Synchronization is a kind of ordinary phenomenon in nature, the study of it includes many mathematical branches. Phase space is one of the most powerful inventions of modern mathematical science. There are two variables, the position and velocity, that can describe the 2-dimensional phase space system. For example, the state of pendulum may be specified by its position and its velocity, so its phase space is 2-dimensional. The state of the system at a given time has a unique corresponding point in the phase space. In order to describe the motion of an oscillator, we can talk about its motion in phase space. Self-sustained oscillators exhibit regular rhythms- they revisit the same points time after time. So the stable oscillation state of a self sustained oscillator can be expressed as some closed curve in phase space, and this closed curve is defined as a limit cycle.
There are two topics in this dissertation: Kuramoto model and FitzHugh-Nagumo (FHN) model. Kuramoto's original analysis of his model gives the critical synchronization value for K (K is the coupling constant ). He also gives an estimate for the value of order parameter r when K is close to critical point Kc. However when we give different initial values for the oscillators, the order parameters are different after a long time. The objective of the first topic is to give the distribution of the value of order parameter r under different initial conditions. We divide the oscillators to synchronized part and unsynchronized part, and find that the order parameter satisfies a Gaussian distribution.
For the second topic, we start with an introduction for oscillatory clusters in the Belousov-Zhabotinsky reaction. The main idea of this topic is to find the phase property of oscillators in the Oregonator and FHN type models with global inhibitory feedback. Numerical simulations suggest that, in many cases, the cubic system has the same phase value as the piecewise linear system. To simplify this model, we reduce the cubic FHN system to piecewise linear system.
In a network of two mutually-coupled neural oscillators, a spike time response curve (STRC) describes the period change of an oscillator given by a perturbation of another oscillator. The STRC is used to predict the phase relations of the two-cell network. We also create a spike time difference map that describes the evolution of the neuron's network based on the STRC.