Bifurcation analysis of hopping behavior in cellular pattern-forming systems

We use symmetry-based arguments to derive normal form equations for studying the temporal behavior of a particular spatio-temporal dynamic cellular pattern, called "hopping" state, which we have recently discovered in computer simulations of a generic example of an extended, deterministic, pattern-forming system in a circular domain. Hopping states are characterized by cellular structures that sequentially make abrupt changes in their angular positions while they rotate, collectively, about the center of the circular domain. A mode decomposition analysis suggests that these patterns are created from the interaction of three steady-state modes. A bifurcation analysis of associated normal form equations, which govern the time-evolution of the steady-state modes, helps us quantify the complexity of hopping patterns. Conditions for their existence and their stability are also derived from the bifurcation analysis. The overall ideas and methods are generic, so they can be readily applied to study other type of spatio-temporal pattern-forming dynamical systems with similar symmetry properties.