Chaotic Behaviors Of One-Dimensional Wave Equations With Van Der Pol Boundary Conditions Containing A Source Term
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS(2020)
摘要
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 (I,f(infinity)) induced by a sequence of continuous maps and its functional envelope (L1(I,I),H-infinity), and show that, under some considerable condition, H-infinity is transitive if and only if f(infinity) is weakly mixing of order 3; H-infinity is Li-Yorke chaotic and sensitive if f(infinity) is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.
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关键词
Wave equation, functional envelope, weak mixing of n order, Li-Yorke chaos, sensitivity
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