Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

Collinear helium under periodic driving: stabilization of the asymmetric stretch orbit

The collinear eZe configuration of helium, with the electrons on opposite sides of the nucleus, is studied in the presence of an external electromagnetic (laser or microwave) field. We show that the classically unstable "asymmetric stretch" orbit, on which doubly excited intrashell states of helium with maximum interelectronic angle are anchored, can be stabilized by means of a resonant driving where the frequency of the electromagnetic field equals the frequency of Kepler-like oscillations along the orbit. A static magnetic field, oriented parallel to the oscillating electric field of the driving, can be used to enforce the stability of the configuration with respect to deviations from collinearity. Quantum Floquet calculations within a collinear model of the driven two-electron atom reveal the existence of nondispersive wave packets localized on the stabilized asymmetric stretch orbit, for double excitations corresponding to principal quantum numbers of the order of N > 10.Comment: 13 pages, 12 figure

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP