##
Intermittency as a codimension-three phenomenon

* 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 **R**^{3} that commute with an action of (**Z**_{2})^{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.