arXiv subjects

Donato Scarcella

Publications and source records attributed to Donato Scarcella.

8 recordsLinked to original sources

arXiv2026

Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases

Many physical phenomena are naturally modeled by dynamical systems subject to non-autonomous perturbations that decay in time, including the interaction of a laser pulse with a molecule, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In the present paper, we consider time-dependent perturbations decaying in time of Hamiltonian systems having a lower-dimensional isotropic normally elliptic (resp. hyperbolic) invariant torus with quasiperiodic solutions. Under suitable rates of decay in time of the perturbation, we prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for H\"older, smooth, and analytic Hamiltonian systems.

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arXiv2026

Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case

Dynamical systems subject to non-autonomous perturbations decaying in time arise naturally in many physical contexts, including laser-molecule interactions, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In this paper, we consider time-dependent perturbations decaying polynomially fast in time of Hamiltonian systems having a lower-dimensional isotropic normally parabolic invariant torus supporting quasiperiodic solutions. We prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for H\"older, smooth, and analytic Hamiltonian systems.

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arXiv2026

Degenerate fixed points of maps in Banach spaces and lattices with decay and their invariant manifolds

Degenerate fixed points of maps appear in many interesting problems in Celestial Mechanics, Economics and Chemistry. Lattice systems, that is, dynamical systems consisting in an infinite array of finite dimensional subsystems interacting locally among them appear in many models of Biology, Physics and Mathematics. In this work, we extend the results concerning the existence of stable and unstable invariant manifolds of degenerate fixed points to lattice systems with decay properties. As an example, we find such manifolds in perturbations of the Toda lattice.

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arXiv2025

Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio

In this work, we prove that a generic unfolding of an analytic Hamiltonian Hopf singularity (in an open set with codimension 1 boundary) possesses transverse homoclinic orbits for subcritical values of the parameter close to the bifurcation parameter. As a consequence, these systems display chaotic dynamics with arbitrarily large topological entropy. We verify that the Hamiltonian of the restricted planar circular three-body problem (RPC3BP) close to the Lagrangian point $L_4$ falls within this open set. The generic condition ensuring the presence of transversal homoclinic intersections is subtle and involves the so-called Stokes constant. Thus, in the case of the RPC3BP close to $L_4$, our result holds conditionally on the value of this constant.

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arXiv2023

Biasymptotically quasiperiodic solutions for time-dependent Hamiltonians

In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian vector field having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of an asymptotic KAM torus. An asymptotic KAM torus is a time-dependent family of embedded tori converging as time tends to infinity to the invariant torus associated with the unperturbed system. Now, it is quite natural to wonder when we have the existence of a biasymptotic KAM torus. That is a continuous time-dependent family of embedded tori converging in the future and the past to suitable quasiperiodic invariant tori. In this work, we go one step further. We analyze time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast in time, we prove the existence of orbit converging to some quasiperiodic solutions in the future and the past.

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arXiv2023

Asymptotic motions converging to arbitrary dynamics for time-dependent Hamiltonians

In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of an asymptotic KAM torus. That is a time-dependent family of embedded tori converging as time tends to infinity to the quasiperiodic invariant torus of the unperturbed system. In this paper, the dynamic on the invariant torus associated with the unperturbed Hamiltonian is arbitrary. Therefore, we need to assume exponential decay in time in order to prove the existence of a time-dependent family of embedded tori converging in time to the invariant torus associated with the unperturbed system. The proof relies on the implicit function theorem, and the most complicated and original part rests on the solution of the associated linearized problem.

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arXiv2022

Asymptotically quasiperiodic solutions for time-dependent Hamiltonians

In 2015, M. Canadell and R. de la Llave consider a time-dependent perturbation of a vector field having an invariant torus supporting quasiperiodic solutions. Under a smallness assumption on the perturbation and assuming the perturbation decays (when t goes to infinity) exponentially fast in time, they proved the existence of motions converging in time (when t goes to infinity) to quasiperiodic solutions associated with the unperturbed system (asymptotically quasiperiodic solutions). In this paper, we generalize this result in the particular case of time-dependent Hamiltonian systems. The exponential decay in time is relaxed (due to the geometrical properties of Hamiltonian systems) and the smallness assumption on the perturbation is removed.

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arXiv2022

Weakly asymptotically quasiperiodic solutions for time-dependent Hamiltonians with a view to celestial mechanics

We consider the planar three-body problem perturbed by a celestial body modeled as a time-dependent perturbation that decays in time. We assume that the motion of the celestial body is given and is unbounded with a non-zero asymptotic velocity. We prove the existence of orbits converging in time to some motions that are ``close'' to the quasiperiodic solutions associated with the Hamiltonian of the planar three-body problem. The proof relies on an abstract theorem that contains a substantial portion of the mathematical complexities presented in this work. This theorem is flexible and can be applied to many other physical phenomena. It considers Hamiltonian vector fields that are the sum of two components. The first possesses quasiperiodic solutions, and the second decays polynomially fast as time tends to infinity. We prove the existence of orbits converging in time to some motions that are ``close'' to the quasiperiodic solutions associated with the unperturbed system. It generalizes a previous work where a stronger polynomial decay in time was considered, and solutions converging in time to the quasiperiodic orbits associated with the unperturbed system were proved. In the abstract theorem contained in the present paper, the too-weak decay in time of the perturbation strongly modifies the dynamic at infinity. This serious difficulty requires a deep modification of the proof. This new strategy relies on the application of a Nash-Moser implicit function theorem (the previous result was proved with the fixed point theorem) and the introduction of weak solutions (in this case, the orbits do not converge to the quasiperiodic solutions associated with the unperturbed system).

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