Symmetric Liapunov center theorem (Q526924)

From the text: The goal of this paper is to prove a symmetric version of the Liapunov center theorem. Let \(\Omega \subset \mathbb{R}^n\) be an open and \(\Gamma\)-invariant subset of \(\mathbb{R}^n\) equipped with a linear action of a compact Lie group \(\Gamma.\) Assume that \(q_0 \in \Omega\) is a critical point of a \(\Gamma\)-invariant potential \(U : \Omega \to \mathbb{R}\) of class \(C^2\) with an isotropy group \(\Gamma_{q_0}=\{\gamma \in \Gamma : \gamma q_0=q_0\}.\) Since the gradient \(\nabla U : \Omega \to \mathbb{R}^n\) is \(\Gamma\)-equivariant, the \(\Gamma\)-orbit \(\Gamma(q_0)=\{\gamma q_0 : \gamma \in \Gamma\}\) consists of critical points of the potential \(U\), i.e., \(\Gamma(q_0) \subset (\nabla U)^{-1}(0),\) and therefore \(\dim \ker \nabla^2 U(q_0) \geq \dim \Gamma(q_0).\) The main result of this article is the following. Theorem. (Symmetric Liapunov center theorem). Let \(U: \Omega \to \mathbb{R}\) be a \(\Gamma\)-invariant potential of the class \(C^2\) and \(q_0 \in \Omega\) be a critical point of \(U\). Assume that \((1)\) the isotropy group \(\Gamma_{q_0}\) is trivial, \((2)\) \(\dim \ker \nabla^2 U(q_0) = \dim \Gamma(q_0),\) \((3)\) there exists at least one positive eigenvalue of the Hessian \(\nabla^2 U(q_0)\), i.e., \(\sigma(\nabla^2 U(q_0)) \cap (0,+\infty) = \{\beta_1^2,\ldots, \beta_m^2\}\) and \(m\geq 1\). Then for any \(\beta_{j_0}\) such that \(\beta_j/\beta_{j_0}\not\in\mathbb{N}\) for all \(j\neq j_0\) there exists a sequence \(\{q_k(t)\}\) of non-stationary periodic solutions of the system \(\ddot{q}(t)=-\nabla U(q(t))\) with minimal periods \((T_k)\) such that dist\((\Gamma(q_0), q_k([0, T_k]))\to 0\) and \(T_k\to 2\pi/\beta_{j_0}\) as \(k\to\infty\). To prove this theorem we apply the infinite-dimensional version of the \((\Gamma \times S^1)\)-equivariant Conley index theory due to \textit{M. Izydorek} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51, No. 1, 33--66 (2002; Zbl 1035.37009)].