Differential Equations and Linear Algebra, 4th Edition

9.2 Linear and Almost Linear Systems

We now discuss the behavior of solutions of the autonomous system

$${\displaystyle \frac{dx}{dt}}=f(x,y),{\displaystyle \frac{dy}{dt}}=g(x,y)$$ (1)

near an isolated critical point $(x_0,y_0)$ where $f(x_0,y_0)=g(x_0,y_0)=0.$ A critical point is called isolated if some neighborhood of it contains no other critical point. We assume throughout that the functions f and g are continuously differentiable in a neighborhood of $(x_0,y_0)$.

We can assume without loss of generality that $x_0=y_0=0.$ Otherwise, we make the substitutions $u=x-x_0,v=y-y_0.$ Then $dx/dt=du/dt$ and $dy/dt=dv/dt,$ so (1) is equivalent to the system

$$\begin{array}{l}{\displaystyle \frac{du}{dt}}=f(u+x_0,v+y_0)=f_1(u,v),\\ {\displaystyle \frac{dv}{dt}}=g(u+x_0,v+y_0)=g_1(u,v)\end{array}$$ (2)

that has (0, ...