LEXICON OF CHAOS THEORY
- Dynamical System
- A mathematical object with two basic components: a state space and a dynamical rule.
- State Space
- A geometric space of finite dimension, finite or infinite in extent, points of which correspond to the observable states of a natural system.
- Dynamical Rule
- At each point in a state space an instruction is attached specifying a finite motion to (or an infinitesimal motion towards) a new point in the same space. In this sketch we consider only the finite motion type of rule, which is called a discrete dynamical rule.
- Given a specific initial point and dynamical system, following the rule repeatedly defines a motion through the state space called a trajectory, leading in the long run to a small part of the state space called an attractor. These are of three types: a single point, a finite set of points, or any other (infinite) set of points. These are called respectively: a static, periodic, or chaotic attractor.
- In case there is more than one attractor in a dynamical system, all initial points tending to one of them is called its basin of attraction. The state space is decomposed into the union of different basins, which may be very intertwined, as in the fractals frequently seen on book covers. The portrait of the state space decomposed into basins is very valuable in applications, being the basis of predictions of the long run behavior.
- Dynamical Systems Theory
- From its beginnings ca 1880 with Poincare' until the computer revolution and the advent of computer graphics in the 1970s, dynamical systems theory consisted in the analysis of a given dynamical system: determining its attractors and basins, and their transformations (called bifurcations) as the dynamical rule is gradually changed.
- Nonlinear Time Series Analysis
- After the computer simulation of dynamical systems along with the computer graphic representation of their trajectories was developed by John von Neumann and associates in the 1940s, scientists became interested in the inverse process: given an attractor-basin portrait (that is, experimental data) find a dynamical rule fitting the data. Such a dynamical model would then be the basis for prediction of new data.
- New Science
- This inverse process has been seen as a new type of science, a shift from the 300 year old Newtonian paradigm, in books eg by James Gleick, and Steve Wolfram. A whole generation of new scientists since 1980 have been applying the new methods to data from the physical, biological, and social sciences: earthquakes, power grid failures, heart attacks, prison riots, etc.
- Attractor Reconstruction
- One of the clever tricks of nonlinear time series analysis assumes that a time series (say, a simple list of numbers, eg, daily stock prices) comes from an unknown dynamical system. Using a trick called time-lag embedding, the time series is transformed into a sequence of points in a higher-dimensional space. Experience with known dynamical systems shows that an analogue of a chaotic attractor is constructed in this way.
- Recurrence Plots
- Within the process of attractor reconstruction, a large array of numbers must be created. These numbers are the distances between pairs of points of the embedded time series. Early pioneers of the attractor reconstruction technique would use a color code to turn this array into a visible image and thus, around 1987, discovered the recurrence plot. Reading the plot requires experience, but can be turned into a predictive skill. In this way, some clever day-traders are making money in the stock market. In fact, the Prediction Company was formed to do just this by two of the early pioneers of the technique.
- Trajectory Fitting
- Another strategy for time series forecasting creates a specific dynamical system that mimics the given time series very approximately. This is an adaptation of a well-known polynomial curve fitting process, least squares approximation. A family of dynamical systems is chosen, for example polynomials of degree three. Then the coefficients of the family are adjusted to minimize the errors of an entire trajectory. This method has been used for power systems and the global economy.
- Complex Dynamical Systems
- A complex dynamical system is a directed graph or network of simpler dynamical systems with parameters, linked by output-to-input functions. For example, techniques of spatial econometrics (or systems dynamics) model each source of raw materials, factory, warehouse, retail outlet, etc, separately, then links them with transportation systems taking the delivery times and costs into account. Making such models requires large numbers of time series: complex data. The attractor reconstruction and trajectory fitting methods have been adapted to complex dynamical systems. See the Complexity Digest and the International Journal of Bifurcations and Chaos for the current state of the art.
Revised Monday 17 June 2002 by Ralph H Abraham