Effects of the eccentricity on the primames in the gravitational capture phenomenon

Gravitational capture is a characteristic of some dynamical systems in celestial mechanics, as in the elliptic restricted three-body problem that is considered in this paper. The basic idea is that a spacecraft (or any particle with negligible mass) can change a hyperbolic orbit with a small positive energy around a celestial body into an elliptic orbit with a small negative energy without the use of any propulsive system. The force responsible for this modification in the orbit of the spacecraft is the gravitational force of the third body involved in the dynamics. In this way, this force is used as a zero cost control, equivalent to a continuous thrust applied in the spacecraft. One of the most important applications of this property is the construction of trajectories to the Moon. The objective of the present paper is to study in some detail the effects of the eccentricity of the primaries in this maneuver.

3-Dimensional hopf bifurcation via averaging theory

We consider the Lorenz system ẋ = σ(y - x), ẏ = rx - y - xz and ż = -bz + xy; and the Rössler system ẋ = -(y + z), ẏ = x + ay and ż = b - cz + xz. Here, we study the Hopf bifurcation which takes place at q± = (±√br - b,±√br - b, r - 1), in the Lorenz case, and at s± = (c+√c2-4ab/2, -c+√c2-4ab/2a, c±√c2-4ab/2a) in the Rössler case. As usual this Hopf bifurcation is in the sense that an one-parameter family in ε of limit cycles bifurcates from the singular point when ε = 0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two 2-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other 3- or n-dimensional differential systems.

Path formulation for multiparameter D3-equivariant bifurcation problems

We implement a singularity theory approach, the path formulation, to classify D3-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a Ba-miniversal unfolding f0 of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of F0 onto its unfolding parameter space. We apply our results to degenerate bifurcation of period-3 subharmonics in reversible systems, in particular in the 1:1-resonance.

Some remarks on bifurcation analysis of a nonlinear vibrating system excited by a shape memory alloy material (SMA)

In this paper, the dynamical response of a coupled oscillator is investigated, taking in consideration the nonlinear behavior of a SMA spring coupling the two oscillators. Due to the nonlinear coupling terms, the system exhibits both regular and chaotic motions. The Poincaré sections for different sets of coupling parameters are verified. © 2011 World Scientific Publishing Company.

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set. © 2012 Elsevier Ltd.

Ciclos limites e a equação de van der Pol

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Estudo das propriedades dinâmicas e sincronização no mapa de rede de Zaslavsky

Synchronization in nonlinear dynamical systems, especially in chaotic systems, is field of research in several areas of knowledge, such as Mechanical Engineering and Electrical Engineering, Biology, Physics, among others. In simple terms, two systems are synchronized if after a certain time, they have similar behavior or occurring at the same time. The sound and image in a film is an example of this phenomenon in our daily lives. The studies of synchronization include studies of continuous dynamic systems, governed by differential equations or studies of discrete time dynamical systems, also called maps. Maps correspond, in general, discretizations of differential equations and are widely used to model physical systems, mainly due to its ease of computational. It is enough to make iterations from given initial conditions for knowing the trajectories of system. This completion of course work based on the study of the map called ”Zaslavksy Web Map”. The Zaslavksy Web Map is a result of the combination of the movements of a particle in a constant magnetic field and a wave electrostatic propagating perpendicular to the magnetic field. Apart from interest in the particularities of this map, there was objective the deepening of concepts of nonlinear dynamics, as equilibrium points, linear stability, stability non-linear, bifurcation and chaos

Involutions whose fixed set has three or four components: a small codimension phenomenon

Let T : M → M be a smooth involution on a closed smooth manifold and F = n j=0 F j the fixed point set of T, where F j denotes the union of those components of F having dimension j and thus n is the dimension of the component of F of largest dimension. In this paper we prove the following result, which characterizes a small codimension phenomenon: suppose that n ≥ 4 is even and F has one of the following forms: 1) F = F n ∪ F 3 ∪ F 2 ∪ {point}; 2) F = F n ∪ F 3 ∪ F 2 ; 3) F = F n ∪ F 3 ∪ {point}; or 4) F = F n ∪ F 3 . Also, suppose that the normal bundles of F n, F 3 and F 2 in M do not bound. If k denote the codimension of F n, then k ≤ 4. Further, we construct involutions showing that this bound is best possible in the cases 2) and 4), and in the cases 1) and 3) when n is of the form n = 4t, with t ≥ 1.

Hopf bifurcation in the full repressilator equations

In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation.

The Hopf bifurcation in the Shimizu-Morioka system

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Characterization of grazing bifurcation in airfoils with control surface freeplay nonlinearity

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

Flashover phenomenon: an analysis with influence of the thickness of the layer pollution of the high voltage polluted insulators

Results of the analysis of dynamic behavior of flashover phenomenon on the high voltage-polluted insulators are presented. These results were taken from a mathematical and an experimental model that introduce the variable thickness influence of the layer pollution deposited on the high-voltage insulator surface. Analysis of the flashover was done by way of introducing a variation in the thickness of the channel of Obenaus' model, simulating a layer pollution of variable thickness. The objective was to obtain a better reproduction of the real layer pollution deposited on the insulator that works in the polluted regions. Two types of thickness variations were used: a sudden variation, using a step; and a soft variation, using a ramp; that were put along the way of the discharge. Comparison between the mathematical and experimental models showed that introduction of a ramp makes Obenaus' model more efficient in analyzing behavior of flashover phenomenon.