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.

An experimental survey of chaos and symmetry breaking in coupled and driven logistic maps

“…We use this system first to verify that the symmetry-broken states do manifest and can be reliably observed in this circuit, and to map out their basins of attraction as a function of coupling strength. In other words, we begin by experimentally verifying predictions from a recent paper [10]. The agreement between theory and experimental measurement is very good.…”

Flip and Neimark–Sacker Bifurcations in a Coupled Logistic Map System

“…For larger r-values, beyond those explored in [8], another bifurcation can be seen at even larger coupling values.…”

“…roughout this work, we consider only (x * , y * ) and not (y * , x * ), its flipped counterpart. Following [8], we determine conditions for a fixed point to be classified as a hyperbolic or nonhyperbolic fixed point, and to determine the stability type of hyperbolic fixed points, we compute the Jacobian of our map F:…”

“…Recent years, however, have seen a renewed interest in the dynamics of coupled logistic maps. At least two developments have spurred this re-examination: (a) the realization that discrete coupled maps could be usefully exploited in digital encryption schemes [5][6][7] and (b) success with their experimental implementation using electronic circuits [8][9][10]. Both of these recent threads have revealed intricate and nonintuitive behavior of these coupled maps.…”

“…Both of these recent threads have revealed intricate and non-intuitive behavior of these coupled maps. One such behaviorspontaneous symmetry breaking -was recently highlighted and explored [7]. That reference, however, did not attempt to analyze the chaotic regime in this coupled system (i.e., for large values of r), focusing primarily on symmetry breaking and its basin of attraction pertaining to n-cycles.…”