Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

Abstract

This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.

Automated computation of autonomous spectral submanifolds for nonlinear modal analysis

“…Another advantage of the method is its suitability for algorithmic implementations for arbitrary orders of accuracy in arbitrary dimensions. For applications of the parameterization method to other types of dynamical systems, we refer the reader to the work of Haro et al [19], van den Berg and Mireles James [20] and Mireles James [21].…”

Rigorous numerics for piecewise-smooth systems: A functional analytic approach based on Chebyshev series

“…where B ρ 1 is the open ball in R n s with a small enough radius ρ 1 > 0 so that P (s)  [19][20][21] with the recent results [7,22,23,6,14] which allow computing rigorously stable and unstable manifolds of equilibria of vector fields.…”

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.…”