For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li–Yorke chaos. For this end, we consider a system [Formula: see text] induced by a sequence of continuous maps and its functional envelope [Formula: see text], and show that, under some considerable condition, [Formula: see text] is transitive if and only if [Formula: see text] is weakly mixing of order [Formula: see text]; [Formula: see text] is Li–Yorke chaotic and sensitive if [Formula: see text] is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.

中文翻译：

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范德波尔边界条件下包含源项的一维波动方程的混沌行为

对于具有范德波尔边界条件的一维波动方程，文献中有几种不同的方法来表征其解的复杂性。但是，如果右端 van der Pol 边界条件包含源项，那么如何描述系统的复杂性就会出现相当大的技术难题。在本文中，我们利用拓扑动力学方法来表征系统的动力学行为，包括灵敏度、传递性和 Li-Yorke 混沌。为此，我们考虑一个由一系列连续映射及其功能包络[公式：参见文本]诱导的系统[公式：参见文本]，并表明，在某些相当大的条件下，[公式：参见文本]是传递的，如果并且仅当 [公式：参见文本] 是顺序 [公式：参见文本] 的弱混合；如果 [公式：见文本] 强烈混合，则 [公式：见文本] 是 Li-Yorke 混乱和敏感的。这些抽象的结果有其自身的意义，可以应用于这类方程。