Flow of a non-autonomous vector field

The definition

Consider a non-autonomous differential equation on a state space $M$ (like $R^{n}$ or a manifold):

\dot{x} = f (t, x)

Given an initial state $x_{0}$ at an initial time $t_{0}$ , assume there is a unique solution $x (t)$ to the initial value problem $x (t_{0}) = x_{0}$ .

The non-autonomous flow is the map $ϕ_{t, t_{0}} : M \to M$ defined by:

ϕ_{t, t_{0}} (x_{0}) = x (t)

This map takes an initial point $x_{0}$ at time $t_{0}$ and maps it to its new position at time $t$ .

Fundamental properties

Unlike autonomous flows, which form a one-parameter group satisfying $ϕ_{t} \circ ϕ_{s} = ϕ_{t + s}$ (see flow of a vector field), the non-autonomous flow $ϕ_{t, t_{0}}$ satisfies a two-parameter property known as the cocycle property (or transition property):

Identity: Starting and ending at the exact same time results in no movement.

ϕ_{t_{0}, t_{0}} (x) = x

Composition (Cocycle): Evolving the system from time $t_{0}$ to $t_{1}$ , and then from $t_{1}$ to $t_{2}$ , is identical to evolving it directly from $t_{0}$ to $t_{2}$ .

ϕ_{t_{2}, t_{0}} (x) = ϕ_{t_{2}, t_{1}} (ϕ_{t_{1}, t_{0}} (x))

Invertibility: Flowing backward in time (assuming uniqueness and existence hold) is the inverse of flowing forward.

ϕ_{t_{0}, t}^{- 1} = ϕ_{t, t_{0}}

The extended phase space trick

Mathematically, it is often easier to study non-autonomous flows by converting them into autonomous flows. This is done by treating time $t$ as just another state variable in an "extended" phase space $M \times R$ .