Bifurcations of the periodic solutions in symmetric systems. (English) Zbl 0596.34024

Consider the system (1) \(\dot x=v(x,\mu)\), where \(x\in R^ n\), \(\mu\in R\), the vector field v is of class \(C^{\infty}\) and invariant under the diffeomorphism g for \(\mu\in R\) (g is an involutory mapping of \(R^ n\) onto itself); the set \(\Delta =\{x\in R^ n\), \(g(x)=x\}\) is a smooth connected submanifold of \(R^ n\). A periodic solution of (1) with trajectory \(\gamma_{\mu}\subset \Delta\), \(g(\gamma_{\mu})=\gamma_{\mu}\), is called homogeneous; it is called \(\Delta\)-symmetric if \(\gamma_{\mu}\cap \Delta =\phi\). The author investigates the periodic doubling bifurcations of homogeneous and \(\Delta\)-symmetric solutions of (1) and describes symmetry-breaking bifurcations (the loss of symmetry occurs on the branch of the stable solution).
Reviewer: G.Bojadziev


34C25 Periodic solutions to ordinary differential equations
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