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Title: Growth of energy density for Klein-Gordon equation coupled to a chaotic oscillator
Author: Warner, Christopher
ISNI:       0000 0004 2737 0941
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2013
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A system consisting of a localised object (an oscillator) coupled to a Klein-Gordon field is considered, where the field is, initially, endowed with infinite energy. For the finite energy case, it is known that the oscillator must lose the energy due to coupling to the field, and the system eventually goes into the ground state. It is shown that if the oscillator is chaotic, then energy may be transferred from the wave field to the oscillator, and the particle can undergo an unbounded acceleration. The coupled system gives rise to a slow-fast system with delay term. By means of a reduction to an invariant manifold the problem is reduced to the study of a slow-fast system of ordinary differential equations. By choosing an appropriate potential function for the particle (confining, steep, and scattering), the oscillator can be made arbitrarily close to any scattering billiard. In the frozen system then, there exists a uniformly hyperbolic invariant set, a horseshoe, supported by a pair of hyperbolic periodic orbits connected by transverse heteroclinics, which also persists in the full system. A trajectory can then be constructed, switching between small neighbourhoods of these periodic orbits, such that, over a long-time period, the particle accelerates up to any predetermined finite level. The results give a first example of an extended Hamiltonian dynamical system with positively defined energy density for which local energy density can grow without bounds.
Supervisor: Turaev, Dimitry Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral