标题： 常微分方程最小周期的计算 显示英文标题

标题： Computation of minimal periods for ordinary differential equations

摘要： 我们考虑寻找给定自治常微分方程的精确周期解的最短可能周期的问题。 我们证明，给定定义在相应动力系统状态空间上的两个类似李雅普诺夫的可观测函数，并满足某种逐点不等式，我们可以得到此类周期的全局下界。 我们提供一种方法，适用于仅对在对称变换下不变的解进行周期边界的情况，以及一般周期轨道的边界。 如果控制方程在状态变量中是多项式的，我们可以使用半定规划来计算这样的辅助函数，从而计算可以使用有理数算术严格验证的下界。 我们将我们的方法应用于洛伦兹系统和亨农-海勒斯系统。 对于这两个系统，我们能够给出精确的边界，精确到小数点后几位。 显示英文摘要

摘要： We consider the problem of finding the shortest possible period for an exactly periodic solution to some given autonomous ordinary differential equation. We show that, given a pair of Lyapunov-like observable functions defined over the state space of the corresponding dynamical system and satisfying a certain pointwise inequality, we can obtain a global lower bound for such periods. We give a method valid for the case of bounding the period of only those solutions which are invariant under a symmetry transformation, as well as bounds for general periodic orbits. If the governing equations are polynomial in the state variables, we can use semidefinite programming to find such auxiliary functions computationally, and thus compute lower bounds which can be rigorously validated using rational arithmetic. We apply our method to the Lorenz and Henon-Heiles systems. For both systems we are able to give validated bounds which are sharp to several decimal places.