4.5 Worked Examples

The methods and principles presented in previous sections will now be applied to several examples. An important feature in constructing the phase space trajectories for many systems arising in the mathematical modeling of dynamical phenomena is that only a knowledge of the behavior of the trajectories in a restricted region is needed. This is a consequence of the fact that for some systems the dependent variables satisfy a positivity requirement, i.e.,

Particular examples include chemical reactions, where the concentrations are always non-negative, and problems involving interacting populations, where the number densities are also non-negative.

For the worked examples to follow, we will only investigate the relevant regions of phase space as defined by the particular system of interest.

4.5.1 Example A

Let us examine the behavior of a dynamical system described by the pair of coupled, nonlinear first-order differential equations

The only fixed-point is ( x,y) = (0, 0) and the linear approximation in a neighborhood of this fixed-point is

where

In matrix form, we have

and the eigenvalues are given by the solutions to

and they are

Thus, the linear analysis suggests that the fixed-point is a center. However, this result may or may not be correct for the full nonlinear equations. In fact, for large values of x and y, we have

and this implies that trajectories may become unbounded. If so, then ( x,y) = (0, 0) cannot be a center.

To study this issue in more detail, let us...