Afternoon on
Hamiltonian Dynamical Systems
Wednesday 19 November 2014,
DynamIC, Room Huxley 130, Imperial College London.
13:00-13:50 William Giles (Imperial College) On homoclinic orbits to center manifolds in Hamiltonian systems
13:50-14:40 Amadeu Delshams (Universitat Politècnica de Catalunya) Global instability in the Elliptic Restricted Three Body Problem
14:40-15:10 refreshment break at 5th floor common room
15:10-16:00 James D. Meiss (University of Colorado) Thirty Years of Turnstiles and Transport
Abstracts:
On homoclinic orbits to center manifolds in Hamiltonian systems (William Giles, Imperial College)
Assuming the existence of a homoclinic orbit to a nonhyperbolic equilibrium, we investigate the existence of nearby orbits which are homoclinic to the center manifold. We consider firstly the case in which the nonhyperbolic eigenvalues at the equilibrium consist of pairs of nonzero purely imaginary eigenvalues. Taking an analytical approach, we use a Lyapunov-Schmidt technique to reduce the problem to that of studying the zero set of a real-valued function defined on the center manifold, which has a critical point at the origin. A formula is found for the Hessian matrix at this critical point, involving the so called scattering matrix. Under nonresonance and nondegeneracy conditions, we characterise the possible Morse indices of the Hessian, permitting an application of eg. the Morse lemma to describe the set of homoclinics.
If time permits, we then consider a more geometric approach to the problem, allowing us to define a nonlinear analogue of the scattering matrix using stable and unstable foliations of the invariant manifolds. We use this approach to unfold the system in parametrised families - we consider here as an example the case of a two dimensional center manifold corresponding to zero eigenvalues - bifurcation diagrams are produced for homoclinics to the origin in this case.
Global instability in the Elliptic Restricted Three Body Problem (Amadeu Delshams, UPC)
The (planar) ERTBP describes the motion of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this talk is to show that there exist trajectories of motion such that their angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive trajectories, that is, with a large variation of angular momentum.
The framework for proving this result consists on considering the motion close to the parabolic orbits of the Kepler problem between the comet and the Sun that takes place when the mass of Jupiter is zero. In other words, studying the so-called infinity manifold. Close to this manifold, it is possible to define a scattering map, which contains the map structure of the homoclinic trajectories to it. Since the inner dynamics inside the infinity manifold is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits, which eventually give rise to true trajectories of the system with the help of standard shadowing techniques.
This talk is based on a joint work with Vadim Kaloshin, Abraham de la Rosa and Tere M. Seara
Thirty Years of Turnstiles and Transport (J.D. Meiss, University of Colorado)
The strongest barriers to transport in area-preserving maps are the invariant Cantor sets that are remnants of the invariant circles predicted by KAM theory. These sets, named cantori by Percival, are fractal, but are typically hyperbolic and have zero dimension. Consequently it was a surprise to realize that they gave rise to impediments for the motion of chaotic orbits. Geometrically the flux across a cantorus can be localized to a region called “turnstile," since it resembled a rotating door. Mathematically, the flux is computed as a difference in action between two orbits, a quantity that also is prominent in Aubry-Mather theory. An early success of the theory was the explanation of Chirikov’s well-known power law for the transit time across a region with a newly destroyed circle.
This year, the turnstile is 30 years old. The turnstile has proved to be a fruitful concept in fields as diverse as atomic physics, oceanography, and celestial mechanics.. In this talk I will discuss some of the ideas that arose from this original research and some of the questions that are still unanswered.