Parameterization of invariant manifolds for periodic orbits. II: A posteriori analysis and computer assisted error bounds

DOI10.1007/s10884-017-9609-zzbMath1419.37020OpenAlexW2749417548MaRDI QIDQ1616363

Jason D. Mireles James, Roberto Castelli, Jean-Philippe Lessard

Publication date: 6 November 2018

Published in: Journal of Dynamics and Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10884-017-9609-z

zbMATH Keywords

periodic orbitsparameterization methodcomputer assisted prooftruncation error analysisstable/unstable manifoldsvalidated error bounds

Mathematics Subject Classification ID

Periodic orbits of vector fields and flows (37C27) Invariant manifold theory for dynamical systems (37D10) Numerical nonlinear stabilities in dynamical systems (65P40) Approximation methods and numerical treatment of dynamical systems (37M99)

Related Items (12)

Rigorous enclosures of solutions of Neumann boundary value problems ⋮ The parameterization method for center manifolds ⋮ Saddle-type blow-up solutions with computer-assisted proofs: validation and extraction of global nature ⋮ Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four-body problem ⋮ Validated numerical approximation of stable manifolds for parabolic partial differential equations ⋮ Semianalytical Computation of Heteroclinic Connections Between Center Manifolds with the Parameterization Method ⋮ Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof ⋮ Efficient representation of invariant manifolds of periodic orbits in the CRTBP ⋮ Homoclinic dynamics in a spatial restricted four-body problem: blue skies into Smale horseshoes for vertical Lyapunov families ⋮ Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence ⋮ A new method to compute periodic orbits in general symplectic maps ⋮ Chebyshev–Taylor Parameterization of Stable/Unstable Manifolds for Periodic Orbits: Implementation and Applications

Uses Software

• Taylor

Cites Work

• The parameterization method for invariant manifolds. From rigorous results to effective computations
• Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra
• Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds
• Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation
• Geometric proof for normally hyperbolic invariant manifolds
• Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem
• Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds
• Blow up oscillating solutions to some nonlinear fourth order differential equations
• Computation of whiskered invariant tori and their associated manifolds: new fast algorithms
• A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results
• Symmetric homoclinic solutions to the periodic orbits in the Michelson system
• A parametrization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms
• The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof
• Computation of maximal local (un)stable manifold patches by the parameterization method
• Analytic enclosure of the fundamental matrix solution.
• Introduction to Hamiltonian dynamical systems and the \(N\)-body problem.
• Covering relations, cone conditions and the stable manifold theorem
• Numerical computation of heteroclinic orbits
• Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study
• Rigorous chaos verification in discrete dynamical systems
• Heteroclinic connections between periodic orbits in planar restricted circular three-body problem -- a computer-assisted proof
• Rigorous computer-assisted application of KAM theory: a modern approach
• Wide oscillation finite time blow up for solutions to nonlinear fourth-order differential equations
• Automatic differentiation for Fourier series and the radii polynomial approach
• Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields
• A new mathematical explanation of what triggered the catastrophic torsional mode of the tacoma narrows bridge
• A qualitative explanation of the origin of torsional instability in suspension bridges
• Blow-up profile for solutions of a fourth order nonlinear equation
• The parameterization method for invariant manifolds. III: Overview and applications
• Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach
• Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits
• Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps
• Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence
• Reliable Computation of Robust Response Tori on the Verge of Breakdown
• Weak Stability Boundary and Invariant Manifolds
• Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation
• A Computational and Geometric Approach to Phase Resetting Curves and Surfaces
• CONTINUATION OF CONNECTING ORBITS IN 3D-ODEs (II) : CYCLE-TO-CYCLE CONNECTIONS
• A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods
• A method to rigorously enclose eigendecompositions of interval matrices
• The Numerical Computation of Connecting Orbits in Dynamical Systems
• Validated Continuation for Equilibria of PDEs
• CONTINUATION OF CONNECTING ORBITS IN 3D-ODEs (I): POINT-TO-CYCLE CONNECTIONS
• A computer-assisted proof of the Feigenbaum conjectures
• Numerical Computation and Continuation of Invariant Manifolds Connecting Fixed Points
• A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem
• The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces
• The parameterization method for invariant manifolds II: regularity with respect to parameters
• Connecting orbits and invariant manifolds in the spatial restricted three-body problem
• Rigorous verification of chaotic behaviour of maps using validated shadowing
• A Homoclinic Solution for Excitation Waves on a Contractile Substratum
• Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP
• Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis
• Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form
• Stationary Coexistence of Hexagons and Rolls via Rigorous Computations
• A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi‐Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity
• TRANSPORT IN DYNAMICAL ASTRONOMY AND MULTIBODY PROBLEMS
• Existence of a Center Manifold in a Practical Domain around $L_1$ in the Restricted Three-Body Problem
• Low energy transfer to the Moon
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Parameterization of invariant manifolds for periodic orbits. II: A posteriori analysis and computer assisted error bounds