Workshop Glossary

### Glossary

Definitions of several terms are a matter of some dispute.
For a more technical treatment of some of these terms, see the faq sheet of the sci.nonlinear newsgroup.
attractor The status that a dynamic system eventually "settles down to". An attractor is a set of values in the phase space to which a system migrates over time, or iterations. An attractor can be a single fixed point, a collection of points regularly visited, a loop, a complex orbit, or an infinite number of points. It need not be one- or two-dimensional. Attractors can have as many dimensions as the number of variables that influence its system.
basin of attraction A region in phase space associated with a given attractor. The basin of attraction of an attractor is the set of all (initial) points that go to that attractor.
bifurcation A qualitative change in the behavior (attractor) of a dynamic system associated with a change in control parameter.
bifurcation diagram Visual summary of the succession of period-doubling produced as a control parameter is changed.
chaos Behavior of a dynamic system that has (a) a very large (possibly infinite) number of attractors and (b) is sensitive to initial conditions.
complexity While, chaos is the study of how simple systems can generate complicated behavior, complexity is the study of how complicated systems can generate simple behavior. An example of complexity is the synchronization of biological systems ranging from fireflies to neurons. (From the FAQ sheet of the sci.nonlinear newsgroup).
complex system Spatially and/or temporally extended nonlinear systems characterized by collective properties associated with the system as a whole--and that are different from the characteristic behaviors of the constituent parts.(From the FAQ sheet of the sci.nonlinear newsgroup)
control parameter A parameter in the equations of a dynamic system. If control parameters are allowed to change, the dynamic system would also change. Changes beyond certain values can lead to bifurcations. .
difference equation A function specifying the change in a variable from one discrete point in time to another.
differential equation A function that specifies the rate of change in a continuous variable over changes in another variable (time, in this book).
dimension See embedding dimension, box-counting dimension, correlation dimension, information dimension, dimension of a system.
dimensions of a system The set of variables of a system.
dynamic system A set of equations specifying how certain variables change over time. The equations specify how to determine (compute) the new values as a function of their current values and control parameters. The functions, when explicit, are either difference equations or differential equations. Dynamic systems may be stochastic or deterministic. In a stochastic system, new values come from a probability distribution. In a deterministic system, a single new value is associated with any current value.
embedding dimension Successive N-tuples of points in a time series are treated as points in N dimensional space. The points are said to reside in embedding dimensions of size N, for N = 1, 2, 3, 4, ... etc.
fractal An irregular shape with self-similarity. It has infinite detail, and cannot be differentiated. "Wherever chaos, turbulence, and disorder are found, fractal geometry is at play" (Briggs and Peat, 1989).
fractal dimension A measure of a geometric object that can take on fractional values. At first used as a synonym to Hausdorff dimension, fractal dimension is currently used as a more general term for a measure of how fast length, area, or volume increases with decrease in scale. (Peitgen, Jurgens, & Saupe, 1992a).
Hausdorff dimension A measure of a geometric object that can take on fractional values. (see fractal dimension).
initial condition the starting point of a dynamic system.
iteration the repeated application of a function, using its output from one application as its input for the next.
iterative function a function used to calculate the new state of a dynamic system.
iterative system A system in which one or more functions are iterated to define the system.
limit cycle An attractor that is periodic in time, that is, that cycles periodically through an ordered sequence of states.
limit points Points in phase space. There are three kinds: attractors, repellors, and saddle points. A system moves away from repellors and towards attractors. A saddle point is both an attractor and a repellor, it attracts a system in certain regions, and repels the system to other regions.
linear function The equation of a straight line. A linear equation is of the form y=mx+b, in which y varies "linearly" with x. In this equation, m determines the slope of the line and b reflects the y-intercept, the value y obtains when x equals zero.
logistic difference equation see logistic map
logistic map x(n+1)= rx(n)[1- x(n)]. A concave-down parabolic function that (with 0 Lorenz attractor A butterfly-shaped strange attractor. It came from a meteorological model developed by Edward Lorenz with three equations and three variables. It was one of the first strange attractors studied.
Lyapunov Number (Liapunov number) The value of an exponent, a coefficient of time, that reflects the rate of departure of dynamic orbits. It is a measure of sensitivity to initial conditions.
nonlinear function One that's not linear! y would be a nonlinear function of x if x were multiplied by another variable (non-constant) or by itself (that is, raised to some power.
nonlinear dynamics The study of dynamic systems whose functions specifying change are not linear.
orbit (trajectory) A sequence of positions (path) of a system in its phase space.
period-doubling The change in dynamics in which a N-point attractor is replaced by a 2N-point attractor.
phase portrait The collection of all trajectories from all possible starting points in the phase space of a dynamic system.
phase space (state space) An abstract space used to represent the behavior of a system. Its dimensions are the variables of the system. Thus a point in the phase space defines a potential state of the system. The points actually achieved by a system depend on its iterative function and initial condition (starting point).
recursive process For our purposes, "recursive" and "iterative" are synonyms. Thus recursive processes are iterative processes, and recursive functions are iterative functions.
repellors One type of limit point. A point in phase space that a system moves away from.
return map Plot of a time series values n vs. n+1.
saddle point A point, usually in three-space, that both an attracts and a repels, attracting in one dimension and repelling to another.
self-similarity An infinite nesting of structure on all scales. Strict self- similarity refers to a characteristic of a form exhibited when a substructure resembles a superstructure in the same form.
sensitivity to initial conditions A property of chaotic systems. A dynamic system has sensitivity to initial conditions when very small differences in starting values result in very different behavior. If the orbits of nearby starting points diverge, the system has sensitivity to initial conditions.
starting state see initial condition
state A point in state space designating the current location (status) of a dynamic system.
state space (phase space) An abstract space used to represent the behavior of a system. Its dimensions are the variables of the system. Thus a point in the phase space defines a potential state of the system.
strange attractor N-point attractor in which N equals infinity. Usually (perhaps always) self-similar in form.
time series A set of measures of behavior over time.
Torus An attractor consisting of N independent oscillations. Plotted in phase space, a 2-oscillation torus resembles a donut.
trajectory (orbit) A sequence of positions (path) of a system in its phase space. The path from its starting point (initial condition) to and within its attractor.
vector A two-valued measure associated with a point in the phase space of a dynamic system. Its 1) direction shows where the system is headed from the current point, and its 2) length indicates velocity.
vector field The set of all vectors in the phase space of a dynamic system. For a given continuous system, the vector field is specified by its set of differential equations.