Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

Abstract

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation.1. Introduction. Equilibria, periodic orbits, connecting orbits, and, more generally, invariant manifolds are the fundamental components through which much of the structure of the dynamics of nonlinear differential equations is explained. Thus it is not surprising that there is a vast literature on numerical techniques for approximating these objects. In particular, the last 30 years have witnessed a strong interest in developing computational methods for connecting orbits [5,10,12,14,15,23]. As mentioned in [13], most algorithms for computing heteroclinic or homoclinic orbits reduce the question to solving a boundary value problem (BVP) on a finite interval where the boundary conditions are given in terms of linear or higher order approximations of invariant manifolds near the steady states. We adopt the same philosophy in this paper. The novelty of our approach is that our computational techniques provide existence results and bounds on approximations that are mathematically rigorous. We hasten to add that a variety of authors have already developed methods that involve a combination of interval arithmetic with analytical and topological tools and provide proofs for the existence of homoclinic and heteroclinic solutions to differential equations [28,22,31,6,32]. However, the combination of techniques we propose appears to be unique, perhaps because our approach is being developed with additional goals in mind. We return to this point later.

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

“…The parameterization method is based on formulating certain operator equations (invariance equations) which simultaneously describe both the dynamics on the manifold and its embedding. The method has been implemented numerically to study a variety of problems involving stable and unstable manifolds of equilibria and fixed points (Mireles-James 2013; Mireles-James and Mischaikow 2013; Mireles James and Lomelí 2010 ;Haro 2011;van den Berg et al 2011), stable/unstable manifolds of periodic orbits for differential equations (Guillamon and Huguet 2009;Huguet and de la Llave 2013;Castelli et al 2015), quasiperiodic invariant sets in dynamical systems (Haro and da la Llave 2006) and more recently in order to simultaneously compute invariant manifolds with their unknown dynamics Haro et al 2014), to mention just a few examples.…”

“…The works of van den Berg et al (2011), Mireles-James and Mischaikow (2013) and Mireles-James (2015) exploit this a posteriori analysis and implement mathematically rigorous numerical validation methods for the stable and unstable manifolds. The term "validation" here expresses the fact that the computations provide explicit bounds on all approximation errors involved.…”

“…First, we note the works of van den Berg et al (2011), Mireles-James and Mischaikow (2013) and MirelesJames (2015) assume that certain non-resonance conditions between the eigenvalues are satisfied. The main goal of the present work is to weaken this assumption.…”

“…Third, existing rigorous continuation methods (van den Berg et al 2010;Breden et al 2013) are directly applicable to our novel approach, thus putting the rigorous computation of branches of connecting orbits within direct reach. Of course, from the viewpoint of continuation, all three advantages are crucial.…”

“…Once these givens are fixed the roots of the radii polynomials yield not only existence and uniqueness results, but also tight error estimates and isolation bounds for the problem. Radii polynomial methods have been applied successfully to a number of problems in dynamical systems, partial differential equations and delay equations, and we refer the interested reader to Day et al (2007), Breden et al (2013), van den Berg et al (2010), van den Berg et al (2011), Hungria et al (2016) and van den Berg et al (2015) for more discussion and references.…”