An Introduction to Emergence Dynamics in Complex Systems

Emergence is one of the most essential features of complex systems. This property implies new collective behaviors due to the interaction and self-organization among elements in the system, which cannot be produced by a single unit. It is our task in this Chapter to extensively discuss the basic principle, the paradigm, and the methods of emergence in complex systems based on nonlinear dynamics and statistical physics. We develop the foundation and treatment of emergent processes of complex systems, and then exhibit the emergence dynamics by studying two typical phenomena. The first example is the emergence of collective sustained oscillation in networks of excitable elements and gene regulatory networks. We show the significance of network topology in leading to the collective oscillation. By using the dominant phase-advanced driving method and the function-weight approach, fundamental topologies responsible for generating sustained oscillations such as Winfree loops and motifs are revealed, and the oscillation core and the propagating paths are identified. In this case, the topology reduction is the key procedure in accomplishing the dimension-reduction description of a complex system. In the presence of multiple periodic motions, different rhythmic dynamics will compete and cooperate and eventually make coherent or synchronous motion. Microdynamics indicates a dimension reduction at the onset of synchronization. We will introduce statistical methods to explore the synchronization of complex systems as a non-equilibrium transition. We will give a detailed discussion of the Kuramoto self-consistency approach and the Ott-Antonsen ansatz. The synchronization dynamics of a star-networked coupled oscillators and give the analytical description of the transitions among various ordered macrostates. Finally, we summarize the paradigms of studies of the emergence and complex systems.