Nonlinear Dynamics and Systems Theory, 8 (1) (2008) 49–96

Strange Attractors and Classical Stability Theory

G. Leonov ∗

St. Petersburg State University Universitetskaya av., 28, Petrodvorets, 198504 St.Petersburg, Russia

Received: November 16, 2006; Revised: December 15, 2007

Abstract: Definitions of global attractor, B-attractor and weak attractor are introduced. Relationships between Lyapunov stability, Poincare stability and Zhukovsky stability are considered. Perron effects are discussed. Stability and instability criteria by the first approximation are obtained. Lyapunov direct method in dimension theory is introduced. For the Lorenz system necessary and sufficient conditions of existence of homoclinic trajectories are obtained.

Keywords: Attractor, instability, Lyapunov exponent, stability, Poincar´esection, Hausdorff dimension, Lorenz system, homoclinic bifurcation.

Mathematics Subject Classification (2000): 34C28, 34D45, 34D20.

1 Introduction

In almost any solid survey or book on chaotic dynamics, one encounters notions from classical stability theory such as Lyapunov exponent and characteristic exponent. But the effect of sign inversion in the characteristic exponent during linearization is seldom mentioned. This effect was discovered by Oscar Perron [1], an outstanding German math- ematician. The present survey sets forth Perron’s results and their further development, see [2]–[4]. It is shown that Perron effects may occur on the boundaries of a flow of solutions that is stable by the first approximation. Inside the flow, stability is completely determined by the negativeness of the characteristic exponents of linearized systems. It is often said that the defining property of strange attractors is the sensitivity of their trajectories with respect to the initial data. But how is this property connected with the classical notions of instability? For continuous systems, it was necessary to remember the almost forgotten notion of Zhukovsky instability. Nikolai Egorovich Zhukovsky, one of the founders of modern aerodynamics and a prominent Russian scientist, introduced

∗ Corresponding author: [email protected]

c 2008 InforMath Publishing Group/1562-8353 (print)/1813-7385 (online)/www.e-ndst.kiev.ua 49