Bifurcation Analysis And Synchronization In Systems With Delay-coupled Oscillators

Coupled systems have been widely used to many nonlinear sciences such as neural science, laser, biological systems and climate systems. Investigating the spatio-temporal bifurcations, multiple stability or synchronization in coupled systems with different topo-logical structures has important theoretical significance and various applications. In this dissertation, we mainly consider systems consisting of coupled limit-cycle oscillators or Kuramoto phase oscillators. Each is divided by two situations:local couplings and non-local couplings are studied in limit-cycle oscillator systems; when investigating phase oscillators, two populations of oscillators, as well as an ensemble with distributed delays, are considered.With respect to locally coupled limit-cycle oscillators, we give some conditions which lead to synchronized, phase-locked and anti-phase oscillations, by using the sta-bility and bifurcation analysis. Some bifurcation sets are drawn in appropriate parameter spaces. More importantly, the effect of neutral feedbacks is discussed, which makes our results of more general.For a nonlocally coupled system, we consider a Dn equivariant case with distance-dependent couplings, discuss how the heterogeneity of couplings affects amplitude death region, and find the size of death region remains unchanged in case of small heterogene-ity. When the heterogeneity increases, the size decreases significantly. The boundary of amplitude death region is studied, where we find synchronized oscillations, quasiperiodic oscillations finally stabilizing to two different coexisted periodic oscillations, and several branches of phase-locked oscillations, etc.Infinite many globally coupled Kuramoto phase oscillators are investigated. The product of the mutual couplings between two populations of oscillators will affect the system dynamics. Choosing delay and this product as bifurcation parameters, bifurcation phenomena are studied and the conditions to guarantee partially synchronized states are given and proved. It is found both supercritical and subcritical bifurcations are possible as the coupling strength increases. Meanwhile, partially synchronized states are bifurcated.Finally, we consider the case that delays come from certain probability distribution in a globally coupled Kuramoto model. Some bifurcation sets are obtained when delay follows degenerate, uniform, Gamma and normal distribution, as well as distribution with positive minimal delay. With the aid of center manifold reduction method, we investigate the Hopf bifurcations, and obtain some results relative with the direction and stability of bifurcated solutions. The appearance of hysteresis loop is explained theoretically.