Analysis of Complex Systems

 

The temporal evolution of complex systems and self-organization can be well-understood on the basis of order parameters, whose temporal dynamics govern the behavior of the microscopic degrees of freedom of the complex system. The microscopic degrees of freedom usually irregularly vary on a much
faster time scales. Their back-reaction on the macroscopic order parameter
dynamics can be treated as fluctuations. As a consequence, the order parameters evolve in time as stochastic processes and the corresponding
order parameter equations are nonlinear Langevin equations. Close to nonequilibrium phase transitions the order parameter dynamics can be rigorously derived from the basic equations of the complex systems formulated for the microscopic degrees of freedom.

However, the mathematical scheme underlying processes of self-organization in systems far from equilibrium allows a top-down approach: Based on the identification of order parameters and the assessment of deterministic and stochastic ingredients of the order parameter dynamics it is possible to gain deep insight into the temporal, spatio-temporal, and functional organization underlying selforganization in complex systems.

In recent years we have developed methods which allows one to deal with the following issues:

• Identifcation of order parameters in selforganizing systems
• Estimation of  trends (drift) and characterizing the nature and strength of fluctuations of stochastic processes

The developed methods which allows one to disentangle deterministic dynamics, dynamical noise, and measurement noise has been successfully applied to  a variety of stochastic processes ranging from physics to biology and medicine.

A recent review will appear in the Springer Series of Complexity and System
Science (2008). This review can be found here.