Real-time dynamical system

Definition

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

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

such that:

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

$\Phi(t + t',x) = \Phi(t,\Phi(t',x))$ 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:

$\Phi_x: I(x) \to M$ given as $t \mapsto \Phi(t,x)$.

• $\Phi$ 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 $\Phi_x$ 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 $\Phi$-invariant if $\Phi(t,x) \in S$ for all $(t,x) \in U$ for which $x \in S$.