Regularity properties of the period function near a center of a planar vector field.

*(English)*Zbl 0769.34033In case a planar vector field has a center, the annular region surrounding the center is filled with a one parameter family of periodic solutions. The function that assigns to a parameter value the period of the corresponding periodic solution is called the period function. The qualitative and analytic properties of this function play a role in the study of nonlinear boundary value problems and in bifurcation theory. The subject of the paper under review is the regularity properties of the period function at the center. Necessary and sufficient conditions are given, in terms of the blow up of the center, for the period function of a \(C^ \infty\) vector field to have a continuous extension to the center. For the analytic case when there is an essential singularity at the center, the growth properties of the period function at the center are explained in terms of some geometric properties. These results are applied to the problem of isochronicity of the period annulus. In particular, the results are used to answer a question posed by R. Conti [Processi di controllo lineari in \(\mathbb{R}^ n\) (1985; Zbl 0587.49001)]. He showed that a cubic system of the form
\[
\dot x=y+P_ 2(x,y)+P_ 3(x,y), \dot y=Q_ 2(x,y)+Q_ 3(x,y)
\]
(the subscript denotes the degree of the indicated homogeneous form) where \(P_ 2^ 2+Q_ 2^ 2\neq 0\), \(P_ 3^ 2+Q_ 3^ 2\neq 0\) can have a center. He asked if such a center can be isochronous. This is shown to be impossible in the paper under review.

Reviewer: C.Chicone (Columbia)

##### MSC:

34C25 | Periodic solutions to ordinary differential equations |

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

##### Keywords:

planar vector field; center; periodic solutions; periodic function; regularity properties; isochronicity
PDF
BibTeX
XML
Cite

\textit{M. Villarini}, Nonlinear Anal., Theory Methods Appl. 19, No. 8, 787--803 (1992; Zbl 0769.34033)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Villarini, M., Punti singolari di tipo centro per campi polinomiali nel piano, () |

[2] | Urabe, M., The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity, Archs ration. mech. analysis, 11, 26-33, (1962) · Zbl 0134.07205 |

[3] | Loud, W.S., Behavior of the period of the solutions of certain plane autonomous systemsnear centers, Cont. diff. eqns III, 1, 21-36, (1964) · Zbl 0139.04301 |

[4] | Pleshkan, I.I.; Sibirskii, K.S., Isochronism of systems with quadratic non linearity, (), 140-154, MR 45, 2262. (In Russian.) · Zbl 0244.34017 |

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

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

[8] | Arnold, V.I., Chapitres supplémentaires de la théorie des équations différentielles ordinaires, (1980), MIR, Moscow · Zbl 0455.34001 |

[9] | Conti, R., About centers of cubic planar systems. analytic theory, Quad. ist. mat. “U. dini” firenze, (1985) |

[10] | On centers of polynomial planar systems, 36-43, (1987), ICNO, XI, Budapest |

[11] | Seidenberg, A., Reduction of singularities of the differential equation A dy = B dx, Am. J. math., 90, 248-269, (1968) · Zbl 0159.33303 |

[12] | Boothby, W.M., An introduction to differentiable manifolds and Riemannian geometry, (1975), Academic Press New York · Zbl 0333.53001 |

[13] | Hartmann, Ph., Ordinary differential equations, (1964), Wiley New York |

[14] | Sansone, G.; Conti, R., Equazioni differenziali non lineari, (1964), Pergamon Press Oxford, English translation · Zbl 0075.26803 |

[15] | Il’yashenko, Yu.S., Dulac’s memoir “on limit cycles” and related problems of the local theory of differential equations, Russ. math. survs., 40, 6, 1-49, (1985) · Zbl 0668.34032 |

[16] | Mazzi, L.; Sabatini, M., A characterization of centers via first integrals, J. diff. eqns, 76, 2, 233-237, (1988) · Zbl 0667.34036 |

[17] | Olver, P., Applications of Lie groups to differential equations, (1986), Springer Berlin |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.