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.Read More
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)
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)Read More
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.
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)Read More
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.
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
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
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.Read More
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.Read More
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 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?"