Modules: Dissecting An Equation

Dissecting A Dynamical System Equation:

The underlying theme of the library of equations of Phaser, and the Modules, is the study of qualitative properties of differential and difference equations. There are two main stages in the numerical study of a specific equation: dynamics and bifurcations.

Dynamics: The first stage in this pursuit is to determine the phase portrait of an equation by varying the initial conditions. In particular, one would like to know the limiting behavior of all the solutions of an equation as time increases in forward or backward directions; i.e., one would like to determine the "limit sets" of all the solutions of an equation. In general, these sets grow in complexity as the dimension of the equation increases.

In one dimension, a solution of a differential equation approaches an equilibrium point in forward or backward time. In two dimensions, the possible limit sets are of three types: equilibria, periodic orbits, and the collection of equilibrium points with orbits joining them. In three dimensions and higher, there are extremely complicated examples of limiting behavior, with little hope of a complete classification. A set that attracts nearby solutions in forward time, but is more complicated than an equilibrium point or a periodic orbit, is called a strange or chaotic attractor.

In the case of difference equations, the situation is still more complicated: even in one dimension, there are examples not merely with fixed points and periodic orbits, but also with strange attractors. All these phenomena, and more, can be seen in the equations stored in the libraries of Phaser and the Modules.

Bifurcations: The second stage in this qualitative study is to explore the possible changes in the phase portrait of an equation as the equation itself is varied. In applications, for example, many models contain changeable parameters. Even when this is not the case, it may be necessary to introduce parameters into a model so that by changing them the "robustness" of the system under small perturbations can be investigated.

The study of qualitative changes (for example, variations in the number or the stability type of equilibria) in the phase portraits of dynamical systems as parameters are varied is called bifurcation theory. For given parameter values, a system is called structurally stable if small changes in the parameters do not change the qualitative properties of the phase portrait. Any other value of the parameters is called a bifurcation value. It is important to locate the bifurcation values, and to classify the possible phase portraits around them. Indeed, a substantial number of the examples in the libraries are designed to illustrate many of the "typical" bifurcations.

Experimental Dynamics: The ideal use of computers in dynamical systems is both to observe known dynamical phenomena and to discover new ones in specific examples.

Many of the equations in the libraries and Modules possess complicated dynamics, which cannot be understood simply by plotting orbits at random. Therefore, before undertaking the study of an equation, one should master the precise mathematical formulation of the phenomena one hopes to observe. Remarks and references for each equation in the libraries and Modules are intended to facilitate this by pointing to sources where specific definitions, theorems, or more detailed information about a particular equation can be found.

It is, of course, more difficult to suggest how to discover new phenomena. At the very least, you should keep in mind that theory and experimentation are mutually beneficial, especially when used iteratively. Do not be discouraged, however, if you do not get the "right" picture immediately. Being an experimentalist, even on the computer, can be a time-consuming activity.