Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Abstract

We study reversible, SO(2)-invariant vector fields in R 4 depending on a real parameter = which possess for ==0 a primary family of homoclinic orbits T : H 0 , : # S 1 . Under a transversality condition with respect to = the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values = (n)The existence of cascades of 2 l 3 m -pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincare map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg Landau equation for which a primary homoclinic solution is known explicitly. Academic Press

Homoclinic and Heteroclinic Bifurcations in Vector Fields

“…We can now state the following result: 2,211,267,268]). Assume that Hypotheses 5.26, 5.27(i), 5.31 and 5.32 are met.…”

“…In addition, there is a constant > 0 so that h s (t; µ) = p(µ) + e lies in the strong stable manifold W ss (p; 0) of the equilibrium p, which consists, by definition, of all solutions u(t) to (2.1) that satisfy u(t) − p = O(e λ ss t ); see [394]. 1 Throughout the entire paper, we say that eigenvalues are complex if they are not real 2 If the eigenvalues are not real, then consider the limits in (2.9) using complex coordinates Figure 2.2: Following the unstable manifold backwards along the homoclinic orbit, we expect that this manifold tends towards the strong unstable directions. This property is violated when the unstable manifold is in an inclination-flip configuration.…”

Multi-pulse solutions to the Navier-Stokes problem between parallel plates

“…The analysis of the ODE (2.5) follows that in [AfM99a], see [KaMP96] for an alternative approach. We have to generalize the theory in three major points.…”

“…A further nondegeneracy condition arises from the fact that the single-pulse solution a pulse in (2.8) has to be transversal in a way to be defined later, see transversality fails, see [AfM99a]. Using the relation (2.11) for the critical values this defines an exceptional set of wave numbers…”

“…To see this, one needs to combine the arguments in [AfM99a], [AfM99b] with the discussion of the Poiseuille problem here. Then, it is possible to speak about spatial chaos that appears in the Poiseuille problem near the instability threshold.…”