We follow the quasi-Penning resonances of the hydrogen atom in crossed external fields to high energies and field strengths observing structural changes and uncovering their bifurcation behavior. The appearance of a stability top and a cusp bifurcation are reported. Another orbit B2 of the same structure is found in the observed parameter space becoming completely stable in a region where the c...

Practical techniques for designing transfer trajectories between Libration Point Orbits (LPOs) are presented. Motivation for development of these techniques was provided by a hardware contingency experienced by the Solar Heliospheric Observatory (SOHO), a joint mission of the European Space Agency (ESA) and the National Aeronautics and Space Administration (NASA) orbiting the L1 point of the Su...

In this work we study the dynamics around an elliptic periodic orbit of Hamiltonian systems To this end we have developped an algorithm to compute a normal form up to a nite order around this orbit that gives an accurate description of the dynamics close to it If the remainder of this normal form can be bounded it is not di cult to produce explicit bounds of the di usion time of trajectories st...

Gutzwiller's trace formula allows interpreting the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations, and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex " ghost orbits ". By stu...

We present a novel numerical method to calculate periodic orbits for dynamical systems by an iterative process which is based directly on the action integral in classical mechanics. New solutions are obtained for the planar motion of three equal mass particles on a common periodic orbit with finite total angular momentum, under the action of attractive pairwise forces of the form 1/rp+1. It is ...

Given a Tonelli Hamiltonian H : T ∗M → R of class C, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class C in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then, there exists a potential V : M → R of class Ck−1, small in C topology, for which the Aubry set of the new Hamiltonian H + V is e...

The behaviour of resonances in the spin–orbit coupling in Celestial Mechanics is investigated. We introduce a Hamiltonian nearly–integrable model describing an approximation of the spin–orbit interaction. A parametric representation of periodic orbits is presented. We provide explicit formulae to compute the Taylor series expansion in the perturbing parameter of the function describing this par...

Gutzwiller's trace formula allows interpretation of the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex ``ghost orbits''. By...

There are two possibilities for the orbits: • If the orbit Oφ(x) is finite, we say that x is a preperiodic point. • If the orbit Oφ(x) is infinite, we say that x is a wandering point. A important subset of the preperiodic points consists of those points whose orbit eventually return to its starting point. These are called periodic points. Example. Consider the polynomial φ(z) = z − 1 as a map Z...

An algebraic flip system is an action of the infinite dihedral group by automorphisms of a compact abelian group X. In this paper, a fundamental structure theorem is established for irreducible algebraic flip sytems, that is, systems for which the only closed invariant subgroups of X are finite. Using irreducible systems as a foundation, for expansive algebraic flip systems, periodic point coun...