Bifurcating Periodic Solutions in Rotationally Symmetric Systems

Rotationally symmetric systems governed by nonlinear ordinary differential equations and dependent on a parameter are analyzed for small bifurcating periodic solutions when the equilibrium configuration of the system loses stability at some critical value of the parameter with complex conjugate airs of eigenvalues crossing the imaginary axis. Two distinct possibilities are shown to exist. In one case, the symmetry of the system induces multiplicity of the complex pairs while in the second case, despite the symmetry of the system, only one complex pair crosses the imaginary axis. For the case when two coincident complex pairs of eigenvalues cross the imaginary axis, using the method of alternate problems, it is shown that the system, in general, has two distinct bifurcating periodic solutions. Explicit expressions for the first approximations to these solutions are obtained using the general form of nonlinearities induced by the symmetry of the system. Conditions for stability of these solutions are also given. Motivated by a physical example, a restricted class with additional symmetries in these rotationally symmetric systems is identified and it is shown that for this case there is an additional family of bifurcating periodic solutions. This family is shown to be unstable when subcritical.