Dynamical Systems can be characterized as conservative or dissipative, depending on whether their phase volume stays constant or contracts.
A linearized damped pendulum decays to a single point -- its attractor, and is said to be dissipative. (see Baker and Golub, Chaotic Dynamics)
Roughly speaking, a dissipative system is not conservative but " open," with an external control parameter that can be tuned to critical values causing the transitions to chaos. In physical terms energy flows through a dissipative system and is lost to microscopic degrees of freedom. Entropy "fans out" into irrelevant variables, while the trajectory of "relevant" variables occupies a smaller and smaller region of phase space.
Dissipative Structures: (usage in Prigogine) The interaction of a system with the outside world, its embedding in nonequilibrium conditions may become the starting point for the formation of new dynamic states of matter. A whirlpool, for example, is a dissipative structure requiring a continuous flow of matter and energy to maintain the form. Prigogine uses this term to emphasise the close association, at first paradoxical, in such situations between structure and order on the one hand, and dissipation or waste on the other. cf chaos / philosophy
"We now know that far from equilibrium, new types of structure may originate spontaneously. In far-from-equilibrium conditions we may have transformation from disorder, from thermal chaos, into order. New dynamic states of matter may originate, states that reflect the interaction of a given system with its surroundings. We have called these new structures dissipative structures to emphasize the constructure role of dissipative processes in their formation." (Order out of Chaos, p. 12)