标题： 混沌的级联路径 显示英文标题

标题： The cascades route to chaos

摘要： 在依赖于参数的动力系统中存在周期倍分岔是通向混沌的基本途径之一。很少有人提到，只要存在一个周期倍分岔，几乎总是存在无限多个分岔。我们报告说，在一维和二维相空间中，当参数变化导致从无混沌到混沌的转变时，在一些温和的假设下必须存在无限多个分岔。我们对混沌的定义包括那些不是吸引子的混沌集的情况。数值研究显示，这一结果适用于受迫阻尼摆和受迫杜芬方程，在每次强迫周期内观察解一次。我们进一步表明，在许多情况下，分岔以成对的形式出现，并通过不稳定周期轨道（在联合参数-状态空间中）连接。成对的分岔可以被扰动破坏或创建，而未配对的分岔即使在显著扰动下也保持不变。 显示英文摘要

摘要： The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one of the basic routes to chaos. It is rarely mentioned that there are virtually always infinitely many cascades whenever there is one. We report that for one- and two-dimensional phase space, in the transition from no chaos to chaos -- as a parameter is varied -- there must be infinitely many cascades under some mild hypotheses. Our meaning of chaos includes the case of chaotic sets which are not attractors. Numerical studies indicate that this result applies to the forced-damped pendulum and the forced Duffing equations, viewing the solutions once each period of the forcing. We further show that in many cases cascades appear in pairs connected (in joint parameter-state space) by an unstable periodic orbit. Paired cascades can be destroyed or created by perturbations, whereas unpaired cascades are conserved under even significant perturbations.