Bifurcation of subharmonic solutions in time-reversible systems
The paper describes the bifurcation of subharmonic solutions for periodic time-reversible equations of the form (1) \(\dot x=f(t,x,\lambda)\) with \(x\in {\mathbb{R}}^ n\), \(\lambda\in {\mathbb{R}}\) and \(f(t,0,\lambda)=0\). Time- reversibility means that \(f(-t,Sx,\lambda)=-Sf(t,x,\lambda)\) for some \(S\in {\mathfrak L}({\mathbb{R}}^ n)\) with \(S^ 2=I\). A consequence of this time-reversibility is that if \(\mu\in {\mathbb{C}}\) is a characteristic multiplier for (1), then so is 1/\(\mu\) ; hence if (1) has a pair of simple complex conjugate characteristic multipliers, then they have to lie on the unit circle. One assumes that (1) has such a pair of multipliers \(\exp (\pm i\psi (\lambda))\), with \(\psi (0)=2\pi p/q\) and \(\psi '(0)\neq 0\). Using a \(D_ q\)-equivariant Lyapunov-Schmidt reduction it is shown that at \(\lambda =0\) there is bifurcation of 2q branches of \(2\pi\) q-periodic solutions. Expressions are given from which one can determine whether the branches are sub- or supercritical; also the (weak) stability properties of the bifurcating subharmonic solutions are determined.
scientific article; zbMATH DE number 3973884