Complex Nonchaotic Dynamics and Hidden Chaos in 3D Jerk System Without Equilibrium or Infinitely Many Equilibria

Abstract

This paper systematically analyzes complex dynamics of a 3D quadratic Jerk system without equilibrium or with infinitely many equilibria, focusing on the generation mechanisms of nonchaotic behaviors and hidden chaos. First, Lyapunov stability of a line of nonhyperbolic equilibria is analyzed. Further, the concept of Jacobi stability is extended to the case where the deviation curvature tensor has a zero eigenvalue, thereby obtaining that a line of nonhyperbolic equilibria is Jacobi unstable for certain parameters. Second, we analyze three special cases: (i) a linear Jerk system for which the induced flow is proved to be weakly Li–Yorke chaos in weak topology and which has infinitely many periodic orbits; (ii) a completely integrable Jerk system, which has a homoclinic orbit and a family of periodic orbits; (iii) a simple nonlinear Jerk system, for which we prove that there exist infinitely many singularly degenerate heteroclinic cycles when invariant algebraic surfaces vanish. Third, we analyze the global dynamics on the invariant algebraic surfaces of Jerk system, covering IMSDHCs, Jacobi stability, and dynamics at infinity. Fourth, we use the averaging theory to prove that a hidden periodic orbit bifurcates from a nonisolated zero-Hopf equilibrium at the origin and nonorigin equilibrium, and establish, based on KAM theorem, the existence of hidden nested invariant tori surrounding this periodic orbit. Meanwhile, there are island chains and hidden chaos near nested invariant tori. Finally, Poincaré map shows that hidden chaos is generated via the Feigenbaum period-doubling route, and its existence is verified based on the topological horseshoe theory with a computer-assisted proof.