Unstable Periodic Orbits Analysis in the Qi System


 In this paper, 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 Poincaré maps. Based on several periodic orbits found 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 pitchfork bifurcation, period-doubling bifurcation, periodic orbit bifurcation, saddle-node bifurcation of periodic orbits, and Hopf bifurcation in the Qi system are explored. 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.