Relative rotation rates for driven dynamical systems

Abstract

Relative rotation rates for two-dimensional driven dynamical systems are defined with respect to arbitrary pairs of periodic orbits. These indices describe the average rate, per period, at which one orbit rotates around another. These quantities are topological invariants of the dynamical system, but contain more physical information than the standard topological invariants for knots, the linking and self-linking numbers, to which they are closely related. This definition can also be extended to include noisy periodic orbits and strange attractors. A table of the relative rotation rates for a dynamical system, its intertwining matrix, can be used to determine whether orbit pairs can undergo bifurcation and, if so, the order in which the bifurcations can occur. The relative rotation rates are easily computed and measured. They have been computed for a simple model, the laser with modulated parameter. By comparing these indices with those of a zero-torsion lift of a horseshoe return map, we have been able to determine that the dynamics of the laser are governed by the formation of a horseshoe. Additional stable periodic orbits, besides the principal subharmonics previously reported, are predicted by the dynamics. The two additional period-five attractors have been located with the aid of their logical sequence names, and their identification has been confirmed by computing their relative rotation rates.