Topological Analysis

Abstract

Most nonlinear systems cannot be solved exactly so we must resort to a variety of approaches in order to obtain an approximate solution. Where applicable, the phase plane portrait can serve as a valuable tool for qualitatively determining the types of possible solutions before resorting to numerical or (usually approximate) analytical methods for specific initial conditions. In this chapter, the concept of phase plane analysis will be examined in some depth, not for a specific problem, but for a wide class of physical problems described by the following system of first order equations:

$$\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = P(x,y),} & {\frac{{dy}}{{dt}} = Q(x,y),} \\ \end{array}$$

(4.1)

where P, Q are, in general, nonlinear functions of x and y and the independent variable has been taken here to be time t. In the laser competition equations (2.32), t would, of course, be replaced with z, the spatial coordinate The mathematician would refer to this set of equations as being autonomous, meaning that P and Q do not explicitly depend on t. Why it is desirable to restrict the discussion for the moment to autonomous equations will become readily apparent.¹

For all men strive to grasp what they do not know, while none strive to grasp what they already know: and all strive to discredit what they do not excel in, while none strive to discredit what they do excel in. This is why there is chaos.

Chuang-tzu (3613–286 BC)