Real-time dynamical system

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Definition

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

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.