We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing an universe filled with a scalar field which possesses three parameters. The main results are the following. First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system ha...

In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k, 2, 2, 2). A 5-dimensional contact SO(3)-manifol...

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a ...

The orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential to the study of symmetric k-varieties and their representations. This paper gives an algorithm to compute these orbits and most of the combinatorial structure of the orbit decomposition. There are several ways to describe these orbits, see for example [22, 28, 35]. Fundamental in all these descriptions ar...

Periodic orbits are studied using generating functions. We develop necessary and sufficient conditions for existence of periodic orbits of a given period or going through a given point in space. These conditions reduce the search to either solving a set of implicit equations, which can often be done graphically, or finding the roots of an equation of one variable only, independent of the Hamilt...

We discuss the influence of periodic orbits on the dissociation of a model diatomic molecule driven by a strong bichromatic laser fields. Through the stability of periodic orbits, we analyze the dissociation probability when parameters, such as the two amplitudes and the phase lag between the laser fields, are varied. We find that qualitative features of dissociation can be reproduced by consid...

We consider a dynamical system that exhibits transition chains of invariant tori alternating with Birkhoff zones of instability in a 2-dimensional center manifold. It is known that there exist orbits that shadow the transition chains. It is also known that there exist orbits that cross the Birkhoff zones of instability. We describe a topological mechanism that allows one to join together the tw...

An important step in the quest for low-cost fusion power is the ability to perform and analyze experiments in prototype fusion reactors. One of the tasks in the analysis of experimental data is the classification of orbits in Poincaré plots. These plots are generated by the particles in a fusion reactor as they move within the toroidal device. In this paper, we describe the use of graph-based m...

Abs t rac t , Sufficient conditions are found so that a family of smooth Hamiltonian flows limits to a billiard flow as a parameter e --~ 0. This limit is proved to be C 1 near non-singular orbits and C o near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby ...

Periodic orbits of nonlinear mappings play a central role in the study of dynamical systems. Traditional root finding algorithms, such as the Newton–family algorithms, have been widely applied for the detection of periodic orbits. However, in the case of discontinuous/nondifferentiable mappings and mappings with poorly behaved partial derivatives, this approach is not valid. In such cases, stoc...