Phase Volume

The way the phase space volume changes in time is an important property of systems with continuous or discrete dynamics. Select a subset of the phase space with a positive finite (hyper-) volume, and evolve the points in this subset in time. If the volume defined by the new subset is always equal to the initial volume, the dynamics under investigation belongs to a conservative system, such as a Hamiltonian system. If, on the other hand, that volume is changing in time, we have a nonconservative system. If the phase volume of the system always increases, the system will be structurally unstable, and the trajectories will diverge to infinity. Thus we cannot observe such systems for long, and they are not of much interest. The class of systems with shrinking phase volume in time are called dissipative systems. They axe structurally stable, and the methods introduced in this chapter axe directed at studying such systems.

The rate of change of the phase space volume for a continuous flow defined by Eq (5.1) is given by the trace of the tangent flow matrix, (or the Jacobian matrix evaluated along the flow),

If this rate is positive (negative), then the phase space volume grows (shrinks) in time.

Example 1 Phase volume change of a flow

Consider the "deterministic non-periodic flow" of Lorenz [350], given by, dx\ ~~dt dx 2 ~df dx 3 dt

with a, p and /3 axe real positive constants. The tangent flow matrix of the system can be found as df_ 9x

—<7 |
a |

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