arXiv subjects

Vitor M. de Oliveira

Publications and source records attributed to Vitor M. de Oliveira.

5 recordsLinked to original sources

arXiv2023

Multistability and Gibbs entropy in the planar dissipative spin-orbit problem

In this work, we numerically investigate and visually illustrate the dynamical properties of the dissipative spin-orbit problem such as the co-existence of multiple periodic and quasi-periodic attractors, and the complexity of the corresponding basins of attraction. Our model is composed by a triaxial satellite (planet) orbiting a planet (star) in a fixed Keplerian orbit with zero obliquity. A dissipative tidal torque that is proportional to its rotational angular velocity is assumed to be acting on the satellite. We use Hyperion as a toy model to characterize the methodology used, since this system has a very rich conservative dynamical scenario, and we later apply our methodology to the Moon and Mercury. Our results show that the basins of attraction may possess an intricate structure in all cases which changes with the orbital eccentricity, and that the Gibbs entropy is a good measure on how dominant one basin is over the others in the phase space.

nlin.CD↗

arXiv2021

Measure, dimension, and complexity of the transient motion in Hamiltonian systems

Hamiltonian systems that are either open, leaking, or contain holes in the phase space possess solutions that eventually escape the system's domain. The motion described by such escape orbits before crossing the escape threshold can be understood as a transient behavior. In this work, we introduce a numerical method to visually illustrate and quantify the transient motion in Hamiltonian systems based on the transient measure, a finite-time version of the natural measure. We apply this method to two physical systems: the single-null divertor tokamak, described by a symplectic map; and the Earth-Moon system, as modeled by the planar circular restricted three-body problem. Our results portray how different locations for the ensemble of initial conditions may lead to different transient dynamical scenarios in both systems. We show that these scenarios can be properly quantified from a geometrical aspect, the transient correlation dimension, and a dynamical aspect, the transient complexity coefficient.

nlin.CD↗

arXiv2021

Dynamical trapping in the area-preserving Hénon map

Stickiness is a well known phenomenon in which chaotic orbits expend an expressive amount of time in specific regions of the chaotic sea. This phenomenon becomes important when dealing with area-preserving open systems because, in this case, it leads to a temporary trapping of orbits in certain regions of phase space. In this work, we propose that the different scenarios of dynamical trapping can be explained by analyzing the crossings between invariant manifolds. In order to corroborate this assertion, we use an adaptive refinement procedure to approximately obtain the sets of homoclinic and heteroclinic intersections for the area-preserving Hénon map, an archetype of open systems, for a generic parameter interval. We show that these sets have very different statistical properties when the system is highly influenced by dynamical trapping, whereas they present similar properties when stickiness is almost absent. We explain these different scenarios by taking into consideration various effects that occur simultaneously in the system, all of which are connected with the topology of the invariant manifolds.

nlin.CD↗

arXiv2020

Order-chaos-order and invariant manifolds in the bounded planar Earth-Moon system

In this work, we investigate the Earth-Moon system, as modeled by the planar circular restricted three-body problem, and relate its dynamical properties to the underlying structure associated with specific invariant manifolds. We consider a range of Jacobi constant values for which the neck around the Lagrangian point $L_1$ is always open but the orbits are bounded due to Hill stability. First, we show that the system displays three different dynamical scenarios in the neighborhood of the Moon: two mixed ones, with regular and chaotic orbits, and an almost entirely chaotic one in between. We then analyze the transitions between these scenarios using the Monodromy matrix theory and determine that they are given by two specific types of bifurcations. After that, we illustrate how the phase space configurations, particularly the shapes of stability regions and stickiness, are intrinsically related to the hyperbolic invariant manifolds of the Lyapunov orbits around $L_1$ and also to the ones of some particular unstable periodic orbits. Lastly, we define transit time in a manner that is useful to depict dynamical trapping and show that the traced geometrical structures are also connected to the transport properties of the system.

nlin.CD↗

arXiv2017

A map for heavy inertial particles in fluid flows

We introduce a map which reproduces qualitatively many fundamental properties of the dynamics of heavy particles in fluid flows. These include a uniform rate of decrease of volume in phase space, a slow-manifold effective dynamics when the single parameter $s$ (analogous of the Stokes number) approaches zero, the possibility of fold caustics in the "velocity field", and a minimum, as a function of $s$, of the Lyapunov (Kaplan-Yorke) dimension of the attractor where particles accumulate.

physics.flu-dyn↗