Growth of energy density for Klein-Gordon equation coupled to a chaotic oscillator
Author(s)
Warner, Christopher
Type
Thesis
Abstract
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.
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.
Date Issued
2013
Date Awarded
2013-03
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Turaev, Dimitry
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)