Bifurcation and chaos in discrete-time BVP oscillator

In this paper, we investigate the discrete-time Bohöffer–Van der Pol (BVP) oscillator obtained by Euler method. We provide the sufficient conditions of existence, asymptotic stability of the fixed points, then give theoretical analysis for local bifurcations of the fixed points, and derive the conditions under which the local bifurcations such as pitchfork, saddle-node, flip and Hopf occur at the fixed points. Furthermore, we prove that the fixed point eventually evolves into a snap-back repeller which generates chaotic behavior in the sense of Marotto's chaos when certain conditions are satisfied. Finally, several numerical simulations are provided to demonstrate the theoretical results of the previous and to show the new complex dynamical behaviors of the system.