sacker bifurcation under some certain parameter conditions. The center manifold theorem and bifurcation theory are the main tools in the analysis of the local bifurcations. Furthermore, we prove rigorously there exists Marotto's chaos in this discrete BVP system, which means the fixed point eventually evolves into a snap-back repeller. Finally, numerical simulation evidences are provided not only to further demonstrate our theoretical analysis, but also to exhibit the complex dynamical phenomena, such as the period-9, -17, -18 orbits, attracting invariant cycles, quasi-periodic orbits, ten-coexisting chaotic attractors, etc. These phenomena illustrate relatively rich dynamical behaviors of the discrete BVP oscillator.