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