Generalized Floquet Exponent, Attractiveness Portrait and Structure Hidden in an Attractor

Abstract

The generalized Floquet exponent and the attractiveness portrait (or A-portrait for short) of the attractor and of the smallest invariant closed set are suggested to be used for the study of dynamical systems. Based on the A-portrait, some simple structures hidden in a complicated attractor may emerge from an attractor with complicated structure. The hidden structure plays important role in the bifurcation phenomena of the invariant sets. The examples of A-portraits for the Van der Pol limit cycle, for Lorenz attractor, for the closed limit orbits of different rotation numbers and complicated attractors of Silnikov equation, and for three interlocked smallest invariant closed set of the new improved Nos\'e-Hoover oscillator are given.