Jean-Philippe Lessard, Jason Mireles-James and Jan Bouwe van den Berg
Submit Time
2014-02-27 05:45:54
Session
Special Session 117: Rigorous and numerical methods for invariant manifolds
Contents
In the first part of this talk we present a computational method based on Chebyshev series to rigorously compute solutions
of initial and boundary value problems of analytic nonlinear vector fields. The idea is to recast solutions as fixed points of an operator on the Banach space of geometrically decaying
Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point nearby an approximate solution. We illustrate the method by presenting a sample application to the solution of initial value problems in the Lorenz system.
In the second part of the talk we show how an analogue approach can be used to rigorously solve the invariance equation occurring in the application of the parametrization method to the computation of the local stable manifold of a hyperbolic fixed point of an analytic ODE. We derive an equivalent fixed point problem on the space of geometrically decaying power series coefficients that we again tackle using the method of radii polynomials. In particular the resulting fixed point equation shares many structural features with the ODE case. We finish the talk with applications showing the performance of our approach.