4 An Introduction to Emergence Dynamics in Complex Systems
191
invariances such as invariant integrals and constants of motion, which are
responsible for low-dimensional motions. Another example is the reduction
problem in hydrodynamics, where the hydrodynamic collision invariance leads
to conservation laws and furthermore the deduction of fluid dynamics such as
Navior-Stokes equations and Euler equations.
(2) Distinct time/space scales
The slaving principle we discussed in this chapter is closely related to the separation of time scales of different degrees of freedom, which also provides a
scheme in picking up order parameters from many state variables. This reminds
us the emergence of order at the onset of phase transition in statistical physics,
at which the slowing down effect can be observed and only a few stable modes
become unstable and dominate the global behavior of the system.
Physically, the relaxation time scale is related to the correlation time scale, and the
spatial diffusion scale is related to the spatial correlation scale. A large distinction
time or space scale naturally leads to the separation of state variables. An impressive
example is the theoretical foundation of Brownian motion, and the Langevin equation
was proposed, which is a stochastic equation of motion including both deterministic
and random forces due to the time-scale separation of the relaxational time and
the rapid thermal fluctuation. Statistical dynamics of Brownian-related processes
distributed from physics to chemistry and biology becomes an important subject.
The above criteria for a reducible system in many cases are constructive. The
invariance and conservation related to symmetry can bring forth various ways
in accomplishing the reduction procedure in terms of transformation invariance,
invariant group, invariant manifold, and invariant subspace. Therefore the most
important mission is to seek for these invariant elements and symmetries. The slaving
principle can be facilitated in terms of adiabatic eliminations, time averaging, or the
central-manifold theorem. In statistical physics, one often applies the projection
operators to obtain the dynamics of a lower-dimensional distribution function.
In this Chapter we also applied the dimension-reduction schemes to study the
emergence of sustained oscillation in networks of excitable units and gene-regulatory
networks. Because collective rhythms are generated by the feedback mechanism, the
topological motifs should be an important source of this feedback. For excitable
networks, the feedback is organized by the loop structure (Winfree loop). One
can apply the DPAD scheme to determine the phase order of units and reveal the
embedded loops in a highly-complicated network. For gene-regulatory networks,
the feedback is formed by an appropriate match of the active and repressive regulations among genes. A computation of functional weights can be applied to evaluate
the dynamical contributions of topological networked links, where small functionalweight links can be eliminated by keeping dominant links. We should emphasize
that these two techniques can be well applied to practical systems, where only data
or time series are available to measure.
Synchronization of coupled oscillators is another vivid example of dynamical
emergence, which covers a large extent of different topics. The ordered emergence is
accompanied with a dimension reduction of microdynamics, so one may introduce
191
invariances such as invariant integrals and constants of motion, which are
responsible for low-dimensional motions. Another example is the reduction
problem in hydrodynamics, where the hydrodynamic collision invariance leads
to conservation laws and furthermore the deduction of fluid dynamics such as
Navior-Stokes equations and Euler equations.
(2) Distinct time/space scales
The slaving principle we discussed in this chapter is closely related to the separation of time scales of different degrees of freedom, which also provides a
scheme in picking up order parameters from many state variables. This reminds
us the emergence of order at the onset of phase transition in statistical physics,
at which the slowing down effect can be observed and only a few stable modes
become unstable and dominate the global behavior of the system.
Physically, the relaxation time scale is related to the correlation time scale, and the
spatial diffusion scale is related to the spatial correlation scale. A large distinction
time or space scale naturally leads to the separation of state variables. An impressive
example is the theoretical foundation of Brownian motion, and the Langevin equation
was proposed, which is a stochastic equation of motion including both deterministic
and random forces due to the time-scale separation of the relaxational time and
the rapid thermal fluctuation. Statistical dynamics of Brownian-related processes
distributed from physics to chemistry and biology becomes an important subject.
The above criteria for a reducible system in many cases are constructive. The
invariance and conservation related to symmetry can bring forth various ways
in accomplishing the reduction procedure in terms of transformation invariance,
invariant group, invariant manifold, and invariant subspace. Therefore the most
important mission is to seek for these invariant elements and symmetries. The slaving
principle can be facilitated in terms of adiabatic eliminations, time averaging, or the
central-manifold theorem. In statistical physics, one often applies the projection
operators to obtain the dynamics of a lower-dimensional distribution function.
In this Chapter we also applied the dimension-reduction schemes to study the
emergence of sustained oscillation in networks of excitable units and gene-regulatory
networks. Because collective rhythms are generated by the feedback mechanism, the
topological motifs should be an important source of this feedback. For excitable
networks, the feedback is organized by the loop structure (Winfree loop). One
can apply the DPAD scheme to determine the phase order of units and reveal the
embedded loops in a highly-complicated network. For gene-regulatory networks,
the feedback is formed by an appropriate match of the active and repressive regulations among genes. A computation of functional weights can be applied to evaluate
the dynamical contributions of topological networked links, where small functionalweight links can be eliminated by keeping dominant links. We should emphasize
that these two techniques can be well applied to practical systems, where only data
or time series are available to measure.
Synchronization of coupled oscillators is another vivid example of dynamical
emergence, which covers a large extent of different topics. The ordered emergence is
accompanied with a dimension reduction of microdynamics, so one may introduce