Reversibility and branching of periodic orbits
The authors consider reversible systems of the form \[ \dot{x} = F (x,\lambda ), \] \(x \in\mathbb R^{2 n}, \;\lambda \in \mathbb R^2 \) and \(F :\mathbb R^{2 n} \times\mathbb R^2 \to\mathbb R^{2 n}\) of class \(C^k\) (with \(k \geq 2\)) such that \[ R ( F (x , \lambda)) \;= \;-RF (R x , \lambda) \] for some reversing linear involution \(R\), i.e., with \(R^2 = I\), \(R \not= I\) (for details on and applications of reversible dynamical systems, see e.g. [\textit{J. S. Lamb} and \textit{J. A. G. Roberts}, Physica D 112, No. 1--2, 1--39 (1998; Zbl 1194.34072)]). It is also assumed that the linear space \(\operatorname{Fix} (R)\) of \(x\) such that \(R x = x\) has dimension \(n\). When the linear part of the system satisfies non-resonance conditions, it is known (see [\textit{R. L. Devaney}, Trans. Am. Math. Soc. 218, 89--113 (1976; Zbl 0363.58003)]) that results analogous to the Lyapunov center theorem hold. The resonant case has been studied in previous works [\textit{A. Jacquemard} et al., Ann. Mat. Pura Appl. (4) 187, No. 1, 105--117 (2008; Zbl 1206.37011); \textit{M. F. S. Lima} and \textit{M. A. Teixeira}, Bull. Braz. Math. Soc. (N.S.) 40, No. 4, 511--537 (2009; Zbl 1189.34076)] for the cases of \(0:p:q\), \(1:1:1\), \(1:1:2\) and \(1:2:2\) resonances and \(n=3\). Here, the authors consider again systems with \(n=3\), focusing on some special structure. That is, first, a system \(F^{(0)}\) corresponding to \( x^{(vi)} + \lambda_1 x^{(iv)} + \lambda_2 x^{(ii)} + x = 0\) is studied, and, then, one considers systems of the form \(F = F^{(0)} + F^{(1)}\) with \(F^{(1)} = (0,0,0,0,0,f)\), i.e., corresponding to \[ x^{(vi)} + \lambda_1 x^{(iv)} + \lambda_2 x^{(ii)} + x + f(x) = 0. \] The analysis is based on the Belitskii normal form and the Lyapunov-Schmidt reduction, and provides conditions for the existence of families of periodic solutions. The systems analyzed are either non-resonant or resonant with a resonance of the types \(\alpha : 1 :p\) (with \(\alpha\) rational, \(p \geq 2\) a positive integer).
scientific article; zbMATH DE number 6155714