# Techniques, Concepts and Examples

## 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.

*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

## Keywords

Phase Portrait Stagnation Point Integral Curve Gradient Vector Polar System## Preview

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