Symmetry breaking in symmetrically coupled logistic maps

Abstract

Generally speaking, symmetry breaking refers to the phenomenon where a system manifests a solution that does not exhibit a symmetry obeyed by that system. It has long been considered a fundamental mechanism for pattern formation, as well as a kind of precursor along the route to complexity within physical systems. Since the concept of symmetry breaking in its various forms is quite ubiquitous across physics, it is instructive to find a simple system that clearly illustrates the phenomenon. For this reason we examine a pair of symmetrically coupled logistic maps. While this discrete map is characterised by perfect parity symmetry, we show that time-series solutions can break this symmetry by selecting different steady-state values for the two individual map variables. We examine the symmetry-broken states that arise from the 2-, 4-, 8- and 6-cycle of the individual logistic map. We then show that these symmetry-broken states are subtly connected to phase-shifted cycles that exist in the uncoupled logistic map pair via a global invariance. This correspondence, finally, leads us to view the symmetry-broken states as arising due to a bifurcation parameterised by the coupling strength.