Alternative paths to Earth-Moon transfer

The planar, circular, restricted three-body problem predicts the existence of periodic orbits around the Lagrangian equilibrium point L1. Considering the Earth-lunar-probe system, some of these orbits pass very close to the surfaces of the Earth and the Moon. These characteristics make it possible for these orbits, in spite of their instability, to be used in transfer maneuvers between Earth and lunar parking orbits. The main goal of this paper is to explore this scenario, adopting a more complex and realistic dynamical system, the four-body problem Sun-Earth-Moon-probe. We defined and investigated a set of paths, derived from the orbits around L1, which are capable of achieving transfer between low-altitude Earth (LEO) and lunar orbits, including high-inclination lunar orbits, at a low cost and with flight time between 13 and 15 days.

Resonance and chaos .2. Exterior resonances and asymmetric libration

The motion of a test particle in the vicinity of exterior resonances is examined in the context of the planar, circular, restricted three-body problem. The existence of asymmetric periodic orbits associated with the 1 : n resonances (where n = 2, 3, 4, 5) is confirmed; there is also evidence of asymmetric resonances associated with larger values of n. A detailed examination of the evolution of the family of orbits associated with the 1:2 resonance shows the sequence that leads to asymmetric libration. on the basis of numerical studies of the phase space it is concluded that the existence of asymmetric libration means that the region exterior to the perturbing mass is more chaotic than the interior region. The apparent absence of 'particles' in 1 : n resonances in the solar system may reflect this inherent bias.

Three-body problem, its Lagrangian points and how to exploit them using an alternative transfer to L4 and L5

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the limit cycles of a class of piecewise linear differential systems in R-4 with two zones

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Integrable approximation to the overlap of resonances

We study the interaction of resonances with the same order in families of integrable Hamiltonian systems. This can occur when the unperturbed Hamiltonian is at least cubic in the actions. An integrable perturbation coupling the action-angle variables leads to the disappearance of an island through the coalescence of stable and unstable periodic orbits and originates a complex orbit plus an isolated cubic resonance. The chaotic layer that appears when a general term is added to the Hamiltonian survives even after the disappearance of the unstable periodic orbit. © 1992.

Resonance and chaos: II. Exterior resonances and asymmetric libration

The motion of a test particle in the vicinity of exterior resonances is examined in the context of the planar, circular, restricted three-body problem. The existence of asymmetric periodic orbits associated with the 1 : n resonances (where n = 2, 3, 4, 5) is confirmed; there is also evidence of asymmetric resonances associated with larger values of n. A detailed examination of the evolution of the family of orbits associated with the 1:2 resonance shows the sequence that leads to asymmetric libration. On the basis of numerical studies of the phase space it is concluded that the existence of asymmetric libration means that the region exterior to the perturbing mass is more chaotic than the interior region. The apparent absence of 'particles' in 1 : n resonances in the solar system may reflect this inherent bias.

Interplanetary natural routes via Moon and Lagrangiam points L1 and L2

Swing-by techniques are extensively used in interplanetary missions to minimize fuel consumption and to raise payloads of spaceships. The effectiveness of this type of maneuver has been proven since the beginning of space exploration. According to this premise, we have explored the existence of a natural and direct links between low Earth orbits and the lunar sphere of influence, to obtain low-energy interplanetary trajectories through swing-bys with the Moon and the Earth. The existence of these links are related to a family of retrograde periodic orbits around the Lagrangian equilibrium point L1 predicted for the circular, planar, restricted three-body Earth-Moon-particle problem. The trajectories in these links are sensitive to small disturbances. This enables them to be conveniently diverted reducing so the cost of the swing-by maneuver. These maneuvers allow us a gain in energy sufficient for the trajectories to escape from the Earth-Moon system and to stabilize in heliocentric orbits between the Earth and Venus or Earth and Mars. On the other hand, still within the Earth sphere of influence, and taking advantage of the sensitivity of the trajectories, is possible to design other swing-bys with the Earth or Moon. This allows the trajectories to have larger reach, until they can reach the orbit of other planets as Venus and Mars.(3σ)Broucke, R.A., Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses, JPL Technical Report 32-1168, 1968.

Swarm control designs applied to a micro-electro-mechanical gyroscope system (MEMS)

This paper analyzes the non-linear dynamics of a MEMS Gyroscope system, modeled with a proof mass constrained to move in a plane with two resonant modes, which are nominally orthogonal. The two modes are ideally coupled only by the rotation of the gyro about the plane's normal vector. We demonstrated that this model has an unstable behavior. Control problems consist of attempts to stabilize a system to an equilibrium point, a periodic orbit, or more general, about a given reference trajectory. We also developed a particle swarm optimization technique for reducing the oscillatory movement of the nonlinear system to a periodic orbit. © 2010 Springer-Verlag.

Stable regions around Pluto

In a previous work, GiuliattiWinter et al. found several stable regions for test particles in orbit around Pluto associated with families of periodic orbits obtained in the circular, restricted three-body problem. They have shown that a possible eccentricity of the Pluto-Charon binary slightly reduces but does not destroy any of these stable regions. In thiswork, we extended their results by analysing the cases with the orbital inclination (I) equal to zero and considering the argument of pericentre (w) equal to 90°, 180° and 270°. We explore the influence of the orbital inclination of the particles in these stable regions. In this case, the initial inclination varies from 10° to 170° in steps of 10°. We also present a sample of results for the longitude of the ascending node Ω = 90°, considering the cases I = 20°, 50°, 130° and 180°. Our results show that stable regions are present in all of the inclined cases, except when the initial inclination of the particles is equal to 110°. A sample of 3D trajectories of quasi-periodic orbits were found related to the periodic orbits obtained in the planar case by Giuliatti Winter et al. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.

Scaling invariance of the diffusion coefficient in a family of two-dimensional Hamiltonian mappings

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island. © 2013 American Physical Society.

Periodicity in a delayed version of the kaldor trade cycle model

We are concerned with the Kaldor's trade cycle model under the effect of a delay which represents a gestation lag between a decision of investment and its effect on the capital stock. Taking the adjustment coefficient in the goods market as a bifurcation parameter, we achieve global branches of periodic solutions. In our setting the delay is a constant inherent to the specific economy. Copyright © 2013 Watam Press.

Ciclos limites de sistemas lineares por partes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Estudo de sistemas lineares por parte com três zonas e aplicação na análise de um circuito elétrico envolvendo um memristor

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A peculiar stable region around Pluto

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)