Homogeneous Linear System of Coupled Differential Equations

This Demonstration shows the solution paths, critical point, eigenvalues, and eigenvectors for the following system of homogeneous first-order coupled equations:

,

.

The origin is the critical point of the system, where and . You can track the path of the solution passing through a point by dragging the locator. This is not a plot in time like a typical vector path; rather it follows the and solutions. A variety of behaviors is possible, including that the solutions converge to the origin, diverge from it, or spiral around it.

This example comes from the discussions given in Chapter 7 of [1] on critical points using a phase plot.

Reference

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.