Lie algebras, structure of nonlinear systems and chaotic motion.

*(English)*Zbl 0996.37035The authors deal with a large scale structure theory for systems of the form

$$\dot{x}=A\left(x\right)x\xb7\phantom{\rule{2.em}{0ex}}\left(1\right)$$

They define a Lie algebra associated with this system to be the Lie subalgebra of $\mathrm{\U0001d524\U0001d529}(n,\u2102)$ generated by all the matrices $A\left(x\right)$, $x\in {\mathbb{R}}^{n}$, 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.

Reviewer: Messoud Efendiev (Berlin)

##### MSC:

37D45 | Strange attractors, chaotic dynamics |

34A34 | Nonlinear ODE and systems, general |

34A26 | Geometric methods in differential equations |

37J99 | Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems |

37C99 | Smooth dynamical systems |

17B66 | Lie algebras of vector fields and related (super)algebras |