scholarly journals Quadratic systems with an invariant conic having Darboux invariants

2018 ◽  
Vol 20 (04) ◽  
pp. 1750033
Author(s):  
Jaume Llibre ◽  
Regilene Oliveira
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Complete Characterization ◽  
Phase Portraits ◽  
Affine Transformations ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Vector Fields ◽  
Polynomial Differential Systems

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [Formula: see text]: [Formula: see text], and a Darboux invariant of the form [Formula: see text] with [Formula: see text].

PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

2009 ◽  
Vol 19 (03) ◽  
pp. 765-783 ◽  
Author(s):  
BARTOMEU COLL ◽  
ANTONI FERRAGUT ◽  
JAUME LLIBRE
Keyword(s):  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

scholarly journals Isochronicity of bi-centers for symmetric quartic differential systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wilker Fernandes ◽  
Viviane Pardini Valério ◽  
Patricia Tempesta
Keyword(s):  
Normal Forms ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Quartic Polynomial ◽  
Different Types ◽  
Simultaneous Existence ◽  
Isochronous Centers

<p style='text-indent:20px;'>In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.</p>

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 Centers and limit cycles for a generalized kukles differential systems of degree 8

2021 ◽  
Author(s):  
Loubna Damene ◽  
Rebiha Benterki
Keyword(s):  
Limit Cycles ◽  
Main Tool ◽  
Upper Bounds ◽  
First Integrals ◽  
Averaging Theory ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.

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).

THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER

2006 ◽  
Vol 16 (11) ◽  
pp. 3127-3194 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Second Order ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
The Family

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass [Formula: see text] which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class [Formula: see text] also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for [Formula: see text], 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class [Formula: see text] reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

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