9.2 Linear and Almost Linear Systems

We now discuss the behavior of solutions of the autonomous system

dxdt=f(x,y),dydt=g(x,y) (1)

near an isolated critical point (x0,y0) where f(x0,y0)=g(x0,y0)=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 (x0,y0).

We can assume without loss of generality that x0=y0=0. Otherwise, we make the substitutions u=xx0, v=yy0. Then dx/dt=du/dt and dy/dt=dv/dt, so (1) is equivalent to the system

dudt=f(u+x0,v+y0)=f1(u,v),dvdt=g(u+x0,v+y0)=g1(u,v) (2)

that has (0, ...

Get Differential Equations and Linear Algebra, 4th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.