# 9.2 Linear and Almost Linear Systems

We now discuss the behavior of solutions of the autonomous system

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 $u=x-{x}_{0},\text{}v=y-{y}_{0}.$ Then $dx/dt=du/dt$ and $dy/dt=dv/dt,$ so (1) is equivalent to the system

that has (0, ...

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