Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

Abstract

The Lorenz family is a class of dynamic systems which is parameterized in one variable and includes both the classic Lorenz system and the Chen system. Both systems (and therefore the Lorenz family) can show chaotic behaviour. For different questions of system analysis, e.g. for the estimation of various types of dimensions, it is helpful to be able to determine bounds on the attractor. One typical approach ist the usage of Lyapunov functions to describe positive invariant sets. However, this approach usually requires a good knowledge of the system to perform the necessary calculations. For polynomial systems, these calculations can be carried out using a computation technique known as quantifier elimination. In this contribution we describe this approach and employ it to calculate bounds on the Lorenz family.