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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mischaikow, Konstantin Mireles James, J. D.
Copyright Year	2013
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.
Starting Page	957
Ending Page	1006
Page Count	50
File Format	PDF
DOI	10.1137/12088224X
e-ISSN	15360040
Issue Number	2
Volume Number	12
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-06-19
Access Restriction	Subscribed
Subject Keyword	parameterization method Homoclinic and heteroclinic orbits validated computation computer assisted proof high order methods Invariant manifold theory stable manifolds connecting orbits Algorithms with automatic result verification Numerical chaos
Content Type	Text
Resource Type	Article

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