A Study Of Several Complex & Nonlinear Dynamical Problems In Coupled Dynamical Systems

Recently, with the development in nonlinear science, and the rising of interest in complexity science around the world, collective behaviors of coupled dynamic system have been the focus of the academic field. Because coupled dynamical system can be regarded as the combination of many dynamical units via certain means, it is a kind of complex systems. And the complexity of coupled dynamical system is mainly embodied in structure complexity (such as complex topology, spatial randomness, etc.) and process complexity (such as internal fluctuation, environment fluctuation, etc.). It is because of these complex factors that the whole system exhibits fruitful dynamic behaviors (chaos, stochastic resonance, etc.) and there by accomplishing complicated functions. However, there is not yet a universal agreement on how the complexity and nonlinearity interact with each other, and thereby regulating the collective behaviors. Thus, the effects of noise and topology on synchronization and coherence in neural system and chaotic chemical reaction system are investigated in this dissertation. The results are believed to make contributions for getting a better understanding in dynamic mechanism in coupled complex systems.Noise (internal fluctuation or environment fluctuation) could play rather counterintuitive positive roles in certain circumstances, currently is a common idea in non-equilibrium statistical & nonlinear dynamical societies. Stochastic resonance (SR) is the most famous phenomenon, and it refers to the response of the system to a periodic force, which may reach maximal order at an optimal noise level. And there are still several other examples like molecular motor (brownian rachet), coherent resonance (CR), noise induced phase transition, and noise enhanced synchronization etc., reminding us that new phenomena are being discovered. SR still has another extending form, the System-Size Resonance (SSR). If the system is composed by coupled units, it has already been known that the effective noise intensity on the mean field of the system is actually modulated by the number of these units. And, for a single unit in the system, internal noise level is closely related with its volume or area. Therefore, the response of the mean field of the system is maximally ordered at an optimal system size (the number of the elements in coupled system, volume or area of a single element, etc.). Such phenomenon is termed as SSR. These noisy processes are of great importance in understanding meso-scopic dynamics, statistics.Nearly any complex system in nature can be described by networks, coupled system is neither an exception. Complex network has received much attention due to its fundamental importance in physics, chemistry and life science, especially in neuroscience. So far, studies on complex networks can be divided into two main categories: one is studying the topological properties of complex networks and various mechanisms to determine the topology; the other is studying how their complex connectivity can affect the dynamic features of the system. Previous researches show that complex connectivity plays a crucial role in the dynamics of the system.In the first part of dissertation, we investigate the dynamics of coupled Hodgkin-Huxley neurons which are subjected to external noise or internal noise. System-Size Resonance and Double System-Size Resonance have been founded: there are an optimal number of coupled neurons with an optimal noise intensity at which the system's collective behavior has maximal order. Then, we investigate the effects of complex networks' topology structure on the neurons' synchronization and coherence in the second part of this dissertation. It is found that the synchronization of the coupled neurons can be enhanced by increasing the random links' number. Moreover, we find that there is an optimal number of random links at which the neurons' motion has maximal order, which means appropriate number of random links can tame spatiotemporal chaos of chaotic neural networks and induce coherent resonance in excitable neural networks. Next the effect of inherent randomness in coupled chemical chaotic oscillators on phase synchronization is studied in the third part of this dissertation. The results show that internal noise can enhance the phase synchronization and there is an optimal internal noise level at which the best phase synchronization can be achieved.