Attractors are **geometric** forms that characterize long-term behaviour in the **phase space**. Roughly speaking, an attractor is what the behaviour of a systems settles down to, or is attracted to. They are globally stable in the sense that the system will return if perturbed off the attractor, as long as it remains within the **basin of attraction**.

dynamics

# basin of attraction

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)

# bifurcation

A bifurcation occurs when an **attractor** changes **qualitatively** with the **smooth** variation of a control parameter. Physically, bifurcations denote phase transitions from a state of equilibrium to new possible states of equilibria. (see also **singularities**)

# 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.

# feedback

In all feedback systems, some portion of the output system is used as input. Positive feedback adds the output to the input, leading to "vicious cycles." Negative feedback is self-regulating, inducing the system to approach equilibrium or steady-state. (In communications engineering, these two modes are also called regenerative and degenerative cycles.)

Read More# 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.

Read More# Mandlebrot set

The Mandlebrot set has been described as the "most complicated object in the world." The figure represents the boundary of the **domain of attraction** of the behaviour of a simple equation in the complex plane. It is not the domain of attraction of a single system but rather a map of a family of systems, based on a single criterion.

# non-linearity

In **dynamical** terms, such as in the study of **chaos**, a non-linear situtation is one where the result is not proportional to the cause. For instance "the straw that broke the camel's back" (eg. the elastic/plastic limit in building structures) introduces non-linearity. Up until that point, deformation had been proportional to load. Suddenly it loses all proportionality.

# phase boundary

A phase boundary separates different areas of **phase space**, for example the region of ordered **dynamics** from the region of **chaotic** dynamics. The region at or near this boundary is described as the **complex** area or regime.

# Sensitivity to Intial conditions

Sensitivity to Initial Conditions: An extremely small change in the initial conditions of a **chaotic** or **non-linear** system leads to extremely differing results. Any arbitrarily small interval of initial values will be enlarged significantly by iteration. This is the so-called "butterfly effect" in which the flapping wings of a single butterfly could theoretically make the difference whether or not a hurricane occurred in another place and time. (The title of a paper by Edward N. Lorentz was "*Can the flap of a butterfly's wing stir up a tornado in Texas*?"