One of the most important tasks in a study of dynamical systems is the numerical calculation of the trajectories. Thus far we have considered the integration method to be a black box into which we pop the system, initial conditions, method and time range and out pops a plot of the trajectories. Although this approach is common in courses on dynamical systems it obscures many of the pitfalls of numerical integration.
It is not possible at the present state of the art to choose a ‘best’ algorithm for the calculation of trajectories. There are several types of numerical algorithm, each with their own advantages and disadvantages. We shall consider a class of methods known as discrete variable methods. These methods approximate the continuous time dynamical system by a discrete time system. This means that we are not really simulating the continuous system but a discrete system which may have different dynamical properties. This is an extremely important point.
The discrete variable methods which we consider fall into two main types, Runge-Kutta methods and Linear Multistep methods. Maple has implementations of both types of method as well as a number of more sophisticated techniques designed to overcome some of the pitfalls of numerical solution. The more sophisticated methods still fall into the discrete variable category.