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Existence of nonstationary periodic solutions of \(G\)-symmetric asymptotically linear autonomous Newtonian systems with degeneracy. (English) Zbl 1203.37101

The purpose of this paper is to study, in the presence of \(\Gamma\)-symmetry, the existence of the nonstationary periodic solutions \(x:\mathbb{R} \rightarrow V\) of the following autonomous Newtonian system \(\ddot{x}=- \nabla \varphi(x), x(0)=x(2\pi),\dot{x}(0)=\dot{x}(2\pi),\) where \(\varphi :V \rightarrow \mathbb{R}\) is a \(C^2\)-differentiable \(\Gamma\)-invariant function such that \((\nabla \varphi)^{-1}(0)=\{0\}\) and \(\nabla\varphi\) is asymptotically linear at infinity, i.e., there exists a symmetric \(\Gamma\)-equivariant linear map \(B:V\rightarrow V\) such that \(\nabla\varphi (x)=Bx+o(\|x\|)\) as \(\|x\| \rightarrow \infty\). Here \(\Gamma\) is a finite group, which is a symmetry group of certain regular polygon or polyhedra in \(\mathbb{R}^n\) acting on \(V=\mathbb{R}^n\) by permuting the coordinates of the vectors \(x\in V\).
The authors obtain similar results as in the paper of H. Ruan and S. Rybicki [Nonlinear Anal., Theory Methods Appl. 68, No. 6, A, 1479–1516 (2008; Zbl 1154.47048)] for the above system allowing \(0\) and \(\infty\) to be isolated degenerate critical points of \(\varphi\). The main result is contained in Theorem 4.2, where the authors discuss the existence and nonexistence of certain maximal orbit types appearing in several degenerate cases. Computational examples are provided with \(\Gamma\) being the dihedral groups \(D_6,D_8,D_{10}\) and \(D_{12}\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
58E40 Variational aspects of group actions in infinite-dimensional spaces

Citations:

Zbl 1154.47048
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References:

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