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.

scholarly journals On the number of limit cycles of a class of polynomial differential systems

2012 ◽  
Vol 468 (2144) ◽  
pp. 2347-2360 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Generalized Polynomial

We study the number of limit cycles of polynomial differential systems of the form where g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 ( x )= f 1 ( x )=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

scholarly journals On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550131 ◽  
Author(s):  
Fangfang Jiang ◽  
Junping Shi ◽  
Jitao Sun
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = -y + g1(x) + f1(x)y and y′ = x + g2(x) + f2(x)y, where fi(x) and gi(x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

scholarly journals Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

2020 ◽  
Vol 30 (04) ◽  
pp. 2050051
Author(s):  
Jaume Llibre ◽  
Arefeh Nabavi ◽  
Marzieh Mousavi
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Seventh Order ◽  
Whole Class

Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).

scholarly journals Centers and Limit Cycles of Generalized Kukles Polynomial Differential Systems: Phase Portraits and Limit Cycles

2020 ◽  
pp. 378-398
Author(s):  
Ahlam Belfar ◽  
Rebiha Benterki
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Sixth Order ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

In this work, we give the seven global phase portraits in the Poincar´e disc of the Kukles differential system given by x˙ = −y, y˙ = x + ax8 + bx4y4 + cy8, where x, y ∈ R and a, b, c ∈ R with a2 + b2 + c2 ̸= 0. Moreover, we perturb these system inside all classes of polynomials of eight degrees, then we use the averaging theory up sixth order to study the number of limit cycles which can bifurcate from the origin of coordinates of the Kukles differential system

scholarly journals On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ziguo Jiang
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Isochronous Center ◽  
Discontinuous Systems ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial

We study the number of limit cycles for the quadratic polynomial differential systemsx˙=-y+x2,y˙=x+xyhaving an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.

scholarly journals Limit Cycles for the Class ofD-Dimensional Polynomial Differential Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Zouhair Diab ◽  
Amar Makhlouf
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System

We perturb the differential systemx˙1=-x2(1+x1),x˙2=x1(1+x1), andx˙k=0fork=3,…,dinside the class of all polynomial differential systems of degreeninRd, and we prove that at mostnd-1limit cycles can be obtained for the perturbed system using the first-order averaging theory.

scholarly journals Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Amar Makhlouf ◽  
Lilia Bousbiat
Keyword(s):  
Small Parameter ◽  
Real Number ◽  
Periodic Solutions ◽  
Differential System ◽  
Sufficient Conditions ◽  
Periodic Functions ◽  
Third Order ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Existence Of Periodic Solutions

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential systemx.=-y+εP(x,y,z)+h1(t),  y.=x+εQ(x,y,z)+h2(t),  and  z.=az+εR(x,y,z)+h3(t), whereP,Q, andRare polynomials in the variablesx,y, andzof degreen,  hi(t)=hi(t+2π)withi=1,2,3being periodic functions,ais a real number, andεis a small parameter.

scholarly journals Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1346 ◽  
Author(s):  
Amor Menaceur ◽  
Salah Boulaaras ◽  
Salem Alkhalaf ◽  
Shilpi Jain
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
New Class ◽  
Extended Work

In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order.

scholarly journals On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1137
Author(s):  
Maoan Han ◽  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Equilibrium Points ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
3 Dimensional ◽  
Volterra Systems

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

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