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**Finiteness for critical periods of planar analytic vector fields.**
*(English)*
Zbl 0773.34032

The authors treat analytic vector fields on the plane and on the unit sphere which exhibit one-parameter families of periodic orbits. To any such “period annulus”, there corresponds a “period function” \(P(\xi)\), where \(\xi\in I\subset\mathbb{R}\) is a local parameter specifying the periodic orbits and \(P(\xi)\) denotes their period. \(\overline\xi\in I\) is said to be a critical period if \(P'(\overline\xi)=0\); and if \(P'(\xi)\equiv 0\), the period annulus is called isochronous. The main question investigated in this paper is to determine the cardinality of the set of critical periods. The basic result: If a relatively compact period annulus is not isochronous, then there are only finitely many critical periods. For the proof, previous work by the second author is used to desingularize the boundary of a period annulus (Blowing-up procedures transform a neighborhood of any boundary point to one of three canonical forms). The case of unbounded period annuli remains open in general. But for the special case of Hamiltonian vector fields with Hamiltonian \(H(x,y)={1\over 2}y^ 2+V(x)\), \(V\) a polynomial, the authors also prove the existence of at most finitely many critical periods unless \(V(x)\) is quadratic. The proof is by introducing appropriate hyperelliptic coordinates. The latter result is immediately applicable to the Neumann boundary value problem \(\ddot x+g(x)=0\), \(\dot x(0)=\dot x(T)=0\), where \(g(x)\) is a polynomial. It is proved that, unless \(g(x)\) is linear, this problem has at most a finite number of solutions whose number of zeros in \((0,T)\) is uniformly bounded. This result also holds for the polynomial Dirichlet problem \(\ddot x+g(x)=0\), \(x(0)=x(T)=a\); but its proof requires some modifications, since the solutions need no longer be periodic.

Reviewer: J.Hainzl (Kassel)

### MSC:

34C25 | Periodic solutions to ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

### Keywords:

planar vector field; period annulus; period function; critical period; Hamiltonian vector fields; Neumann boundary value problem; polynomial Dirichlet problem
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\textit{C. Chicone} and \textit{F. Dumortier}, Nonlinear Anal., Theory Methods Appl. 20, No. 4, 315--335 (1993; Zbl 0773.34032)

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### References:

[1] | Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. am. math. soc., 312, 433-486, (1989) · Zbl 0678.58027 |

[2] | Andronov, A.A., Theory of bifurcations of dynamical systems on a plane, (1973), Wiley New York |

[3] | Ecalle, J., Finitude des cycles limites et accéléro-sommation de l’application de retour, Lecture notes in mathematics, Vol. 1455, 74-159, (1991) |

[4] | Il’yasenko, J., Finiteness theorems for limit cycles, Russ. math. survs, 40, 143-200, (1990) |

[5] | Chicone, C., The monotonicity of the period function for planar Hamiltonian vector fields, J. diff. eqns, 69, 310-321, (1987) · Zbl 0622.34033 |

[6] | Chicone, C., Geometric methods for nonlinear two point boundary value problems, J. diff. eqns, 72, 360-407, (1988) · Zbl 0693.34014 |

[7] | Chicone, C.; Dumortier, F., A quadratic system with a non monotonic period function, Proc. am. math. soc., 102, 706-710, (1988) · Zbl 0651.34043 |

[8] | Chow, S.N.; Sanders, J.A., On the number of critical points of the period, J. diff. eqns, 64, 51-66, (1986) · Zbl 0594.34028 |

[9] | Chow, S.N.; Wang, D., On the monotonicity of the period function of some second order equations, Čas. Pěst. mat., 111, 14-25, (1986) · Zbl 0603.34034 |

[10] | Rothe, F., Periods of oscillation, nondegeneracy and specific heat of Hamiltonian systems in the plane, () |

[11] | Schaaf, R., A class of Hamilton systems with increasing periods, J. reine angew. math., 363, 96-109, (1985) · Zbl 0565.34037 |

[12] | Schaaf, R., Global solution branches of two point boundary problems, () · Zbl 0780.34010 |

[13] | Smoller, J.; Wasserman, A., Global bifurcation of steady state solutions, J. diff. eqns, 39, 269-290, (1981) · Zbl 0425.34028 |

[14] | Waldvogel, J., The period in the Lotka-Volterra system is monotonic, J. math. analysis applic., 114, 178-184, (1986) · Zbl 0588.92018 |

[15] | Wang, D., The critical points of the period function of x″ −x2(x − α)(x− 1) = 0 (0 ≤ α < 1), Nonlinear analysis, 11, 1029-1050, (1987) · Zbl 0641.34029 |

[16] | Chicone, C., Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. math. analysis, 23, 6, 1577-1608, (1992) · Zbl 0765.58018 |

[17] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001 |

[18] | {\scDumortier} F., Singularities of vector fields on the plane,J. diff. Eqns 23, 53-106. · Zbl 0346.58002 |

[19] | Perko, L.M., On the accumulation of limit cycles, Proc. am. math. soc., 99, 515-526, (1987) · Zbl 0626.34022 |

[20] | Brieskorn, E.; Knörrer, H., Plane algebraic curves, (1986), Birkhäuser Boston, J. Stillwell (Trans.) · Zbl 0588.14019 |

[21] | Hartman, P., On local homeomorphisms of Euclidean spaces, Bol. soc. math. mexicana, 5, 220-241, (1960) · Zbl 0127.30202 |

[22] | Dumortier, F.; Rodrigues, P.; Roussarie, R., Germs of diffeomorphisms in the plane, () · Zbl 0502.58001 |

[23] | Andronov, A.A., Qualitative theory of second-order dynamic systems, (1973), Wiley New York · Zbl 0282.34022 |

[24] | Arnold, V.I., Ordinary differential equations, (1978), MIT Press Massachusetts, R. A. Silverman (Trans.) |

[25] | Lyapunov, A., Problème général de la stabilité du mouvement, () |

[26] | Lyapunov, A., Mathematics in science and engineering, Vol. 30, (1966), Academic Press New York |

[27] | Brüll, L.; Mawhin, J., Finiteness of the set of solutions of some boundary-value problems for ordinary differential equations, Arch. math., 24, 163-172, (1988), (Brno) · Zbl 0678.34023 |

[28] | Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer New York · Zbl 0508.35002 |

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