# Techniques, Concepts and Examples

• David Betounes
Chapter

## Abstract

In this chapter we look at a number of examples of dynamical systems (systems of DEs) in detail and use this as an opportunity to introduce many concepts, such as gradient vector fields, stable/unstable fixed points, separatrices, limit cycles, transformations of DEs, and conservation laws, which will be studied more formally later. Our goal is to give the reader some experience with looking at, working with, studying, and analyzing some typical examples of systems that can occur. The computer exercises here, in the previous chapter, and on the electronic component are intended to aid the student in obtaining this experience and to help establish an intuitive feel for the concepts well before we study the underlying theory for these concepts. Waiting until after the development of the theory to begin computer analyses of dynamical systems is too long to wait.

A basic understanding of the concepts is easily obtained with a good computer and a computer algebra system (CAS), like Maple, which we will use here and throughout the text. Maple will be used to numerically plot the integral curves for the systems we study and not as a tool for finding closed-form formulas for these integral curves (which rarely is possible). In essence the vector field X defining the system
$$x^\prime = X(t, x)$$
contains all the information we need to numerically study the integral curves of the system via their plots. It also contains much geometric information about the phase portrait of the system as well.

## Keywords

Phase Portrait Stagnation Point Integral Curve Gradient Vector Polar System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.