Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

Abstract

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

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

“…As a new feature of the present work in comparison with the analysis in van den Berg et al (2011), Mireles-James andMischaikow (2013) and Mireles-James (2015) we are able to incorporate resonant cases directly in our novel framework for solving and validating the parametrization of the (un)stable manifold. In particular, we focus on the two types of co-dimension one resonances, namely a single regular resonance and an algebraically double, geometrically simple eigenvalue.…”

“…The recursive approach shows that there is (a priori) a unique solution of (8) satisfying the constraints (9), although the decay of the sequence is not guaranteed a priori. The validation in van den Berg et al (2011), Mireles-James andMischaikow (2013) and Mireles-James (2015) relies on analysis in function spaces of so-called N -tails.…”

“…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.…”

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

“…For this we follow the previous work of authors one and two with J.B. van den Berg and K. Mischaikow in [3,26] and exploit the so-called parameterization method for invariant manifolds [5,6,7]. This method facilitates the computation of polynomial approximations of the chart maps to any desired finite order and provides rigorous error bounds on the truncation errors.…”