Unstable periodic orbits analysis in the Qi system

We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length. The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits, Lyapunov exponents, and Poincare maps. Based on several periodic orbits with different sizes and shapes, they are encoded systematically with two letters or four letters for two different sets of parameters. The periodic orbits outside the attractor with complex topology are discovered by accident. In addition, the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively. In this process, the rule of orbital period changing with parameters is also investigated. The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.