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La traversée de l’axe imaginaire n’a pas toujours lieu là où l’on croit l’observer. (The traverse through the imaginary axis has not always been where supposed). (French) Zbl 0687.58029
Mathématique finitaires & analyse non standard, Publ. Math. Univ. Paris VII 31, 45-51 (1989).
[For the entire collection see Zbl 0667.00002.]
The authors show by specific examples that if the bifurcation parameter $$\mu$$ of a system $$dx/dt=f_{\mu}(x)$$ is replaced by a slowly varying monotone function $$\mu$$ (t), and if a Hopf bifurcation of some critical point of this system occurs at say $$\mu =0$$ then the resulting non- autonomous system may not exhibit the same “bifurcation” at a value of t for which $$\mu (t)=0$$. They give a theorem which states in effect that if $$| \mu (t)|$$ is sufficiently small for t in an interval containing where $$\mu (t_ 0)=0$$, then the bifurcation for the nonautonomous system will coincide with that of the autonomous system.
Reviewer: G.Seifert

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior