Resonance Zones and Lobe Volumes for Volume-Preserving Maps

We study exact, volume-preserving diffeomorphisms that have heteroclinic connections between a pair of normally hyperbolic invariant manifolds. We develop a general theory of lobes, showing that the lobe volume is given by an integral of a generating form over the primary intersection, a subset of the heteroclinic orbits. Our definition reproduces the classical action formula in the planar, twist map case. For perturbations from a heteroclinic connection, the lobe volume is shown to reduce, to lowest order, to a suitable integral of a Melnikov function.Comment: ams laTeX, 8 figure

Generic Twistless Bifurcations

We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area preserving map, there is generically a bifurcation that creates a ``twistless'' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created, and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.Comment: 29 pages, 9 figure

Integrability and Ergodicity of Classical Billiards in a Magnetic Field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a "bouncing map". We compute a general expression for the Jacobian matrix of this map, which allows to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.Comment: Explanations added in Section 5, Section 6 enlarged, small errors corrected; Large figures have been bitmapped; 40 pages LaTeX, 15 figures, uuencoded tar.gz. file. To appear in J. Stat. Phys. 8

Canonical Melnikov theory for diffeomorphisms

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure

Chaos and Semiclassical Limit in Quantum Cosmology

In this paper we present a Friedmann-Robertson-Walker cosmological model conformally coupled to a massive scalar field where the WKB approximation fails to reproduce the exact solution to the Wheeler-DeWitt equation for large Universes. The breakdown of the WKB approximation follows the same pattern than in semiclassical physics of chaotic systems, and it is associated to the development of small scale structure in the wave function. This result puts in doubt the ``WKB interpretation'' of Quantum Cosmology.Comment: 14 pages in LaTex (RevTex), 6 figure

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps

Molecular dynamics approach: from chaotic to statistical properties of compound nuclei

Statistical aspects of the dynamics of chaotic scattering in the classical model of $\alpha$-cluster nuclei are studied. It is found that the dynamics governed by hyperbolic instabilities which results in an exponential decay of the survival probability evolves to a limiting energy distribution whose density develops the Boltzmann form. The angular distribution of the corresponding decay products shows symmetry with respect to $\pi/2$ angle. Time estimated for the compound nucleus formation ranges within the order of $10^{-21}$s.Comment: 11 pages, LaTeX, non

The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: I. Basic Results

The problem of finding the exact energies and configurations for the Frenkel-Kontorova model consisting of particles in one dimension connected to their nearest-neighbors by springs and placed in a periodic potential consisting of segments from parabolas of identical (positive) curvature but arbitrary height and spacing, is reduced to that of minimizing a certain convex function defined on a finite simplex.Comment: 12 RevTeX pages, using AMS-Fonts (amssym.tex,amssym.def), 6 Postscript figures, accepted by Phys. Rev.

Visual Explorations of Dynamics: the Standard Map

The Macintosh application \textit{StdMap} allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program.Comment: Corrections in a couple of equations, and updated to the latest version of StdMa