Bifurcation from relative periodic solutions (Q2716658)

This paper concerns relative periodic solutions \({\mathcal P}=\{\gamma \Phi_t(u_0): \gamma \in \Gamma\), \(t \in \mathbb R\}\) of a \(\Gamma\)-equivariant ODE \(u'={\mathcal F}(u).\) Here \(\Phi_t\) is the flow corresponding to the ODE, \(\Gamma\) is an finite-dimensional algebraic Lie group acting smoothly and properly on the phase space, and the isotropy subgroup \(\Delta\) of \(u_0\) is supposed to be compact. \textit{J. S. W. Lamb} and \textit{C. Wulff} [Phys. Lett., A 267, 167--173 (2000; Zbl 0946.37042)] presented a systematic approach to local bifurcation from \({\mathcal P}\) in the case that \(\dim \Gamma = \dim \Delta\) (then generically \({\mathcal P}\) is an isolated periodic orbit). In the present paper these results are generalized to the general case \(\dim \Gamma \geq \dim \Delta\), using a center bundle approach of \textit{B. Sandstede, A. Scheel} and \textit{C. Wulff} [J. Nonlinear Sci. 9, No. 4, 439--478 (1999; Zbl 0951.35014)]. This approach says, roughly speaking, that, in a comoving frame, \({\mathcal P}\) can be transformed into a group orbit of ordinary periodic solutions, and that bifurcation from \({\mathcal P}\) reduces, modulo drifts along \(\Gamma\)-orbits, to bifurcation from the ordinary periodic solution.