1,538 research outputs found
Some results on homoclinic and heteroclinic connections in planar systems
Consider a family of planar systems depending on two parameters and
having at most one limit cycle. Assume that the limit cycle disappears at some
homoclinic (or heteroclinic) connection when We present a method
that allows to obtain a sequence of explicit algebraic lower and upper bounds
for the bifurcation set The method is applied to two quadratic
families, one of them is the well-known Bogdanov-Takens system. One of the
results that we obtain for this system is the bifurcation curve for small
values of , given by . We obtain
the new three terms from purely algebraic calculations, without evaluating
Melnikov functions
Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials
In this paper, we consider some cubic near-Hamiltonian systems obtained from
perturbing the symmetric cubic Hamiltonian system with two symmetric singular
points by cubic polynomials. First, following Han [2012] we develop a method to
study the analytical property of the Melnikov function near the origin for
near-Hamiltonian system having the origin as its elementary center or nilpotent
center. Based on the method, a computationally efficient algorithm is
established to systematically compute the coefficients of Melnikov function.
Then, we consider the symmetric singular points and present the conditions for
one of them to be elementary center or nilpotent center. Under the condition
for the singular point to be a center, we obtain the normal form of the
Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic
Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating
from the center using the algorithm to compute the coefficients of Melnikov
function. Finally, perturbing the symmetric hamiltonian system by symmetric
cubic polynomials, we consider the number of limit cycles near one of the
symmetric centers of the symmetric near-Hamiltonian system, which is same to
that of another center
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