scholarly journals Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1128
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Limit Cycles ◽  
The Other ◽  
Differential Systems ◽  
Other Hand ◽  
Straight Line ◽  
Line Of Discontinuity ◽  
Straight Lines

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.

scholarly journals The acyclicity of a quadratic differential system

2021 ◽  
pp. 32-41
Author(s):  
Адам Дамирович Ушхо ◽  
Вячеслав Бесланович Тлячев ◽  
Дамир Салихович Ушхо
Keyword(s):  
Differential Equations ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Qualitative Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Differential System ◽  
Straight Line

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

scholarly journals Detour Extra Straight Lines in the Euclidean Plane

2020 ◽  
pp. 237-249
Author(s):  
I. Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Plane ◽  
Euclidean Geometry ◽  
The Other ◽  
Real Numbers ◽  
Other Hand ◽  
Straight Line ◽  
Axiom Systems ◽  
Straight Lines

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we explode the Euclidean plane. Exploding the Euclidean straight lines we get super straight lines. The extra straight line is the window phenomenon of super straight line. In general, the extra straight lines are curves in Euclidean sense, but they have more similar properties to Euclidean straight lines. On the other hand, with respect of parallelism we find a surprising property: there are detour straight lines.

scholarly journals Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibria separated by reducible cubics

2021 ◽  
pp. 1-38
Author(s):  
Rebiha Benterki ◽  
Jeidy Jimenez ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian Systems ◽  
Differential System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Main Role ◽  
Differential Systems ◽  
Cubic Curve ◽  
Straight Line ◽  
Linear Hamiltonian Systems ◽  
Linear Differential

Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.

scholarly journals Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

2007 ◽  
Vol 79 (4) ◽  
pp. 563-575 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias
Keyword(s):  
Large Amplitude ◽  
Symmetric Polynomial ◽  
The Other ◽  
Poincaré Compactification ◽  
Heteroclinic Loop ◽  
Differential Systems ◽  
Heteroclinic Cycles ◽  
Polynomial Differential Systems ◽  
Global Study ◽  
Straight Lines

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Qingyi Zhan
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Uniqueness Of Limit Cycles ◽  
Cubic System ◽  
Straight Lines ◽  
Cubic Systems

A class of cubic systems with two invariant straight linesdx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2.is studied. It is obtained that the focal quantities ofO(0,0)are,W0=δ; ifW0=0, thenW1=m(n+l); ifW0=W1=0, thenW2=−nm(b+1); ifW0=W1=W2=0, thenOis a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focalO(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

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