# Real-time dynamical system

## Definition

A **real-time dynamical system** or a **real dynamical system** or **flow** is a triple where:

- is an open nonempty interval in the reals, containing zero
- is a differential manifold
- is a map where

such that:

if both sides are defined

### Terminology

To each let . Then is open in . We can them define a map:

given as .

- is termed the
**evolution function** - is called the
**phase space**or**state space** - is called the
**evolution parameter** - is called the
**initial state**of the system - The map is termed the
**flow**at and its graph the**trajectory**through . Its image is termed the**orbit**of - A subspace of is said to be -invariant if for all for which .