Almost invariance of a slow manifold with bifurcation
- When?
- Wednesday 12 May 2010, 16:00 to 17:00
- Where?
- 24AA04
- Open to:
- Students, Staff
- Speaker:
- Kristian Kristiansen (Surrey)
Abstract: Slow manifolds in singular perturbed ODEs are the set of equilibria of the fast system with $\epsilon=0$, where $\epsilon$ is the small parameter. Fenichel showed that normally hyperbolic slow manifolds do persist together with its stable and unstable manifolds. On the other hand, normally elliptic slow manifolds only persist adiabatically in general. In particular for Hamiltonian systems with only one fast degree of freedom they persist with exponentially small error.
The fast system bifurcates at a border of normally ellipticity and hyperbolicity. The analysis near such a point is complicated by the fact that the time scales become comparable. In this talk we consider a slow-fast, two degree of freedom Hamiltonian system in which the equilibria of the fast system pitchfork bifurcates at a certain value of the slow variables. The system arises when modelling tethered satellites. We use averaging and combine ideas of Chow and Young on separatrix crossing for systems for one and a half degree of freedom with a blow-up near the bifurcation to show that a small set, polynomial in the small parameter, remains close to the union of the normally elliptic slow manifolds as the slow variables drift through the bifurcation point.
