3 research outputs found
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
Period doubling in the Rössler system- a computer assisted proof
Using rigorous numerical methods we validate a part of the bifurcation diagram for a Poincaré map of the Rössler system [26]- the existence of two period doubling bifurcations and the existence of a branch of period two points connecting them. Our approach is based on the Lyapunov-Schmidt reduction and uses the C r-Lohner algorithm [31] to obtain rigorous bounds for the Rössler system