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Classification of quadratic systems admitting the existence of an algebraic limit cycle

Bulletin Des Sciences Mathematiques, 2007
Nowadays people know the existence of seven different families of quadratic systems having algebraic limit cycles. These limit cycles are real ovals of irreducible invariant algebraic curves of degree \(n,\) with \(n\) being 2, 4, 5 or 6. Recall that given a polynomial \(F(x,y)\) it is said that the curve \(F(x,y)=0\) is invariant for the quadratic ...
Jaume Llibre, Grzegorz Swirszcz
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Algebraic Limit Cycles

International Journal of Bifurcation and Chaos
In the qualitative theory of differential equations in the plane [Formula: see text], one of the most difficult objects to study is the existence of limit cycles. Here, we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems.
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Planar Polynomial Systems with Non-Algebraic Limit Cycles

AIP Conference Proceedings, 2009
In this paper, we study the existence of the non‐algebraic limit cycles of the systems dxdt = Pn(x,y)+xRm(x,y) dydt = Qn(x,y)+yRm(x,y) where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with ...
Khalil I. T. Al-Dosary, George Maroulis, Theodore E. Simos   +2 more
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Polynomial Vector Fields with Prescribed Algebraic Limit Cycles

Geometriae Dedicata, 2001
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Hilbert’s 16th Problem for Algebraic Limit Cycles

2016
In this chapter we state Hilbert’s 16th problem restricted to algebraic limit cycles. Namely, consider the set Σ’ n of all real polynomial vector fields \( \chi = \left( {P,\,Q} \right)\) of degree n having real irreducible \( \left( {{\rm on}\, \mathbb{R}\left[ {x,\,y} \right]} \right)\) invariant algebraic curves.
Jaume Llibre, Rafael Ramírez
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Algebraical Limit Cycles of Polynomial Vector Fields on the Plane

Differential Equations, 2001
The existence of limit cycles of algebraic differential equations has been studied and their number has been estimated in numerous papers. In recent decades, much attention was paid to polynomial vector fields with invariant algebraic curves. The statement of the problem goes back to Darboux, Poincaré, and Erugin.
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Algebraic limit cycles in polynomial systems of differential equations

Journal of Physics A: Mathematical and Theoretical, 2007
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree
Jaume Llibre, Yulin Zhao
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NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

International Journal of Mathematics, 2007
In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y)
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Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations

Nonlinearity, 2006
We give a family of planar polynomial differential systems whose limit cycles can be explicitly described using polar coordinates. Moreover, we characterize the multiplicity of each one of the limit cycles whenever they exist. The given family of planar polynomial differential systems can have at most two limit cycles, counted with multiplicity.As an ...
Jaume Giné, Maite Grau
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Coexistence of limit cycles and invariant algebraic curves for a Kukles system

Nonlinear Analysis: Theory, Methods & Applications, 2004
This paper deals with the so-called Kukles systems, \[ \dot x=-y,\qquad \dot y=f(x,y), \] \(f\) being a polynomial of degree \(d.\) Firstly, the authors study the number and distribution of invariant straight lines that this type of systems can have.
Chavarriga, J., Sáez, E., Szántó, I., Grau, M.   +3 more
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