The **phase space** of a particular **dynamical system** is partitioned into one or many basins of attraction , perhaps intimately intertwined, each with its own attractor .

basins of attraction for a pendulum swinging over three magnets. (from Peitgen, Jürgens, and Saupe, Chaos and Fractals)

physics

# dynamics

a dynamical system consists of a **space**, or manifold, where the motion of the system takes place, and a rule of motion, or vector **field**. The starting point is called the initial state, and the path of motion the trajectory. The end point of a trajectory is the system's **attractor**.

# energy

Joule's principle of the conservation of energy is an example of the subsumption of qualitative transformations into a **quantifiable** entity. "Thus it is that **order **is maintained in the universe--nothing is deranged, nothing ever lost, but the entire machinery, complicated as it is, works smoothly and harmoniously." (Joule, quoted in Prigogine, p.108-9)

# Entropy: Second law of

Entropy: Second law of thermodynamics:

Entropy is a measure of the **energy** distribution through a system. As energy becomes more dispersed or more evenly distributed in a system, the possibility of that energy's being used for mechanical **work** is decreased, and entropy increases.

# dissipative systems

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 non-equilibrium 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.

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