Dynamics of Order Parameters

Order parameter are associated with macroscopic variables of a dynamical system whose dynamics can be described by bifurcation theory, i.e. they have control parameters with critical values around which the order parameter dynamics change qualitatively. They can have fixed points that may either be stable or unstable and can even exhibit multi-stability, i.e. multiple fixed points can exist in parallel. Their trajectories can be summarized in flow charts (i.e. phase portraits). In contrast to bifurcation theory, synergetics are concerned with the critical slowing down and fluctuations of the order parameter dynamics near a phase transition point. These are considered important indicators of self-organizing behavior of the system. Fluctuations can for example be externally induced and their effect on the order parameter dynamics is of interest to synergetics. 

While systems with a single order parameter can only be defined by stable and unstable fixed points, two-order-parameter systems can also show stable or unstable limit cycles. Three-order-parameter systems can in addition exhibit a so called torus. A torus can be thought of as a superpositition of two limit cycles with different periods, such that the system dynamics can be described in 3D space as a spiralling in one frequency around a limit cycle of another frequency. Drawing the trajectory of the systems behavior will lead to a donut-shaped form in 3D space. Another attractor that can appear in 3 or higher dimensional systems is a chaotic attractor. Since the order parameters are usually of small magnitude close to a phase transition of a system, their equations of motion can often be expressed by only a few leading powers. This may lead to physically quite different systems being expressed by very similar equations for their order parameters. The theory of normal forms goes a step further even and provides prototypical functional forms for the description of systems that can undergo certain bifurcations. By appropriate changes of variables, every system exhibiting a particular type of bifurcation can be brought into the respective normal form. This is of great value for the analysis of the system, since the control parameters and their critical values are known at this point. A theory of how to establish such normal forms is given by Thom's catastrophe theory.