摘要
应用广义胞映射图论（ＧＣＭＤ）方法研究了非线性强迫Ｍａｔｈｉｅｕ方程的激变、瞬态混 沌、以及随系统参数变化的全局分岔特性．揭示了参数激励常微分系统混沌吸引子的边界激变 是由于混沌吸引子与其吸引域边界上的不稳定周期轨道发生碰撞而产生的，发现了边界激变产 生的瞬态混沌，在Ｐｏｉｎｃａｒé截面上直观地表明了瞬态混沌的几何空间结构，以及瞬态混沌的空 间结构随着系统参数逐渐远离激变临界值的衰变．给出了对自循环胞集进行局部细化的方法．
in this paper, generalized cell mapping digraph (GCMD) method is used to study crises, transient chaos, and global bifurcations with the variation of the system parameter in forced nonlinear Mathieu oscillator. GCMD is a new method to efficiently perform global analysis of nonlinear systems including global transient analysis by using digraphic algorithms on the basis of a strictly theoretical correspondence between generalized cell mapping systems and digraphs. By means of GCMD method, attractors, domains of attraction, basin boundaries and unstable solutions are obtained once through a global analysis at low computational cost. It is directly revealed that a boundary crisis in parametrically forced ordinary differential systems is caused by the collision of a chaotic attractor with an unstable periodic orbit on its basin boundary. In the case the chaotic attractor together with its basin of attraction is eradicated from phase space, and transient chaos created by the boundary crisis is found. The geometrically spatial structure of transient chaos is explicitly shown on the Poincaré section, and the decay of spatial structure of transient chaos is shown as the system parameter is varied gradually away from the critical point of the crisis. It is also demonstrated that the sudden disappearance of a periodic attractor may result from a collision between the periodic attractor and an unstable periodic orbit on its basin boundary. In the whole process of the global bifurcation, the approaches of the chaotic and periodic attractors to the unstable periodic orbit at the basin boundary and the collisions of the chaotic and periodic attractors with the unstable period orbit at the basin boundary as well as the changes of domain of attraction and basin boundary can be explicitly seen. The locally refining process of generalized cell mapping (GCM) method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space.
出处
《力学学报》
EI
CSCD
北大核心
2001年第3期423-429,共7页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(19672046).
