In this case, a sketch of the phase portrait may give qualitative information about the dynamics of the system, such as the limit cycle of the Van der Pol oscillator shown in the diagram.
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Because this constraint is nonholonomic, Liouville's theorem does not apply, and although energy is conserved, the motion is dissipative in the sense that phase space volume is not conserved.
Their predictability usually deteriorates with time and to quantify predictability, the rate of divergence of system trajectories in phase space can be measured (Kolmogorov-Sinai entropy, Lyapunov exponents).
Later with Denis Evans and Gary Morriss in 1990 he proved that for certain classes of thermostatted nonequilibrium steady states the relevant transport coefficient the transport coefficient has a simple relation to the sum of the largest and smallest Lyapunov exponents describing the trajectory of the N-particle steady state system in phase space.
C. K. Zachos, D. B. Fairlie, and T. L. Curtright, "Quantum Mechanics in Phase Space" (World Scientific, Singapore, 2005) ISBN 981-238-384-0.
His research topics include path integral representation of Fractional Brownian motion, ring closing opening dynamics in polymers, first passage time calculation in phase space, energy transfer between molecule and graphene, quantum coherence etc.
Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ★-product.