WIAS Preprint No. 1772, (2013)

On the construction of a class of generalized Kukles systems having at most one limit cycle


  • Schneider, Klaus R.
  • Grin, Alexander

2010 Mathematics Subject Classification

  • 34C05 34C23


  • Kukles system, Duclac-Cherkas function, limit cycle


Consider the class of planar systems $$fracdxdt = y, quad fracdydt = -x + mu sum_j=0^3 h_j(x,mu) y^j$$ depending on the real parameter $mu$. We are concerned with the inverse problem: How to construct the functions $h_j$ such that the system has not more than a given number of limit cycles for $mu$ belonging to some (global) interval. Our approach to treat this problem is based on the construction of suitable Dulac-Cherkas functions $Psi(x,y,mu)$ and exploiting the fact that in a simply connected region the number of limit cycles is not greater than the number of ovals contained in the set defined by $Psi(x,y,mu)=0.$

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