Real-time dynamical system

Definition

A real-time dynamical system or a real dynamical system or flow is a triple $(T,M,\mathrm{\Phi })$ where:

• $T$ is an open nonempty interval in the reals, containing zero
• $M$ is a differential manifold
• $\mathrm{\Phi }:U\to M$ is a map where $U\subseteq T\times M$

such that:

$\mathrm{\Phi }(0,x)=x\mathrm{\forall }x\in M$

$\mathrm{\Phi }(t+t^{\prime },x)=\mathrm{\Phi }(t,\mathrm{\Phi }(t^{\prime },x))$ if both sides are defined

Terminology

To each $x$ let $I(x)=\{t\in T|(t,x)\in U\}$. Then $I(x)$ is open in $T$. We can them define a map:

${\mathrm{\Phi }}_x:I(x)\to M$ given as $t\mapsto \mathrm{\Phi }(t,x)$.

• $\mathrm{\Phi }$ is termed the evolution function
• $M$ is called the phase space or state space
• $t$ is called the evolution parameter
• $x$ is called the initial state of the system
• The map ${\mathrm{\Phi }}_x$ is termed the flow at $x$ and its graph the trajectory through $x$. Its image is termed the orbit of $x$
• A subspace $S$ of $M$ is said to be $\mathrm{\Phi }$-invariant if $\mathrm{\Phi }(t,x)\in S$ for all $(t,x)\in U$ for which $x\in S$.