Collective Behaviors In Interacting Oscillator Systems And Interacting Particle Systems

This thesis is devoted to the spatiotemporal dynamics and collective behaviors in nonlinear systems. It is composed of two parts: continuous systems and discrete systems.In part one, we study the dynamics of interacting oscillating systems which is time continue and spatial discrete. Here the basic oscillator is the classical pendulum equation, or is called Josephson junction equation. Following subjects are studied step by step: systems of periodically driven oscillator; systems of coupled two oscillators; one dimensional chain of coupled multi-oscillators or is called Frenkel-Kontorova model. New nonlinear dynamics and collective behaviors are discovered.Applying the periodical driven on single oscillator, we have a system composed of two competitive frequencies. It will results kinds of synchronization states. In cases of over-damping and under-damping, the invariant curve of Poincarémap on phase cylinder is introduced and its properties are studied; the Arnold tongues in parameter space are used to character the bifurcation properties. In case of under-damping, the onset of chaos is studied.We study the systems composed of two oscillators that have natural frequency respectively. Under diffusive coupling and sinusoidal coupling, the transitions among different states are studied. The onset of chaos is discovered in sinusoidal coupled system.In sinusoidal coupled multi-oscillators chain with periodic boundary condition, a special solution will emerge. In this state, all particles in the chain rotate synchronically with very little driving force. We call it as the "super-rotating" state and reveal the underlying mechanics.In part two, the epidemic spreading in complex networks is studied. The Susceptible-infected-removed(SIR) epidemic model is used in this paper. Firstly we build a Markov chain for this model, and study the properties of this chain and its transition probability matrix. For complex cases, we just build macroscopic models, which are mean-field rate equations. The threshold conditions in these models are analyzed.For the SIR spreading model in networks, we build a time homogeneous Markov chain. Using the transition probability matrix, we prove that the chain is convergent. This method can be used to study the spread processes in some kinds of networks.The potential infectivity of each node is rarely identical in realistic cases, which means the probabilities of different contact processes are different. In networks, this means non-uniform transmission of edges. We study the effects of this factor on epidemic process.There are series theories about the epidemic spreading in networks with degree-degree correlations. Here we make some improvements to these theories. A new solution of these equations are presented; and a measurement of correlations based on connectivity matrix is defined.Our last work is the spreading model in directed networks. Here the threshold conditions are analyzed, and the problem of how to measurement the correlations in directed networks is discussed...