Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction

Details Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction

2009-12-18

arxiv.org/abs/0912.3709v1

Nonlinear Sciences

Pattern Formation and Solitons

25 pages, 12 figures

Scientific paper

Bifurcation with spherical symmetry and mode interaction is known to produce robust heteroclinic cycles between axisymmetric steady-states. The case of mode interaction with representations of $\SO3$ of degrees $\ell=1$ and 2 has been studied in the context of the onset of convection in a spherical shell and central gravity force by \cite{ArCh91} and \cite{ChGuLa99} who explained certain numerical observations made in 1986 by Friedrich and Haken \cite{FriHak86}. This problem has recently regained interest thanks to the launch of an experiment of convection between two concentric spheres in the International Space Station, thus providing a system with (nearly) spherical symmetry (GeoFlow project, see \cite{egbersetal}). In this case however, the onset of convection cannot be excited by modes of degree $\ell$ smaller than 3 due to technical constraints. Motivated initally by this experiment, we have analyzed the occurence of robust heteroclinic cycle in the case of $\ell=3, 4$ mode interaction. This case is highly complex but, applying the methods of equivariant bifurcation theory, we have shown the existence of (generalized) robust heteroclinic cycles involving, not only axisymmetric states, but also and principally states with cubic symmetry. These objects are observable in the numerical simulations of the dynamics on the center manifold. We provide video clips showing the corresponding evolution of the pattern on the sphere.

Affiliated with

Also associated with

No associations

Rating

Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bifurcation of robust heteroclinic cycles in spherically invariant systems with $\ell =3,4$ mode interaction will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-349945