Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds
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In this article, the authors developed a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on hyperbolic invariant manifolds theory, which allows to validate the needed conditions in a finite number of steps, which can be performed by a computer by means of rigorous interval arithmetic computations.About:
This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2022-03-01 and is currently open access. It has received 2 citations till now. The article focuses on the topics: Computer-assisted proof & Mathematical proof.read more
Citations
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Diffusing orbits along chains of cylinders
Marian Gidea,Jean-Pierre Marco +1 more
TL;DR: In this paper , the existence of trajectories that drift along a sequence of cylinders with homoclinic connections was shown to be possible under certain conditions on the dynamics of the trajectories.
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Computer assisted proof of drift orbits along normally hyperbolic manifolds II: Application to the restricted three body problem
TL;DR: In this paper , a computer assisted proof or diffusion in the Planar Elliptic Restricted Three Body Problem (PEBP) is presented, based on shadowing of orbits along transversal intersections of stable/unstable manifolds of a normally hyperbolic cylinder.
References
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An exponential estimate of the time of stability of nearly-integrable hamiltonian systems
TL;DR: The main ideas of the proof of the exponential estimate were discussed in this paper, including steepness conditions and forbidden motions of the discs of fast drift on the steepness of the unperturbed Hamiltonian.
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The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces
TL;DR: In this paper, the authors introduced a method to prove existence of invariant manifolds and to find simple polynomial maps which are congruent to the dynamics on them, and they showed that if X1 is an invariant subspace of A and A satisfies certain spectral properties, then there exists a unique Cr manifold which is invariant under======F and tangent to X1.
Book
Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space
TL;DR: In this article, the existence of invariant manifolds for perturbed semi-low coordinate systems has been studied in the context of local coordinate systems with Cone lemmas and normal hyperbolicity.
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The parameterization method for invariant manifolds III: overview and applications
TL;DR: In this paper, the authors describe a method to establish existence and regularity of invariant manifolds and, at the same time, find simple maps which are conjugated to the dynamics on them.