Hidden attractors, singularly degenerate heteroclinic orbits, multistability and physical realization of a new 6D hyperchaotic system

Abstract

This paper reports a sequential design of linearly controlling a three-dimensional (3D) quadratic system to a simple six-dimensional hyperchaotic system with complex dynamics. By adding three linear dynamical controllers, the resulting 6D system has no equilibrium and a hidden attractor, which has four positive Lyapunov exponents (LEs). This paper focuses on the 6D system, to reveal its unusual dynamics such as infinitely many singularly degenerate heteroclinic cycles and bifurcations from such singular orbits to hidden hyperchaotic attractors. Detailed numerical investigations are carried out, including bifurcation diagram, LE spectrum and phase portrait. Furthermore, the system has multistability corresponding to three types of equilibria, including no equilibrium and infinite non-isolated equilibria. In particular, we find that at least seven different attractors coexist when the system has one equilibrium line. Finally, this 6D hyperchaotic system is verified by 0–1 test and a circuit.