Validated numerics for equilibria of analytic
                    vector fields; Invariant manifolds and connecting
                    orbits

James, J. D. Mireles

doi:10.1090/psapm/074/02

Cited by 10 publications

(28 citation statements)

References 67 publications

(109 reference statements)

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“…The potential of this feature is shown in many preceding works (e.g. [3,8,30,50,59]) for obtaining global feature of dynamical systems. We then extend the locally validated stable manifolds through the time integration of the time-reversal desingularized vector fields.…”

Section: Basic Methodologymentioning

confidence: 92%

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

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Section: Basic Methodologymentioning

confidence: 92%

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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“…We also assume that the eigenvalues are non-resonant, in the sense of Equation (22). We have the following lemma, whose proof is found in [17].…”

Section: Validated Error Bounds For the Lorenz Equationsmentioning

confidence: 99%

“…These methods yield bounds on the errors and on the size of the domain of analyticity, accurate to nearly machine precision, even a substantial distance from the equilibrium. See also the lecture notes [17].…”

Section: Introductionmentioning

confidence: 99%

“…Another advantage of the Lorenz system is that we exploit the discussion of rigorous numerics for stable/unstable manifolds given in the Lecture notes of [17]. Again this helps to minimize technical complications and allows us to focus instead on what is new here.…”

Section: Introductionmentioning

confidence: 99%

“…Hence the methods of [38] apply in a host of situations where the methods of the present work -which are based on the theory of analytic functions of several complex variables -breakdown. Moreover the initial local patch used for the computations in [38] is smaller than the validated local manifold developed in [17] from which we start our computations.…”

Section: Introductionmentioning

confidence: 99%

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Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds

Kalies¹,

Kepley²,

James³

2018

SIAM J. Appl. Dyn. Syst.

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We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilibrium. An important component of our method is that we obtain mathematically rigorous lower bounds on the size of this neighborhood, as well as validated a-posteriori error bounds for the polynomial approximation. In the second step we use a rigorous numerical integrating scheme to propagate the boundary of the local stable/unstable manifold as long as possible, i.e. as long as the integrator yields validated error bounds below some desired tolerance. The procedure exploits adaptive remeshing strategies which track the growth/decay of the Taylor coefficients of the advected curve. In order to highlight the utility of the procedure we study the embedding of some two dimensional manifolds in the Lorenz system. *

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Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

2023

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

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Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

Cited by 10 publications

References 67 publications

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

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