The Third Order Melnikov Function of a Quadratic Center under Quadratic Perturbations
نویسندگان
چکیده
We study quadratic perturbations of the integrable system (1 + x)dH, where H = (x2 + y2)/2. We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles.
منابع مشابه
Bifurcation of limit cycles from a quadratic reversible center with the unbounded elliptic separatrix
The paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of a quadratic reversible and non-Hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the poincar'{e} disk. Attention goes to the number of limit cycles produced by the period annulus under perturbations. By using the appropriate Picard...
متن کاملEight Limit cycles around a Center in quadratic Hamiltonian System with Third-Order perturbation
In this paper, we show that generic planar quadratic Hamiltonian systems with third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. We use higher-order focus value computation to prove this result, which is equivalent to the computation of higher-order Melnikov functions. Previous results have shown, based on first-order and higher-order Melnikov func...
متن کاملQuadratic perturbations of quadratic codimension-four centers
We study the stratum in the set of all quadratic differential systems ẋ = P2(x, y), ẏ = Q2(x, y) with a center, known as the codimension-four case Q4. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I. We show that the orbits of the unpertur...
متن کاملBifurcation of limit cycles in quadratic Hamiltonian systems with various degree polynomial perturbations
Article history: Received 1 May 2011 Accepted 15 February 2012 Available online 16 March 2012 0960-0779/$ see front matter 2012 Elsevier Ltd doi:10.1016/j.chaos.2012.02.010 ⇑ Corresponding author at: Department of Applied E-mail address: [email protected] (P. Y In this paper, we consider bifurcation of limit cycles in planar quadratic Hamiltonian systems with various degree polynomial per...
متن کاملComparison of Kullback-Leibler, Hellinger and LINEX with Quadratic Loss Function in Bayesian Dynamic Linear Models: Forecasting of Real Price of Oil
In this paper we intend to examine the application of Kullback-Leibler, Hellinger and LINEX loss function in Dynamic Linear Model using the real price of oil for 106 years of data from 1913 to 2018 concerning the asymmetric problem in filtering and forecasting. We use DLM form of the basic Hoteling Model under Quadratic loss function, Kullback-Leibler, Hellinger and LINEX trying to address the ...
متن کامل