COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

Bursting oscillations are the archetypes of complex dynamical behaviors in systems with multiple time scales, and the problem related to dynamical mechanisms and classifications of bursting oscillations is one of the important problems in bursting research.However, up to now, most of the structures of bursting revealed by researchers are relatively simple.In this paper, we take the non-autonomous discrete Duffing system as an example to explore novel bursting patterns with complex bifurcation structures, which are divided into two groups, i.e., symmetric bursting induced by Fold bifurcations and asymmetric bursting induced by delayed Flip bifurcations.Typically, the fast subsystem exhibits an Sshaped fixed point curve with two Fold points, and the stable upper and lower branches evolve into chaos by a cascade of Flip bifurcations.When the non-autonomous term (i.e., the slow variable) passes through Fold points, transitions to various attractors (e.g., periodic orbits and chaos) on the stable branches may take place, which accounts for the appearance of Fold-bifurcation-induced symmetric bursting patterns.If the non-autonomous term is not able to pass through Fold points, but to go through Flip points, delayed Flip bifurcations can be observed.Based on this, delayed-Flip-bifurcation-induced asymmetric bursting patterns are obtained.In particular, the bursting patterns reported here exhibit complex structures containing inverse Flip bifurcations, which has been found to be related to the fact that the nonautonomous term slowly passes through Flip points of the fast subsystem in an inverse way.Our results enrich dynamical mechanisms and classifications of bursting in discrete systems.