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Bifurcation of planar vector fields and Hilbert’s sixteenth problem. (English) Zbl 0898.58039
Progress in Mathematics (Boston, Mass.). 164. Basel: Birkhäuser (ISBN 3-7643-5900-5/hbk; 978-3-0348-9778-5/pbk). xvii, 204 p. (1998).
The focus of this book is a study of bifurcations of limit periodic sets in families of planar vector fields. These vector fields $$X_\lambda$$ depend on a parameter $$\lambda$$; the phase space is $$\mathbb{R}^2$$ or, more generally, a surface of genus zero. Important examples of such vector fields include polynomial vector fields $$X_\lambda^n$$, of degree not exceeding $$n$$ whose parameter $$\lambda$$ belongs to the coefficients of the two components $$P$$ and $$Q$$: $X_n^\lambda (x,y)= P(x,y) \frac{\partial}{\partial x}+Q(x,y) \frac{\partial}{\partial y}.$ In order to know the dynamics of the flow associated with $$X_\lambda$$, it is crucial to determine the nature of the singular elements – singular points and periodic orbits – and to know how the dynamics change as the parameter $$\lambda$$ varies.
The author devotes considerable attention to the determination of a bound for the number of limit cycles which bifurcate from a limit periodic set $$\Gamma$$. Such a bound is called the cyclicity of $$\Gamma$$ in $$X_\lambda$$. This bound depends only on the germ of the family along $$\Gamma$$, also called the unfolding $$(X_\lambda,\Gamma)$$. A general conjecture is that the cyclicity is finite for any analytic unfolding. The author proves that this conjecture implies a positive answer to Hilbert’s 16th problem: “For any $$n\geq 2$$, there exists a finite number $$H(n)$$ such that for any polynomial vector field of degree less than or equal to $$n$$ has less than $$H(n)$$ limit cycles.”
Besides addressing this general conjecture, the author also explores the computation of cyclicity for typical explicit unfoldings.

##### MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory