Topological Character of a Periodic Solution in Three-Dimensional Ordinary Differential Equation System

Abstract

The torsion number of a periodic solution of a three-dimensional Ordinary Differential Equation (3-dim. O.D.E.) is defined. It characterizes the local structure of the flow around the periodic solution. The knot type and the torsion number of the periodic solution determine the topological character of the solution which bifurcates from it in the case of subharmonic or pitchfork bifurcation. The relative torsion number is also defined to describe torsion in the neighbourhood of a periodic solution relative to that of the periodic solution. It provides information on the rotation of a variational vector around the periodic solution, which is not contained in a Lyapunov characteristic number. Further, we discuss the mechanism of t: n _ bifurcation in O.D.E. and show numerical results for the torsion number, etc., in the parameter regions in which 2 n -bifurcation cascades are observed in the Lorenz and the forced Lorenz systems.