Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics

Periodic orbits play important roles in the analysis of dynamic behaviors in chaotic systems, and they are fundamental keys to understanding the properties of the strange attractor. In this paper, the unstable periodic orbits of a three-dimensional (3D) autonomous Lorenz-type system, the so-called generalized Lorenz-type system (GLTS), are investigated using the variational method. Taking the parameters of the GLTS as (a,b,c)=(10,100,10.4), we use two cycles as basic building blocks to establish 1D symbolic dynamics, and all short unstable cycles up to topological length 5 are found. With typical parameters (a,b,c)=(10,42.72,1), the correlation between the Burke-Shaw system (BSS) and the GLTS is discussed, symbolic dynamics for four letters are identified, and the periodic orbits are labeled inside or outside of the attractor based on whether the symbol sequence of the cycle contains the building blocks with the self-linking number 1. The variational method verifies the effectiveness of locating the periodic orbits, and topological classification provides a novel way to build appropriate symbolic dynamics. We also utilize the homotopy evolution approach to continuously deform the found periodic orbits while varying different parameters, and analyze the various bifurcations in the GLTS. The current work further expands the approach of establishing symbolic dynamics while analyzing the periodic orbits in chaotic systems, and can be applied to the study of turbulence and other complex problems.