Chapter 4
An Introduction to Emergence Dynamics
in Complex Systems
Zhigang Zheng
Abstract 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.
Keywords Emergence · Order parameter · Slaving principle · Self-sustained
oscillation · Winfree loop · Synchronization · Complex networks
Z. Zheng (B)
Institute of Systems Science and College of Information Science and Engineering,
Huaqiao University, Xiamen 361021, China
e-mail: zgzheng@hqu.edu.cn
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_4
133
An Introduction to Emergence Dynamics
in Complex Systems
Zhigang Zheng
Abstract 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.
Keywords Emergence · Order parameter · Slaving principle · Self-sustained
oscillation · Winfree loop · Synchronization · Complex networks
Z. Zheng (B)
Institute of Systems Science and College of Information Science and Engineering,
Huaqiao University, Xiamen 361021, China
e-mail: zgzheng@hqu.edu.cn
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_4
133