Dynamical Systems and Chaos

Abstract

A dynamical system is a mathematical object to describe the development of a physical, biological or another system from real life depending on time. It is defined by a phase space M, and by a one-parameter family of mappings φ: M → M, where t is the parameter (the time). In the following, the phase space is often Rⁿ, a subset of it, or a metric space. The time parameter t is from R (time continuous system) or from Z or from Z₊ (time discrete system). Furthermore, it is required for arbitrary x ∈ M that

• a) φ0(x) = x and

• b) φt(φs(x)) = φ^t+s(x) for all t, s. The mapping φ¹ is denoted briefly by φ.