Knotted Periodic Orbits and Chaotic Behavior of a Class of Three-Dimensional Flows

Abstract

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

Bifurcations of Invariant Torus and Knotted Periodic Orbits in Generalized Hopf-Langford Type Equations

“…When h = 0, there exist two symmetric heteroclinic orbits Γ 0 1,2 connecting two equilibrium points A and B. To find exact solutions of system (11), from (9) and the first equation of (11), we know that…”

“…Different pairs (m, n) correspond to distinct type of knots. Obviously, m is also the number of strands of a torus knot (see [9] and the references therein).…”

Knotted Periodic Solutions of a Linear Nonautonomous System and Some Related Three-Dimensional Flows

“…Over the past three decades, a large number of articles concerning knotted periodic orbits in threedimensional dynamical systems had been published (see [Li & Chen, 2011] and cited references therein).…”