7 research outputs found
The local period function for Hamiltonian systems with applications
In the first part of the paper we develop a constructive procedure to obtain
the Taylor expansion, in terms of the energy, of the period function for a
non-degenerated center of any planar analytic Hamiltonian system. We apply it
to several examples, including the whirling pendulum and a cubic Hamiltonian
system. The knowledge of this Taylor expansion of the period function for this
system is one of the key points to study the number of zeroes of an Abelian
integral that controls the number of limit cycles bifurcating from the periodic
orbits of a planar Hamiltonian system that is inspired by a physical model on
capillarity. Several other classical tools, like for instance Chebyshev systems
are applied to study this number of zeroes. The approach introduced can also be
applied in other situations.Comment: 23 page
New lower bound for the Hilbert number in low degree Kolmogorov systems
Our main goal in this paper is to study the number of small-amplitude
isolated periodic orbits, so-called limit cycles, surrounding only one
equilibrium point a class of polynomial Kolmogorov systems. We denote by
the maximum number of limit cycles bifurcating from the
equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov
vector field of degree . In this work, we obtain another example such that . In addition, we obtain new lower bounds for proving that and
Estudo de ciclos limites em uma classe de equações diferenciais descontÃnuas
Neste trabalho temos como principal objetivo determinar quota inferior para o número máximo de ciclos limites de um sistema diferencial polinomial de Liénard descontÃnuo de grau n com m zonas, para m=2,4. A principal ferramenta é uma combinação da Teoria da Média de primeira ordem com o processo de regularização de sistemas descontÃnuos. Analisamos detalhadamente um caso particular de um sistema polinomial de Liénard de grau 3 com 4 zonasIn this work our main aim is to determine the lower upper bound for the maximum number of limit cycles of a m-piecewise discontinuous Liénard polynomial differential system of degree n, for m=2,4. The main tool is a combination of the first order Averaging Method with the regularization process of discontinuous systems. We analyzed in details a particular case of a 4-piecewise discontinuous Liénard polynomial differential system of degree 3.Coordenação de Aperfeiçoamento de Pessoal de NÃvel Superior (CAPES
Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centres
These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centres (consequently isochronous) and cubic isochronous centres with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles
The local period function for Hamiltonian systems with applications
In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations
Limit cycles for some families of smooth and non-smooth planar systems
We apply the averaging method in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set Σ = {xy = 0}. They also motivate to consider a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees