Differential Equations and Linear Algebra, 4th Edition

9 Nonlinear Systems and Phenomena

9.1 Stability and the Phase Plane

A wide variety of natural phenomena are modeled by two-dimensional first-order systems of the form

$$\begin{array}{l}{\displaystyle \frac{dx}{dt}}=F(x,y),\\ {\displaystyle \frac{dy}{dt}}=G(x,y)\end{array}$$ (1)

in which the independent variable t does not appear explicitly. We usually think of the dependent variables x and y as position variables in the xy-plane and of t as a time variable. We will see that the absence of t on the right-hand sides in (1) makes the system easier to analyze and its solutions easier to visualize. Using the terminology of Section 2.2, such a system of differential equations in which the derivative values are independent (or “autonomous”) ...