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.     

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.     

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.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

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.     

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.     

一类广义Liénard方程的概周期解

研究广义 Liénard方程: x'=φ(y)-F(x), y'=-g(x)+p(t), 利用 Amerio的结果证明方程的解部分变元的最终有界性意味着概周期解的存在性, 推广了Cieutat^[1]的主要结果.     

The continuation of solutions for certain nonlinear differential equations and its applications

We give some sufficient conditions for the continuation of solutions for some nonlinear differential equations. As an application, we obtain a new criterion for the oscillation of solutions of the Liénard 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.     

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.     

Oscillatory and phase dimensions of solutions of some second-order differential equations

In order to measure fractal oscillatority of solutions at t=∞, we define oscillatory and phase dimensions of solutions of a class of second-order nonlinear differential equations. The relation between these two dimensions is found using formulas for box dimension of chirps and nonrectifiable spirals. Applications include the Liénard equation and weakly damped oscillators.     

Quantization of quadratic Liénard-type equations by preserving Noether symmetries

The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (Nucci 2011) [28] that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in Choudhury and Guha (2013) [6].     

On a Multivalued Version of the Sharkovskii Theorem and its Application to Differential Inclusions

Motivated by the applications to differential equations without uniqueness conditions, we separately prove multivalued versions of the celebrated Sharkovskii and Li–Yorke theorems. These are then applied, via multivalued Poincaré operators, to Carathéodory differential inclusions. Thus, besides another, infinitely many subharmonics of all integer orders can be obtained. Unlike in the single-valued case, for example, period three brings serious obstructions. Three counter-examples, related to these complications, are therefore presented as well. In a multivalued setting, new phenomena are so exhibited.     

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.     

Fixed points,Volterra equations,and Becker’s resolvent

Summary In a recent paper we derived a stability criterion for a Volterra equation which is based on the contraction mapping principle. It turns out that this criterion has significantly wider application. In particular, when we use Becker’s form of the resolvent it readily establishes critical resolvent properties which have been very illusive when investigated by other techniques. First, it enables us to show that the resolvent is L¹. Next, it allows us to show that the resolvent satisfies a uniform bound and that it tends to zero. These properties are then used to prove boundedness of solutions of a nonlinear problem, establish the existence of periodic solutions of a linear problem, and to investigate asymptotic stability properties. We also apply the results to a Liénard equation with distributed delay and possibly negative damping so that relaxation oscillations may occur.     

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.     

无限时滞的随机泛函微分方程解的渐近性质

本文研究了无限时滞随机泛函微分方程解的存在唯一性,矩有界性的问题.利用Lyapunov函数法以及概率测度的引入得到了确保方程解在唯一、矩有界、时间平均矩有界同时成立的一个新的条件.推广了Khasminskii-Mao定理的相关结果.     

Small-amplitude limit cycles of some Liénard-type systems

As we know, the Liénard system and its generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered by most people is the number of limit cycles. In this paper, we investigate two kinds of Liénard systems and obtain the maximal number (i.e. the least upper bound) of limit cycles appearing in Hopf bifurcations by applying some known bifurcation theorems with technical analysis.     

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.