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, ...

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