29 research outputs found
Small- and large-amplitude limit cycles in Kukles systems with algebraic invariant curves
Limit cycles of planar polynomial vector fields have been an active area of
research for decades; the interest in periodic-orbit related dynamics comes
from Hilbert's 16th problem and the fact that oscillatory states are often
found in applications. We study the existence of limit cycles and their
coexistence with invariant algebraic curves in two families of Kukles systems,
via Lyapunov quantities and Melnikov functions of first and second order. We
show center conditions, as well as a connection between small- and
large-amplitude limit cycles arising in one of the families, in which the first
coefficients of the Melnikov function correspond to the first Lyapunov
quantities. We also provide an example of a planar polynomial system in which
the cyclicity is not fully controlled by the first nonzero Melnikov function.Comment: 18 pages, Submitted to Journal of Dynamical and Control System
Modules of Abelian integrals and Picard-Fuchs systems
We give a simple proof of an isomorphism between the two
-modules: the module of relative cohomologies and the module of Abelian integrals corresponding to a regular at
infinity polynomial in two variables. Using this isomorphism, we prove
existence and deduce some properties of the corresponding Picard-Fuchs system.Comment: A separate section discusses Fuchsian properties of the Picard-Fuchs
system, Morse condition exterminated. Few errors were correcte
On a computer-aided approach to the computation of Abelian integrals
An accurate method to compute enclosures of Abelian integrals is developed.
This allows for an accurate description of the phase portraits of planar
polynomial systems that are perturbations of Hamiltonian systems. As an
example, it is applied to the study of bifurcations of limit cycles arising
from a cubic perturbation of an elliptic Hamiltonian of degree four
The th Hilbert problem on algebraic limit cycles
For real planar polynomial differential systems there appeared a simple
version of the th Hilbert problem on algebraic limit cycles: {\it Is there
an upper bound on the number of algebraic limit cycles of all polynomial vector
fields of degree ?} In [J. Differential Equations, 248(2010), 1401--1409]
Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper
bound, in the case of invariant algebraic curves generic for the vector fields,
and they posed the following conjecture: {\it Is the maximal
number of algebraic limit cycles that a polynomial vector field of degree
can have?}
In this paper we will prove this conjecture for planar polynomial vector
fields having only nodal invariant algebraic curves. This result includes the
Llibre {\it et al}\,'s as a special one. For the polynomial vector fields
having only non--dicritical invariant algebraic curves we answer the simple
version of the 16th Hilbert problem.Comment: 16. Journal Differential Equations, 201
Minimality and ergodicity of a generic analytic foliation of 2
It is well known that a generic polynomial foliation of 2 is minimal and ergodic. In this paper we prove an analogous result for analytic foliation
On the number of limit cycles bifurcating from a non-global degenerated center
AbstractWe give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system (1+x)dH=0, where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y2/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis