Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

Abstract

We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a posteriori analysis of truncation errors which, when combined with careful management of round off errors, yields a mathematically rigorous enclosure of the manifold. The main novelty of the present work is that, by conjugating the dynamics on the manifold to a polynomial rather than a linear vector field, the computerassisted analysis is successful even in the case when the eigenvalues fail to satisfy non-resonance conditions. This generically occurs in parametrized families of vector fields. As an example, we use the method as a crucial ingredient in a computational existence proof of a connecting orbit in an amplitude equation related to a pattern formation model that features eigenvalue resonances.

Computing Invariant Sets of Random Differential Equations Using Polynomial Chaos

“…Remark 3.3. In cases where resonant eigenvalues are present, a similar approach can still be used, but the conjugacy condition (9) defining Q has to be adapted [31].…”

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

“…. , d (see Lemma 2.1 in [84] and Lemma 2.6 in [79] for elementary proofs in finite dimensional contexts).…”

“…It must also be noted that the present work builds on a growing body of literature devoted to validated numerical methods for studying stable/unstable manifolds of equilibrium solutions for finite dimensional vector fields. A thorough review is beyond the scope of the present work, and we direct the reader to [6,2,96,93,86,57,21,19,68,79,11,20] for more complete discussion of the literature. This list ignores works devoted to validated numerical methods for stable/unstable manifolds of discrete time dynamical systems and also validated methods for computing other types of invariant manifolds (for example invariant tori and their stable/unstable manifolds).…”

Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

“…In this section we compute an approximate parameterization of the (local) stable manifold at 0, and provide explicit error bounds on this parameterization. This is done by combining the ideas of the parameterization method (first introduced in [5,6,7], see also [13]) and of rigorous computation (following the approach of [27,1]). Having computed the parameterization, we will be able to obtain the homoclinic connection in the next section by taking advantage of the fact that it is now enough to compute an orbit on a finite time interval, i.e., an orbit that ends up in the local stable and unstable manifolds (or rather, we compute and verify an orbit that starts from the symmetric section and ends up, after some finite time, in the local stable manifold, see (1.3)).…”