一类非光滑映射的边界碰撞分岔

BORDER-COLLISION BIFURCATION IN A KIND OF NON-SMOOTH MAPS

对于一类分段映射讨论了非线性幂次z导致的不同非光滑性,推导了周期n解的边界碰撞分岔及光滑flip和fold分岔条件.通过数值仿真验证了这些分岔条件的正确性,发现存在稳定周期窗的加周期分岔序列是非光滑映射的一个普遍现象,根本原因在于边界碰撞分岔和光滑flip或fold分岔相互作用.当z取值不同分岔序列有很大的不同,而参数γ对于分岔序列的结构影响不大,因此令参数γ=0可简化映射的参数分析.

For a kind of piecewise map, its nonsmoothness with the parameter z is discussed, and the conditions for border-collision bifurcation and smooth flip/fold bifurcation are derived. These bifurcation conditions are verified by numerical simulation, and the period-adding bifurcation scenario is a general phenomenon for non-smooth maps. The bifurcation scenario is rooted on the interrelation between border-collision bifurcation and smooth flip/fold bifurcation. The parameter z has important effect on the bifurcation scenario, but the parameter y has a minor effect, so the analysis on bifurcation scenarios with parameters can be simplified by giving y = 0.