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

Lobry, C.; Wallet, G.

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

Keywords:

bifurcation; Hopf bifurcation; critical point; non-autonomous system

Citations:

Zbl 0667.00002