In this thesis, we present a dynamical systems analysis of models of
movement coordination, namely the Haken-Kelso-Bunz (HKB) model
and the Jirsa-Kelso excitator (JKE).
The dynamical properties of the models that can describe various phenomena
in discrete and rhythmic movements have been explored in the
models' parameter space. The dynamics of amplitude-phase approximation
of the single HKB oscillator has been investigated. Furthermore, an
approximated version of the scaled JKE system has been proposed and
analysed.
The canard phenomena in the JKE system has been analysed. A combination
of slow-fast analysis, projection onto the Poincare sphere and
blow-up method has been suggested to explain the dynamical mechanisms
organising the canard cycles in JKE system, which have been
shown to have different properties comparing to the classical canards
known for the equivalent FitzHugh-Nagumo (FHN) model. Different
approaches to de fining the maximal canard periodic solution have been
presented and compared.
The model of two HKB oscillators coupled by a neurologically motivated
function, involving the effect of time-delay and weighted self- and
mutual-feedback, has been analysed. The periodic regimes of the model
have been shown to capture well the frequency-induced drop of oscillation
amplitude and loss of anti-phase stability that have been experimentally
observed in many rhythmic movements and by which the development
of the HKB model has been inspired. The model has also been demonstrated
to support a dynamic regime of stationary bistability with the
absence of periodic regimes that can be used to describe discrete movement
behaviours.This work was supported by The Higher Committee For Education Development in Iraq (HCED) and the University of Mosul
