Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system

This paper deals with the analytical property of the first Melnikov function for general Hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the Hamiltonian value corresponding to the loop. The explicit formulas for the first coefficients of the expansion have been given. We prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions. Then we consider the cyclicity of a cuspidal loop in some Liénard and Hamiltonian systems, and determine the number of limit cycles that can bifurcate from the perturbed system.     

A New Criterion for Controlling the Number of Limit Cycles of Some Generalized Liénard Equations

We consider a class of planar differential equations which include the Liénard differential equations. By applying the Bendixson-Dulac Criterion for ?-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign. As an application, this method is used to give a sharp upper bound for the number of limit cycles of some Liénard differential equations. In particular, we present a polynomial Liénard system with exactly three limit cycles.     

The period function of classical Liénard equations

In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree 2?−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ?, can be reduced to the study of slow-fast Liénard equations close to their limiting layer equations. We show that near the central system of degree 2?−1 the number of critical periods is at most 2?−2. We show the occurrence of slow-fast Liénard systems exhibiting 2?−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2?−2 is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even.     

Slow divergence integral and its application to classical Liénard equations of degree 5

The slow divergence integral is a crucial tool to study the cyclicity of a slow–fast cycle for singularly perturbed planar vector fields. In this paper, we deduce a useful form for this integral in order to apply it to various problems. As an example, we use it to prove that the slow divergence integral along any non-degenerate slow–fast cycle for singular perturbations of classical Liénard equations of degree 5 has at most one zero, and the zero is simple if it exists; hence the cyclicity of any non-degenerate slow–fast cycle in this class of equations is at most 2. Up to now there were many interesting results about Liénard equations of degree 3, 4 and ≥6, but almost nothing is known about degree 5. The result in this paper can be seen as a first stage to study the uniform property for classical Liénard equations of degree 5.     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Compactification and desingularization of spaces of polynomial Liénard equations

The paper deals with polynomial Liénard equations of type (m,n), i.e. planar vector fields associated to a scalar second order differential equation , with f and g polynomials of respective degree m and n. It is shown that, besides compactifying the phase plane, or the Liénard plane, one can also compactify and desingularize the space of Liénard equations of type (m,n) for each (m,n) separately, by adding both singular perturbation problems and Hamiltonian perturbation problems.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Limit cycle bifurcations of some Liénard systems

In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.     

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems , 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.     

A note on existence and uniqueness of limit cycles for Liénard systems

We consider the Liénard equation and we give a sufficient condition to ensure existence and uniqueness of limit cycles. We compare our result with some other existing ones and we give some applications.     

Periodic solutions for a Liénard type p-Laplacian differential equation

By using topological degree theory and some analysis skill, we obtain sufficient conditions for the existence and uniqueness of periodic solutions for Liénard type p$p$-Laplacian differential equation.     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.     

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles.     

Perturbation from an elliptic Hamiltonian of degree four—III global centre

The paper deals with Liénard equations of the form , with P and Q polynomials of degree, respectively, 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a global centre. It is proven that the least upper bound of the number of zeros of the related elliptic integral is four, and this is a sharp one.This result permits to prove the existence of Liénard equations of type (3,2) with a quadruple limit cycle, with both a triple and a simple limit cycle, with two semistable limit cycles, with one semistable and two simple limit cycles or with four simple limit cycles.     

A note on: “Relaxation oscillators with exact limit cycles”

In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work [M.A. Abdelkader, Relaxation oscillators with exact limit cycles, J. Math. Anal. Appl. 218 (1998) 308-312]. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle.     

Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems

The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.     

Small amplitude limit cycles for the polynomial Liénard system

We estimate for the maximal number of limit cycles bifurcating from a focus for the Liénard equation , where f and g are polynomials of degree m and n respectively. These estimates are quadratic in m and n and improve the existing bounds. In the proof we use methods of complex algebraic geometry to bound the number of double points of a rational affine curve.