【摘要】 许多非线性动力系统都有某种对称性,在不同情形下可有不同的表现形式,但始终保持其对称的特点.不同对称形式间的转变导致对称破缺分岔或激变.关于非线性动力系统中相空间运动轨道的对称破缺分岔,已有大量研究工作,但绝大多数是指周期或拟周期相轨的对称破缺,偶尔提到对称系统中的混沌相轨也存在"对偶性".最近,在简谐外激Duffing系统周期轨道对称破缺引发鞍-结分岔的研究中,得到了分岔后由Poincar啨映射点间断流构成的图像,其中包括两个稳定周期结点、一个周期鞍点,及其稳定流形与不稳定流形,均较规则.本工作研究了正弦参激Duffing系统共存的周期与混沌吸引子,及其在初始平面上中心对称分形吸引域的错综复杂图像;并定性探索其邻近发生的对称破缺激变,这些结果正是上述研究的延拓和补充.更多还原

【Abstract】 Most nonlinear dynamic systems may exhibit a certain symmetrical form.Symmetry is a kind of invariance,maybe appearing in different topological forms under different situations,but always keeping the characteristic of symmetry.The transition between different symmetric forms often leads to symmetry breaking bifurcation or crisis.Many studies on symmetry breaking bifurcations of phase trajectories of a nonlinear dynamical system have been reported,most of which are related to periodic or quasi-periodic orbits.Only a few of them ever mentioned that "duality" might also exist for a couple of chaotic attractors in symmetrical nonlinear dynamical systems.Recently,in the study of the saddle-node bifurcation resulting from symmetry breaking of periodic phase orbits in a Duffing oscillator driven by a sinusoidal excitation,an interesting phase portrait of the flow pattern of discrete Poincaré mapping points has been obtained after symmetry breaking bifurcation.Along with the flow pattern,two stable periodic nodes and one periodic saddle,together with its stable and unstable manifolds,are shown,which are all in a regular form yet.In this study,as an extension of the above results,a complicated portrait for attractive basins of coexisting periodic and chaotic attractors in a parametrically driven double-well Duffing system is obtained,which is fractal,interwoven,yet symmetrical.In addition,the neighboring symmetry breaking crises are studied qualitatively.更多还原