arXiv subjects

Naiara V. de Paulo

Publications and source records attributed to Naiara V. de Paulo.

5 recordsLinked to original sources

arXiv2026

Transverse foliations for two-degree-of-freedom mechanical systems

We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-$2$ Lyapunov orbits in the neck region about the rest points and a particular index-$3$ orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above $1/6$. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction.

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arXiv2024

Genus zero transverse foliations for weakly convex Reeb flows on the tight $3$-sphere

A contact form on the tight $3$-sphere $(S^3,ξ_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,ξ_0)$ admitting a prescribed finite set of index-$2$ Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-$2$ orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index $3$. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-$2$ orbits are mutually non-coincident.

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arXiv2024

Reeb flows, pseudo-holomorphic curves and transverse foliations

Pseudo-holomorphic curves in symplectizations, as introduced by Hofer in 1993, and then developed by Hofer, Wysocki, and Zehnder, have brought new insights to Hamiltonian dynamics, providing new approaches to some classical questions in Celestial Mechanics. This short survey presents some recent developments in Reeb dynamics based on the theory of pseudo-holomorphic curves in symplectizations, focusing on transverse foliations near critical energy surfaces.

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arXiv2017

On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\mathbb{R}^4$

We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function $H:\mathbb{R}^4 \to \mathbb{R}$ contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset $S_0\subset H^{-1}(0)$. From previous work [de Paulo-Salomão, Memoirs of the AMS] we know that for any small energy $E>0$, the energy level $H^{-1}(E)$ contains a closed $3$-ball $S_E$ in a neighborhood of $S_0$ admitting a singular foliation called $2-3$ foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit $P_{2,E}$ contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of $S_E$ and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the $2-3$ foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to $P_{2,E}$ in $S_E$. Moreover, if the branches of the stable and unstable manifolds of $P_{2,E}$ inside $S_E$ do not coincide then the Hamiltonian flow on $S_E$ has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems.

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arXiv2016

Systems of transversal sections near critical energy levels of Hamiltonian systems in $\mathbb{R}^4$

In this article we study Hamiltonian flows associated to smooth functions $H:\mathbb{R}^4 \to \mathbb{R}$ restricted to energy levels close to critical levels. We assume the existence of a saddle-center equilibrium point $p_c$ in the zero energy level $H^{-1}(0)$. The Hamiltonian function near $p_c$ is assumed to satisfy Moser's normal form and $p_c$ is assumed to lie in a strictly convex singular subset $S_0$ of $H^{-1}(0)$. Then for all $E>0$ small, the energy level $H^{-1}(E)$ contains a subset $S_E$ near $S_0$, diffeomorphic to the closed $3$-ball, which admits a system of transversal sections $\mathcal{F}_E$, called a $2-3$ foliation. $\mathcal{F}_E$ is a singular foliation of $S_E$ and contains two periodic orbits $P_{2,E}\subset \partial S_E$ and $P_{3,E}\subset S_E\setminus \partial S_E$ as binding orbits. $P_{2,E}$ is the Lyapunoff orbit lying in the center manifold of $p_c$, has Conley-Zehnder index $2$ and spans two rigid planes in $\partial S_E$. $P_{3,E}$ has Conley-Zehnder index $3$ and spans a one parameter family of planes in $S_E \setminus \partial S_E$. A rigid cylinder connecting $P_{3,E}$ to $P_{2,E}$ completes $\mathcal{F}_E$. All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to $P_{2,E}$ in $S_E\setminus \partial S_E$ follows from this foliation.

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