Recognition and analysis of symmetric dynamics and hidden dynamics of a three-dimensional global coupling discrete chaotic system based on the coordinate rotation transformation

Abstract:

A three-dimensional chaotic map is created with global coupling by substituting the mean field with standard 1-D Logistic map. The complex dynamics and hidden dynamics of coupled system are analyzed and identified by coordinate rotation transformation. The attractors and the bifurcation structure together with the coexistent attraction domain of the coupled system, are shown in respective diagram before and after utilizing the coordinate rotation transformation. The comparison among them shows that the projection of the attractors and the coexisting basin of attraction of the original coupled system on the rotated plane are of triangular symmetry. Co-dimension 1 and 2 bifurcation lines are superimposed on the two-parameter steady-state diagrams. It provides a way to recognize possible coexisting attractors in parameter domains. Both two-parameter steady state distribution diagram and bifurcation lines are independent of coordinate rotation transformation. However, their computation speed would be greatly accelerated with the help of orthogonal transformation.