A note on the period function for certain planar vector fields

Abstract

Consider a smooth planar autonomous differential equation having a period annulus, 𝒫. We present a new criterion to ensure that the period function has at most one critical period on 𝒫. Our result has a compact form when the differential equation is written as . It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation , where f(z) and are holomorphic functions and h is a C ² smooth real valuated function.

Keywords:

• meromorphic vector fields
• periodic orbit
• period function
• critical period

AMS Subject Classification::

• 32M25
• 34C25
• 37C27

Acknowledgements

The first author is partially supported by grant MTM2008–03437/Consolider (including a FEDER contribution). The second author is partially supported by MTM2008–01486/Consolider (including a FEDER contribution). The third author is partially supported by MTM2006–05849 and MTM–2008–01486/Consolider (including a FEDER contribution). Finally, all authors are partially supported by 2005SGR–00550.