The Critical Cases for Differential Equations

Abstract

In secs. 13 and 43 we defined the term critical behavior. For a differential equation

$$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}x{\rm{ }} + {\rm{ }}g(x,{\rm{ }}t),{\rm{ }}g(x,{\rm{ }}t) = {\rm{ }}O{\rm{ }}({\left| x \right|^2}), $$

(68.1)

we have a critical case if the stability of the equilibrium is significantly influenced by the terms of higher order and cannot be discussed by means of the reduced equation

$$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}x{\rm{ }}. $$

(68.2)

For autonomous equations the idea can be positively stated: The critical case is given if the matrix A has characteristic roots with negative as well as zero real parts but none with positive real parts. For periodic equations a similar characterization can be made with the aid of the characteristic exponents.