Stability by the linear approximation
Part of the Applied Mathematical Sciences book series (AMS, volume 62)
The oldest method of investigating the stability of a system is to replace the system by its linear approximation. This method was used long before Liapunov, although Liapunov appears to be the first person who justified the method for ordinary differential equations. In 1929 Perron  investigated the question of when the stability of the difference equation
determines the stability of the nonlinear equation,
$$x' = Ax$$
$$x' = Ax + f\left( x \right)$$
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© Springer Science+Business Media New York 1986