Symmetry reduction

Systems with symmetry arise in many applications; examples relevant to control of these systems include rotating spiral waves, traveling pulses, and formations in animals.

We consider systems endowed with a continuous symmetry and use a template-based approach to reduce the equations to a frame in which the symmetry is factored out. Relative equilibria are steady states in the symmetry-reduced frame; an example is traveling waves in systems with periodic boundary conditions.

The control goal is to stabilize unstable relative equilibria and the control design is based on linearization of the reduced equations about these steady states. The key feature of the control design is that the controlled system retains the symmetry of the original system.

In the movies below, we demonstrate the technique using the example of a 1D Kuramoto-Sivashinsky equation with periodic boundary conditions. This system, which models chemical kinetics, exhibits rich dynamics, ranging from simple steady states to unstable traveling waves, and even chaos.

Movie blurb:

Solution of the 1D Kuramoto-Sivashinsky equation with periodic boundary conditions, with the initial condition being in the neighborhood of an unstable traveling wave. The solution gradually diverges away from the traveling wave to a heteroclinic orbit.

The left frame shows the dynamics of the original system, while the right frame shows the same solution in a "frozen" or symmetry/traveling-subtracted frame of reference. The red curve is the template that is used to freeze the traveling wave.

S. Ahuja, I. G. Kevrekidis and C. W. Rowley. Template-based stabilization of relative equilibria in systems with continuous symmetry. Journal of Nonlinear Science, 17(2): 109-143, 2007. (pdf)