number of limit cycles; generalized Li\'enard systems
DulacCherkas functions
systems of linear differential equations
On the construction of a class of DulacCherkas functions for generalized Li\'enard systems
Leonid Cherkas
Cherkas
Leonid
Alexander Grin
Grin
Alexander
Klaus Schneider
Schneider
Klaus
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 8, 120
On the construction of a class of DulacCherkas functions for generalized Li\'enard systems
Leonid Cherkas
,
Alexander Grin
,
Klaus Schneider
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 8, 120
MSC 2000
 34C07 Theory of limit cycles of polynomial and analytic vector fields

34C05 Location of integral curves, singular points, limit cycles
Abstract
DulacCherkas functions can be used to derive an
upper bound for the number of limit cycles of planar autonomous
differential systems including criteria for the nonexistence of limit cycles,
at the same time they provide information
about their stability and hyperbolicity. In this paper, we present a method to
construct a special class of DulacCherkas functions for generalized Li\'enard
systems of the type
$ \frac{dx}{dt} = y, \quad \frac{dy}{dt} = \sum_{j=0}^l h_j(x)
y^j$ with $l \ge 1$ by means of linear differential equations. In
case $1 \le l \le 3$, the described algorithm works generically.
We show that this approach can be applied also to systems with $l
\ge 4$. Additionally, we show that DulacCherkas functions can be
used to construct generalized Li\'enard systems with any $l$
possessing limit cycles.
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