Phase Flow

• A one parameter group of transformations of a set is an action on the set by the group of all real numbers .

  We denote the one parameter group of transformations parameterized by as

  For any

  We call the transformation as the transformation in time .

  In the group, we define in the obvious way — as the “do nothing transformation”.

  We also call a one parameter group of transformations a phase flow within the phase space . ¹.

• One parameter groups of transformations can be thought of as equivalents to two sided deterministic processes where we map the phase space onto itself. In particular, we map states to future states (i.e., for , the state of the system at time starting from )

  • It is easy to show that the group properties hold for such a transformation defined on the phase space.

• The orbits of phase flow are called the phase curves.

• If the surface is a smooth manifold, then we additionally require that the phase flow be a Diffeomorphism.

• The phase velocity vector of the flow at the point is the velocity with which leaves. That is

  The phase velocity vectors of all points form a smooth vector field called the phase velocity field.

  • The fixed points of the flow are equilibrium points of the phase velocity field. Conversely, the equilibrium points of the phase velocity field are precisely the fixed points of the flow.
  • (Arnold 4.4.1) Consider the mapping
    This mapping is a solution to the equation
    With initial condition .
  • Under the action of the phase flow, the phase point moves so that its velocity vector at any instant equals the phase velocity vector at the point of the phase space at which the moving point is located.

• The phase flow of the differential equation is the one-parameter diffeomorphism group for which is the phase velocity vector field.

  To find the phase flow, it suffices to solve with initial condition .

  • Not every smooth vector field is a phase velocity vector field of a flow. In particular, we require the smooth vector field to be defined on a compact vector space.
  • Every smooth vector field on the line that has at most linear growth at infinity (i.e., ) is the phase velocity field of a one-parameter group of diffeomorphisms on the line
    • Proof: It can be verified that . Thus, it is bounded and is continuous on the entire -axis. Clearly a phase flow exists by solving for .

Links §

• Ordinary Differential Equations by Arnold
• Differential Calculus

Footnotes §

1. Imagine the phase space as a fluid map. A phase flow maps each particle in the fluid to its future state. ↩