Abstract:Multiple-time scale problems are ubiquitous in both science and engineering, while the slow varying parameter is one of the iconic feature of multiple-time scale. However, up till now, most of bifurcation structures and oscillation patterns revealed by literatures are relatively simplex. In this paper, we take the non-autonomous Duffing map as a example to explore family of complex relaxation oscillation patterns, which are little concerned by previous study. The fast subsystem exhibits an S-shaped fixed point curve, and the stable upper and lower branches evolve into chaos by a cascade of Flip bifurcations. What's more, we can observe a pair of critical parameter values under some parameter conditions, which lead to the catastrophe vanish of chaotic attractors. When the bifurcation parameter reaches these values, chaotic attractors may contact with the unstable fixed point or just stay in a distance apart. By simulating the distribution of basins of attraction owned by fast subsystem, we show that there exist critical points of boundary crisis, nearby which chaotic attractor evolved from stable fixed points can coexist with period-2 ⁿ( n=0, 1, 2, …) attractor or even another chaotic attractor. When the non-autonomous term (i.e., the slow variable) passes through critical points, distruction of bi-stability may lead to the transition from chaotic attractor in pre-crisis stage to the coexisting attractor, thus the boundary crisis motivates different patterns of symmetric relaxation oscillation. In particular, patterns here show structures containing different number of delay flip bifurcations, owe to the fact that delay quantities of Flip points in different level take disparate magnitude. Our results enrich dynamical mechanisms of multiple-time scale in discrete systems.