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Lie algebras, structure of nonlinear systems and chaotic motion. (English) Zbl 0996.37035

The authors deal with a large scale structure theory for systems of the form

x ˙=A(x)x·(1)

They define a Lie algebra associated with this system to be the Lie subalgebra of 𝔤𝔩(n,) generated by all the matrices A(x), x n , and is denoted by L A . The main goal of the paper is to demonstrate that the classical structure theory of this Lie algebra has important consequences for stability theory and chaotic motion. The authors show that the well-known chaotic systems of Lorenz and Chua have a natural representation in terms of the Lie algebra L A and lead to an immediate extension to higher-dimensional chaotic structures.

37D45Strange attractors, chaotic dynamics
34A34Nonlinear ODE and systems, general
34A26Geometric methods in differential equations
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
37C99Smooth dynamical systems
17B66Lie algebras of vector fields and related (super)algebras