J. Dyn. Diff. Eqn. 1 (1989) 347-367
Ian Melbourne
Abstract
We analyze the interaction of three Hopf modes and show that locally a bifurcation gives rise to intermittency between three periodic solutions. This phenomenon can occur naturally in three-parameter families. Consider a vector field f with an equilibrium and suppose that the linearization of f about this equilibrium has three rationally independent complex conjugate pairs of eigenvalues on the imaginary axis. As the parameters are varied, generically three branches of periodic solutions bifurcate from the steady-state solution. Using Birkhoff normal form, we can approximate f close to the bifurcation point by a vector field commuting with the symmetry group of the three-torus. The resulting system decouples into phase amplitude equations. The main part of the analysis concentrates on the amplitude equations in R3 that commute with an action of (Z2)3. Under certain conditions, there exists an asymptotically stable heteroclinic cycle. A similar example of such a phenomenon can be found in recent work by Guckenheimer and Holmes. The heteroclinic cycle connects three fixed points in the amplitude equations that correspond to three periodic orbits of the vector field in Birkhoff normal form. We can consider f as being an arbitrarily small perturbation of such a vector field. For this perturbation, the heteroclinic cycle disappears, but an `invariant' region where it was is still stable. Thus, we show that nearby solutions will still cycle around among the three periodic orbits.