Results 11 to 20 of about 155,898 (247)

Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers [PDF]

International Journal of Bifurcation and Chaos, 2018
In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic ...
J. Llibre, Yun Tian
semanticscholar   +5 more sources

The algebraic limit cycles of planar cubic systems

Sibirskie Elektronnye Matematicheskie Izvestiya, 2020
Algebraic limit cycles of differential systems of the form \[\dot{x}=x+P_3(x,y),\quad \dot{y}=y+Q_3(x,y), \] where \(P_3(x,y)\) and \(Q_3(x,y)\) are homogeneous cubic polynomials, are studied. Note that the results of Theorem 1 were obtained much earlier in the monograph [\textit{V. N. Gorbuzov} and \textit{A. A.
E. Volokitin, V. Cheresiz
semanticscholar   +2 more sources

Algebraic limit cycles for quadratic polynomial differential systems [PDF]

Discrete and Continuous Dynamical Systems - B, 2017
We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle.
J. Llibre, C. Valls
semanticscholar   +7 more sources

A family of planar differential systems with hyperbolic algebraic limit cycles

Electronic Journal of Qualitative Theory of Differential Equations
In this paper, we characterize a family of planar polynomial differential systems of degree greater or equal than $n+1$, by presenting polynomial curves of degree $n,$ which generally contain closed components.
Maroua Ghelmi, Aziza Berbache
doaj   +2 more sources

The $16$th Hilbert problem on algebraic limit cycles [PDF]

Journal of Differential Equations, 2014
For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m?
Zhang Xiang
semanticscholar   +4 more sources

Rational Limit Cycles on Abel Polynomial Equations

Mathematics, 2020
In this paper we deal with Abel equations of the form d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 , where A 1 ( x ) , A 2 ( x ) and A 3 ( x ) are real polynomials and A 3 ≢ 0 .
Claudia Valls
doaj   +3 more sources

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

Expositiones Mathematicae, 2020
In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic
J. Llibre, Xiang Zhang
semanticscholar   +5 more sources

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

Qualitative Theory of Dynamical Systems, 2022
For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic ...
A. Gasull, H. Giacomini
semanticscholar   +4 more sources

Uniqueness of Algebraic Limit Cycles for Quadratic Systems

Journal of Mathematical Analysis and Applications, 2001
All known quadratic systems (QS) having an algebraic limit cycle are contained in five families. The degree of the algebraic curve containing the limit cycle is 4 in four of these families and 2 for the other one. Furthermore, it is also known that if there is another QS having an algebraic limit cycle, it should have at least degree 5. The main result
Jaume Llibre, Hector Giacomini
exaly   +3 more sources

Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields [PDF]

Geometriae Dedicata, 2005
In Geometriae Dedicata 79 (2000), 101{108, Rudolf Winkel conjectured: For a given algebraic curve f = 0 of degree m > 4 there is in general no polynomial vector ¯eld of degree less than 2m ¡ 1 leaving invariant f = 0 and having exactly the ovals of f = 0 as limit cycles. Here we show that this conjecture is not true.
Chara Pantazi, Jaume Llibre
exaly   +3 more sources