Symmetric Liapunov center theorem

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• 作者：Ernesto Pérez-Chavela ; Sławomir Rybicki…
• 关键词：Mathematics Subject ClassificationPrimary ; 37G15 ; Secondary ; 37G40
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：603KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider a system \((*)\; \ddot{q}= -\nabla U(q)\), where U(q) is a \(\Gamma \)-invariant potential and \(\Gamma \) is a compact Lie group acting linearly on \({\mathbb {R}}^n\). If system \((*)\) possess a non-degenerate orbit of stationary solutions \(\Gamma (q_0)\) with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian \(\nabla ^2 U(q_0)\), then in any neighborhood of \(\Gamma (q_0)\) there is a non-stationary periodic orbit of solutions of system \((*)\).