Duffing系统共存周期解的稳定性与分岔演化

Stability and Bifurcation Evolution of Coexisting Periodic Solutions of Duffing System

以Duffing系统为研究对象,将延续算法和打靶法相结合,对系统共存的周期解进行延拓追踪,研究周期解的稳定性与分岔。结合吸引域、相图和Poincaré映射图揭示共存吸引子的吸引域随系统参数变化的侵蚀演变过程。结果表明:Duffing系统普遍存在多周期解的共存现象;鞍结分岔是共存吸引子产生或消失的重要原因,并且会导致其吸引域的拓扑结构发生突变;周期解的叉式分岔产生一个不稳定的周期解和两个稳定的反对称周期解;反对称吸引子的吸引域互相缠绕,导致系统的稳定性降低;当反对称吸引子经倍周期分岔进入或退出混沌时,会发生混沌吸引子的内部激变。

The Duffing system is considered as an object.Based on the combination of the continuation algorithm and shooting method,the coexisting periodic solutions of the system were tracked,and the stabilities and bifurcations of the periodic solutions were studied.The erosions and evolutions of the basins of attraction with the variation of system parameters were revealed by combining the basins of attraction,phase portraits and Poincarémapping of the periodic solutions.The results show that the coexistence of multiple periodic solutions is universal in the Duffing system;The saddle-node bifurcation leads to the topological change of the basin of attraction,which is an important reason for the appearance or disappearance of coexisting attractors;The pitchfork bifurcation produces an unstable periodic solution and two stable antisymmetric periodic solutions;The stability of the system is reduced since the basins of attraction of antisymmetric attractors entangle each other.Interior crisis of chaotic attractors occurs as the antisymmetric attractors enter or exit the chaos through period-doubling bifurcation.