WIAS Preprint No. 1525, (2010)

A new approach to study limit cycles on a cylinder


  • Cherkas, Leonid
  • Grin, Alexander
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 34C05 34C07


  • planar system, cylindrical phase space, location and number of limit cycles of first and second kind, Dulac-Cherkas function


We present a new approach to study limit cycles of planar systems of autonomous differential equations with a cylindrical phase space $Z$. It is based on an extension of the Dulac function which we call Dulac-Cherkas function $Psi$. The level set $W:=vf,y) in Z: Psi(vf,y)=0$ plays a key role in this approach, its topological structure influences existence, location and number of limit cycles. We present two procedures to construct Dulac-Cherkas functions. For the general case we describe a numerical approach based on the reduction to a linear programming problem and which is implemented by means of the computer algebra system Mathematica. For the class of generalized Liénard systems we present an analytical approach associated with solving linear differential equations and algebraic equations.

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