Algebraic Limit Cycles Research Articles

Algebraic limit cycles for quadratic polynomial differential systems

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. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.

The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

Abstract In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity

Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having two pairs of diametrally opposite equilibrium points at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

The 16th Hilbert problem restricted to circular algebraic limit cycles

We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S−1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S−1 algebraic limit cycles given by circles, and this number is reached.

On the limit cycles bifurcating from an ellipse of a quadratic center

It is well known that invariant algebraic curves of polynomial differential systems play an important role in questions regarding integrability of these systems. But do they also have a role in relation to limit cycles? In this article we show that not only they do have a role in the production of limit cycles in polynomial perturbations of such systems but that algebraic invariant curves can even generate algebraic limit cycles in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.

Rational Lyapunov Functions and Stable Algebraic Limit Cycles

The main goal of this technical note is to show that the class of systems described by a planar differential equation having a rational proper Lyapunov function has asymptotically stable sets which are either locally asymptotically stable equilibrium points, stable algebraic limit cycles or asymptotically stable algebraic graphics. The use of the Zubov equation is then an adapted tool to investigate the study of an upper bound on the number of stable limit cycles and asymptotically stable graphics and their relative positions for this class of systems.

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems $$\mathop x\limits^. $$ = λ x−y +P n+1(x, y)+xF 2n (x, y), $$\mathop y\limits^. $$ = x+λ y +Q n+1(x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.

Limit Cycle Identification in Nonlinear Polynomial Systems

We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.

On the Multiplicity of Algebraic Limit Cycles

In the present paper we tackle the problem of determining the multiplicity of periodic orbit as limit cycle of a planar differential system. We consider the particular case of a circumference as periodic orbit. We show that the conditions of multiplicity can be almost algebraically solvable. There are parameters in which these conditions depend transcendentally, as happens in the degenerate center-focus problem. Even though this difficulty, these transcendental dependence can be, in some sense, controlled because only a basis of fundamental functions appear. The appearance of this fundamental basis opens the path to approach these types of problems. We present several examples of families for which these conditions can be computed.

Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse

In this paper, for a certain class of Kukles polynomial systems of arbitrary degree n with an invariant ellipse, we show that for certain values of the parameters, the system has an upper bound of limit cycles, where one of the limit cycle is given by an invariant ellipse as an algebraic limit cycle. Writing the system as a perturbation of a Hamiltonian system, we show that the first Poincaré–Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simple zeros of this polynomial, gives the maximum number of the global limit cycles and the multiplicity of the origin as a root of the polynomial, minus one, gives the maximum weakness that may have the weak focus at the origin.

The 16th Hilbert problem on algebraic limit cycles

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? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401–1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is 1 + ( m − 1 ) ( m − 2 ) / 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have? In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.ʼs as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems x˙=y, y˙=−fm(x)y−gn(x) with fm and gn real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681–1701] the author showed that if m⩾3 and m+1<n<2m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011–2022] proved that the Liénard systems with m=3 and n=5 have no hyperelliptic limit cycles and that there exist Liénard systems with m=4 and 5<n<8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m>4 and m+1<n<2m. In this paper we will prove that there exist Liénard systems with m=5 and m+1<n<2m which have hyperelliptic limit cycles.

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.

On the 16th Hilbert problem for algebraic limit cycles

For a polynomial planar vector field of degree n ⩾ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + ( n − 1 ) ( n − 2 ) / 2 when n is even, and ( n − 1 ) ( n − 2 ) / 2 when n is odd. Furthermore, these upper bounds are reached.

Planar polynomial vector fields having first integrals and algebraic limit cycles

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.

On the algebraic limit cycles of Liénard systems

For the Liénard systems with fm and gn polynomials of degree m and n, respectively, we present explicit systems having algebraic limit cycles in the cases m ⩾ 2 and n ⩾ 2m + 1 and m ⩾ 3 and n = 2m. Also we prove that the Liénard system for m = 3 and n = 5 has no hyperelliptic limit cycles. This shows that the result of theorem 1(c) of Zoladek (1998 Trans. Am. Math. Soc. 350 1681–701) on the existence of algebraic limit cycles of the Liénard system is not correct. Moreover, we characterize all hyperelliptic limit cycles of the Liénard systems for m = 4 and n = 6 or n = 7. For m > 4 and n = 2m − 1 or n = 2m − 2 we prove that there are Liénard systems which have [m/2] − 1 algebraic limit cycles, where the [·] denotes the integer part function.

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

A class of Kukles differential systems of degree five having an invariant conic is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, under perturbations of the coefficients of the systems.

Algebraic limit cycles in polynomial systems of differential equations**The first author is partially supported by a DGICYT/FEDER grant number MTM2005-06098-C02-01 and by a CICYT grant number 2005SGR 00550. The second author is partially supported by the Spanish grant SAB2005-0029, NSF of China (no 10571184) and SRF for ROCS, SEM.

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 that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4.

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

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of ⩽5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincaré compactification of R 2 . We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.

Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.