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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.

In this paper we report on the experimentally measured dynamics exhibited by a system comprised of two coupled circuits whose input–output relation follow the logistic-map function. The circuit takes in two external voltages that control the initial conditions, and we employ this capability to examine the phenomenon of symmetry breaking and to submit theoretical/numerical results on this dynamical system to experimental test. We demonstrate that symmetry-broken solutions manifest in this circuit for appropriately chosen initial conditions, and proceed to investigate experimentally the basins of attraction of these solutions, as well as their dependence on the coupling strength, ϵ. We illustrate the full power of this circuit by investigating the chaotic regime and by constructing experimental bifurcation diagrams. One intriguing phenomenon captured here involves the transition from synchronized chaos to decoherent chaos as the coupling is increased. Finally, we experimentally implement uni-directional coupling and explore the dynamics of a driven logistic map.

This paper focuses on a nonlinear equation from thin plate theory of the formWe obtain maximum principles for certain functions defined on the solution of this equation using P -functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles,

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