## Abstract

The dynamical system is a mathematical concept motivated first by Newtonian mechanics. The state of the system is generally denoted by a point in an appropriately defined geometrical space. A dynamical system operates in time. Typically, we take the time set *T* to be the real line *R* (a continuous-time system) or the set of integers *Z* (a discrete-time system). We then formalize an *autonomous* system as a ordered pair (*Q, g*), where *Q* is the state space, and *g : T × Q → Q* is a function that assigns to each initial state *x*_{0} ∈ *Q* the state *x* = *g*(*t*, *x*_{0}), in which the system will be after a time interval t if it started in state *x*_{0}. A fundamental property of *g*, then, is the validity of the identity *g*(*t* + *s*, *x*_{0}) ≡ *g*(*s*, *g*(*t*, *x*_{0})) (4.1) for all states *x*, and times *t*, *s*. Loosely speaking *g* is a fixed rule which governs the motion of the system.