4 An Introduction to Emergence Dynamics in Complex Systems
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of a system composed of these elements still cannot be simply predicted based on
individual properties. All these are called complex systems.
One can list some common properties of complex systems. When looked at in
detail, such as cooperation or self-organization, emergence, and adaption [5]. The
process of organized behavior arising without an internal or external controller or
leader in a system is called the self organization. Since simple rules produce complex
behavior in hard-to-predict ways, the macroscopic behavior of such systems is sometimes called the emergence. A complex system is then often defined as a system that
exhibits nontrivial emergent and self-organizing behaviors. The central question of
the sciences of complexity is how these emergent self-organized behaviors come
about [6].
Emergence is one of the most essential features of complex systems. In this
Chapter, we will discuss extensively 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 selfconsistency approach and the Ott-Antonsen ansatz. The synchronization dynamics of
a star-networked coupled oscillators gives the analytical description of the transitions
among various ordered macrostates.
We will summarize the paradigms of studies of the emergence and complex
systems based on the above discussions.
4.2 Emergence: Research Paradigms
Emergence implies the self-organized behavior in a complex system, which occurs
under the physically non-equilibrium condition. This collective feature comes from
the cooperation of elements through coupling, and it cannot be observed at the
135
of a system composed of these elements still cannot be simply predicted based on
individual properties. All these are called complex systems.
One can list some common properties of complex systems. When looked at in
detail, such as cooperation or self-organization, emergence, and adaption [5]. The
process of organized behavior arising without an internal or external controller or
leader in a system is called the self organization. Since simple rules produce complex
behavior in hard-to-predict ways, the macroscopic behavior of such systems is sometimes called the emergence. A complex system is then often defined as a system that
exhibits nontrivial emergent and self-organizing behaviors. The central question of
the sciences of complexity is how these emergent self-organized behaviors come
about [6].
Emergence is one of the most essential features of complex systems. In this
Chapter, we will discuss extensively 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 selfconsistency approach and the Ott-Antonsen ansatz. The synchronization dynamics of
a star-networked coupled oscillators gives the analytical description of the transitions
among various ordered macrostates.
We will summarize the paradigms of studies of the emergence and complex
systems based on the above discussions.
4.2 Emergence: Research Paradigms
Emergence implies the self-organized behavior in a complex system, which occurs
under the physically non-equilibrium condition. This collective feature comes from
the cooperation of elements through coupling, and it cannot be observed at the