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.

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.

Reviewer: William J.Satzer jun.(St.Paul)

##### 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 |