Complex Behaviour in Coupled Oscillators, Coupled Map Lattices and Random Dynamical Systems.
Abstract
This thesis consists of three detailed studies on complex behaviour of deterministic and random
dynamical systems.
In the rst study, coupled nonlinear oscillator systems are proposed to model a future detection
scheme of axionic dark matter particles in a Josephson junction environment. By studying initial
value problems we observe rich phase space structures under variations of parameters such as the
coupling constant, the strength of an external magnetic eld and the initial conditions. In the
limit of small elongations, we obtain analytic solutions to the linearised equations of motion, with
and without dissipation, and prove a time-shifted synchronisation between the two oscillators, with
comparisons to the nonlinear case.
The second study focuses on distinguished correlation properties of a family of shifted Chebyshev
maps (TN;a). We present analytic results for two-point and higher-order correlation functions and
show that TN;0 are most random-like among all smooth one-dimensional maps conjugated to an Nary
shift, in the sense that they have least higher-order correlations. We discuss the eigenfunctions
of the Perron-Frobenius operator for TN;a. The spectrum is degenerate for odd N. We then consider
coupled map lattices of TN;a and numerically investigate zeros of the spatial and temporal nearestneighbour
correlations.
The third study concerns a random dynamical system that samples between a contracting and a
chaotic map with a certain probability p in time. We rst derive an explicit expression for the
invariant density. Due to long memory of history we consider a Markovian approximation for this
problem and study two-point correlation functions when varying the parameter p, with emphasis on
transitions between an exponential decay (at p = 1) and a power-law decay (when p ! 1=2). Finally
we work towards determining the type of di usion generated by sums of iterates of this random map.
Authors
Yan, JinCollections
- Theses [4125]