Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

Abstract

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the NewtonKantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

“…Other studies which utilize high order expansions of stable/unstable manifolds in order to study connecting orbits between fixed points and equilibria of discrete time dynamical systems and differential equations are found in [56,57]. This idea can also be exploited in order to obtain computer assisted proof of the existence of connecting orbits [60,63,61]. In this section we discuss some numerical computations for homoclinic connections of periodic orbits exploiting the high order parameterization of the present work.…”

“…A useful feature of methods based on Parameterization is that they admit natural a-posteriori error indicators. This notion can be used in order to obtain mathematically rigorous error bounds on the numerical approximation of the invariant manifolds by computer assisted analysis [63,60,61].…”

“…The study of invariant manifolds for periodic orbits also plays a role in the study of biological and chemical oscillations, and we refer for example to the work of [15,13,17,16,72]. The study of connecting orbits can be made mathematically rigorous using computer assisted proof, and we refer to [53,54,63,61,60] [56,57,72] and also in the present work, are methods for computing accurate local representations of the stable/unstable manifolds of invariant objects. In order to understand the global dynamics of a system it is natural to try to extend the local manifold via numerical integration.…”

“…for all θ, s, t. This is the same idea used in [56,57,61] in order to parameterize real invariant manifolds associated with fixed points when there are complex conjugate eigenvalues. See the works just cited for more thorough discussion.…”

“…See also the discussion in Section 5 of [47]. Once one has rigorous a-posteriori bounds on the Fourier-Taylor expansions of the local manifolds then it is possible to extend the methods of [63,60,61] in order to obtain computer assisted proof of transverse cycle-to-cycle and cycle-to-point connections for differential equations. Computer assisted validation for stable/unstable manifolds for periodic orbits is the topic of paper II.…”