Limit cycles and zeroes of Abelian integrals satisfying third order picard — Fuchs equations

1990 ◽  
pp. 160-186 ◽  
Author(s):  
Ljubomir Gavrilov ◽  
Emil Horozov
Keyword(s):  
Limit Cycles ◽  
Abelian Integrals ◽  
Third Order

QUADRATIC PERTURBATIONS OF A CLASS OF QUADRATIC REVERSIBLE LOTKA–VOLTERRA SYSTEMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1350137
Author(s):  
YI SHAO ◽  
A. CHUNXIANG
Keyword(s):  
Limit Cycles ◽  
Positive Answer ◽  
Abelian Integrals ◽  
Volterra System ◽  
Bifurcation Of Limit Cycles ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Volterra Systems

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system [Formula: see text] with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

2004 ◽  
Vol 56 (2) ◽  
pp. 310-343 ◽  
Author(s):  
Jaume Llibre ◽  
Dana Schlomiuk
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Topological Classification ◽  
Phase Portraits ◽  
Third Order ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Affine Invariants ◽  
Geometric Concepts

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

scholarly journals Limit Cycles for a Class of Third-Order Differential Equations

2010 ◽  
Vol 40 (2) ◽  
pp. 581-594 ◽  
Author(s):  
Jaume Llibre ◽  
Jiang Yu ◽  
Xiang Zhang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Third Order

BIFURCATIONS OF LIMIT CYCLES FROM A QUINTIC HAMILTONIAN SYSTEM WITH A FIGURE DOUBLE-FISH

2013 ◽  
Vol 23 (07) ◽  
pp. 1350116 ◽  
Author(s):  
MINGHUI QI ◽  
LIQIN ZHAO
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Liénard Systems

In this paper, we consider Liénard systems of the form [Formula: see text] where 0 < ∣∊∣ ≪ 1 and (α, β, γ) ∈ ℝ3. We prove that the least upper bound of the number of isolated zeros of the related Abelian integrals [Formula: see text] is 4 (counting the multiplicity) and this upper bound is a sharp one.

scholarly journals ON THE NUMBER OF LIMIT CYCLES FOR A GENERALIZATION OF LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350048 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLAUDIA VALLS
Keyword(s):  
Small Parameter ◽  
Periodic Orbits ◽  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
Polynomial Differential Systems

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

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