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Zbl 0628.58035
Vanderbauwhede, A.
Hopf bifurcation in symmetric systems. (English)
[J] Arch. Math., Brno 22, 29-53 (1986). ISSN 0044-8753; ISSN 1212-5059/e

This paper is a very nice, explanatory approach to the problems of bifurcations that arise for symmetric systems. The symmetry of the system makes it usually impossible to apply the classical Hopf bifurcation theorem which describes how cycles bifurcate from a singular point. The symmetry affects the properties of eigenvalues, essential for the Hopf theorem. The author shows how the Hopf theorem can be modified to suit symmetric systems in some special, but important for applications, cases. Namely, he considers the particular symmetries such as rotational symmetries in the plane and time-reversibility. \par Part I of the paper is a quite self-contained introduction to the problem. Part II deals with concrete bifurcation problems. It is shown, among other things, that under some generic conditions there are no bifurcation periodic solutions other than the collinear and circular solutions the author constructs. \par A full bifurcation analysis is done for the equation $$ \ddot x+g(x,\dot x,\lambda)\dot x+f(x,\dot x,\lambda)x=0\quad, $$ with $x\in {\bbfR}\sp 2$, $\lambda\in {\bbfR}\sp a $small parameter, f and g smooth real functions, $f(0,0,0)=1$, $g(0,0,0)=0$, f and g rotationally invariant and satisfying either the transversality condition (H1) or the condition for time-reversibility (H2).
[Z.Denkowska]

MSC 2000:: *37G99 Bifurcation theory
34C25 Periodic solutions of ODE

Keywords: symmetric systems; Hopf bifurcation; rotational symmetries in the plane; time-reversibility

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