Basic terminology

Definition: A dynamic system consists of two ingredients: a rule or "dynamic", which specifies how a system evolves for each possible state, and an initial condition or "state" from which the system starts.

(This notion will do here, although there many ways of defining this notion. See Tim van Gelder's essay for Brain and Behavioral Sciences, `The dynamical hypothesis in cognition'.

A geometric interpretation of the theory of differential equations says that a differential equation is a vector field on a manifold. A manifold is any smooth geometric space (line, surface, solid, etc.). The smoothness condition ensures that the manifold cannot have any sharp edges or corners. A vector field is a rule that smoothly assigns a vector (a directed line segment) to each point of a manifold. This rule is often written as a system of first-order differential equations.

A linear system is one in which the dynamic rule is linearly proportional to the system variables, otherwise the system is non-linear.

Examples:

dx/dt = v or dv/dt = -x define (first order) linear systems

d^2x/dt^2 = -x defines a (second order) linear system

d^2x/dt^2 = -x^2 defines a (second order) non-linear system

Each point in the manifold represents an individual state, or possible initial condition, of the system. The collection of all possible states is called the state space of the system. A process is said to be deterministic if both its future and past states are uniquely determined by its present state.

A system of first order differential equations assigns to each point of the manifold a vector, thereby forming a vector field on the manifold. A solution to a differential equation is called trajectory (or integral curve, since it results from integrating the differential equations of motion). An individual vector in the vector field determines how the solution behaves locally. It tells the trajectory "where to go". The collection of all solutions is called the flow.

In a planar vector field, only four motions are possible: source (or repellor), sink (or stable fixed point), saddle, and limit cycle. (For chaotic motion one needs at least three dimensions). The asymptotic motions of a flow are characterized by four types of behavior: equilibrium points, periodic solutions, quasiperiodic solutions, and chaos. All of the stable asymptotic motions (or limit sets), e.g., sink, stable limit cycle, are examples of attractors. The unstable limit sets, e.g., sources, are examples of repellers. The term strange attractor (strange repellor) is used to describe attracting (repelling) limit sets that are chaotic.

Maps are the discrete time analogs of flows. While flows are specified by differential equations, maps are specified by difference equations (they are easier to compute than flows).

Some demos:

Java applets for all kinds of dynamic systems
Nonlinear Pendulum Demo
Phase Line Demo
Doug Eck's gallery of oscillators (with animation)

Papers, sites:

T. van Gelder, BBS paper in prepublication form: The Dynamical Hypothesis in Cognitive Science
Nonlinear Dynamics and Topological Time Series Analysis Archive
Surveys in Dynamical systems

R. Port for Cognitive Science Q500,
Indiana University