Revealing the true and pseudo-singularly degenerate heteroclinic cycles

Through revisiting the four-dimensional chaotic system discussed in Wang et al. (Dyn Syst Control 8:129, 2019), its hidden dynamical behaviors that were not reported previously can be uncovered, such as the distribution and local stability of equilibrium points, generic and degenerate pitchfork bifurcation, Hopf bifurcation, the coexistence of true and pseudo-singularly degenerate heteroclinic cycles, and the existence of globally exponentially attractive sets as well as heteroclinic orbits. The innovation of the paper lies in the following results: (1) Coexisting pseudo-singularly degenerate heteroclinic cycles (the solution approaching infinity with a short-duration transient of singularly degenerate heteroclinic cycles) and true cycles consisting of normally hyperbolic saddle foci (or saddle nodes) and stable node foci are numerically illustrated with nearby hyperchaotic attractors, verifying a kind of mechanism for formulating hyperchaos. (2) Globally exponentially attractive sets with different exponential rates are located with the aid of the Lyapunov function. (3) The existence of a pair of symmetrically heteroclinic orbits is rigorously proven by the Lyapunov function; and the definitions of both the a-limit set and x-limit set. Since these findings improve and complement the known results, we expect to provide a reference for real-world applications.

Automated summary

By revisiting a four-dimensional chaotic system, this paper uncovers hidden dynamical behaviors that were not previously reported, including the distribution and stability of equilibrium points, bifurcations, coexistence of different cycles, and the existence of attractive sets and heteroclinic orbits. The paper presents innovative results in the formulation of hyperchaos, the identification of attractive sets, and the proof of symmetric heteroclinic orbits. These findings improve and complement the known results and have implications for real-world applications.