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Maciej J. Capinski

Publications and source records attributed to Maciej J. Capinski.

15 recordsLinked to original sources
arXiv2025

Arnold Diffusion in the Full Three-Body Problem

The full three-body problem, on the motion of three celestial bodies under their mutual gravitational attraction, is one of the oldest unsolved problems in classical mechanics. The main difficulty comes from the presence of unstable and chaotic motions, which make long-term prediction impossible. In this paper, we show that the full three-body problem exhibits a strong form of instability known as Arnold diffusion. We consider the planar full three-body problem, formulated as a perturbation of both the Kepler problem and the planar circular restricted three-body problem. We show that the system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- of an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We demonstrate that the approach can be applied to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the diffusion time.

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arXiv2023

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a-posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's assymptotic periodic orbits (transverse homoclinics for $L_{4,5}$), families of periodic orbits passing through collision, and orbits connecting $L_4$ to ejection or collision.

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arXiv2021

Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us 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. We apply our method to the generalized standard map, obtaining diffusion over an explicit range of actions.

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arXiv2020

Arnold Diffusion, Quantitative Estimates and Stochastic Behavior in the Three-Body Problem

We consider a class of autonomous Hamiltonian systems subject to small, time-periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non-zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter $\varepsilon>0$. The method yields quantitative estimates: (i) The existence of orbits along which the energy drifts by an amount independent of $\varepsilon$. The time required by such orbits to drift is $O(1/\varepsilon)$; (ii) The existence of orbits along which the energy makes chaotic excursions; (iii) Explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; (iv) The existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as $\varepsilon$ tends to $0$. For each $\varepsilon$ fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches zero. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion, for appropriate sets of initial conditions. A key feature of our topological method is that it can be implemented in computer assisted proofs. We give an application to a concrete model of the planar elliptic restricted three-body problem, on the motion of an infinitesimal body relative to the Neptune-Triton system.

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arXiv2020

Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions

We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system exhibits some topological properties, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. Our method is not perturbative. It can be applied to establish invariant sets within a prescribed neighbourhood also in the absence of a normally hyperbolic invariant manifold prior to perturbation. The work is in the setting of nonorientable Banach vector bundles, without needing to assume invertibility of the map.

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arXiv2020

Computer Assisted Proofs of Attracting Invariant Tori for ODEs

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $C^0$ and neither star-shaped nor Lipschitz

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arXiv2018

Beyond the Melnikov method II: multidimensional setting

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are `small enough', as is the case in the classical Melnikov approach.

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arXiv2016

Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in [25]. This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an `outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an `inner dynamics', given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, [25] does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems. As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough). The code of this verifications is available. We hope that some full computer assisted proofs will be obtained in a near future since there are packages (CAPD) designed for problems of this type.

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arXiv2016

Validated Computation of Heteroclinic Sets

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of $\mathbb{R}^3$.

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arXiv2016

Computer assisted proof of Shil'nikov homoclinics: with application to the Lorenz-84 model

We present a methodology for computer assisted proofs of Shil'nikov homoclinic intersections. It is based on geometric bounds on the invariant manifolds using rate conditions, and on propagating the bounds by an interval arithmetic integrator. Our method ensures uniqueness of the parameter for which the homoclinic takes place. We apply the method for the Lorenz-84 atmospheric circulation model, obtaining a sharp bound for the parameter, and also for where the homoclinic intersection of the stable/unstable manifolds takes place.

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arXiv2016

Beyond the Melnikov method: a computer assisted approach

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are `small enough', as is the case in the classical Melnikov approach.

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arXiv2011

Lyapunov Orbits at L2 and Transversal Intersections of Invariant Manifolds in the Jupiter-Sun Planar Restricted Circular Three Body Problem

We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from L2 up to half the distance to the smaller primary in the Jupiter-Sun planar restricted circular three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

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arXiv2011

Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds

We present a topological proof of the existence of a normally hyperbolic invariant manifold for maps. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. But a non-rigorous, good enough, guess is necessary. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method for a driven logistic map, for which non-rigorous numerical simulation in plain double precision suggests the existence of a chaotic attractor. We prove that this numerical evidence is false and that the attractor is a normally hyperbolic invariant curve.

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arXiv2011

Transition Tori in the Planar Restricted Elliptic Three Body Problem

We consider the elliptic three body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L2, a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations.

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