Period Annulus Research Articles

Every Period Annulus is Both Reversible and Symmetric

We prove that for every planar differential system with a period annulus there exists a unique involution \(\sigma \) such that the system is \(\sigma \)-symmetric. We also prove that, given a system with a period annulus and a global section \(\delta \), there exist a unique involution \(\sigma \) such that the system is \(\sigma \)-reversible and \(\delta \) is the fixed points curve of \(\sigma \). As a consequence, every system with a period annulus admits infinitely many reversibilities.

Limit Cycles Near a Piecewise Smooth Generalized Homoclinic Loop with a Nonelementary Singular Point

In this paper, we deal with limit cycle bifurcations by perturbing a piecewise smooth Hamiltonian system with a generalized homoclinic loop passing through a nonelementary singular point. We first give an expansion of the first Melnikov function corresponding to a period annulus near the generalized homoclinic loop. Then, based on the first coefficients in the expansion we obtain a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered.

Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.

Equivalence of the Melnikov Function Method and the Averaging Method

There is a folklore about the equivalence between the Melnikov method and the averaging method for studying the number of limit cycles, which are bifurcated from the period annulus of planar analytic differential systems. But there is not a published proof. In this short paper, we prove that for any positive integer k, the kth Melnikov function and the kth averaging function, modulo both Melnikov and averaging functions of order less than k, produce the same number of limit cycles of planar analytic (or $$C^\infty $$ ) near-Hamiltonian systems.

Cubic perturbations of elliptic Hamiltonian vector fields of degree three

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field XεXε:{x˙=Hy+εf(x,y)y˙=−Hx+εg(x,y),H =12y2 +U(x) which bifurcate from the period annuli of X0 for sufficiently small ε. Here U is a univariate polynomial of degree four without symmetry, and f,g are arbitrary cubic polynomials in two variables.We take a period annulus and parameterize the related displacement map d(h,ε) by the Hamiltonian value h and by the small parameter ε. Let Mk(h) be the k-th coefficient in its expansion with respect to ε. We establish the general form of Mk and study its zeroes. We deduce that the period annuli of X0 can produce for sufficiently small ε, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of X0 of higher degrees are also studied.

Non-integrability of measure preserving maps via Lie symmetries

We consider the problem of characterizing, for certain natural number m, the local Cm-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.

A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system

In this paper we study a class of planar piecewise smooth quadratic integrable non-Hamiltonian systems, which have a center. By using the averaging method, we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n. Our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems.

Essential perturbations of polynomial vector fields with a period annulus

Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincare--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.

On the limit cycles of planar polynomial system with non-rational first integral via averaging method at any order

In this paper, we consider a class of cubic planar polynomial differential system with non-rational first integral. Using the averaging method at any order, we bound the maximum number of limit cycles which bifurcate from the periodic annulus of the origin when we perturb them inside the class of all polynomial systems of degree n.

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka–Volterra type with a rational first integral of degree two

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree n. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using Chebyshev criterion, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for n=3. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations (u,v) with 0≤u,v≤3, u+v≤5 are realizable.

Limit cycles for a class of discontinuous planar quadratic differential system

In this paper, we consider the number of limit cycles for a class of discontinuous planar quadratic integrable non-Hamiltonian system under quadratic perturbation. Using the rst order averaging method, we obtain that there are at least 5 limit cycles which can bifurcate from the period annulus of the center for this system. Our result also shows that the discontinuous quadratic system can have at least 2 more limit cycles than the smooth one.

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

In this paper, we deal with the following differential system [Formula: see text] where p, q are positive integers, and P(x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n - 1)p1 + (t + 1)q - 1 + 2rp1q1(q + 3) + 2tqrp1q1, where t = [n/2q] + 2, (p, q) = r(p1, q1), p1 and q1 are coprime.

The cyclicity of period annuli of a quadratic reversible Lotka-Volterra system with two centers

The cyclicity of period annuli of a quadratic reversible Lotka-Volterra system with two centers

Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe-α|x-ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Maple.

Perturbations of symmetric elliptic Hamiltonians of degree four in a complex domain

The cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator is a well known problem extensively studied in the literature. In the present paper we provide a complete bifurcation diagram for the number of the zeros of the associated Melnikov function in a suitable complex domain.

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.

Algebraic and analytical tools for the study of the period function

In this paper we consider analytic planar differential systems having a first integral of the form H(x,y)=A(x)+B(x)y+C(x)y2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results.

Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible Lotka-Volterra system with two centers of genus one under small quadratic perturbations. It shows that the cyclicities of each period annulus and two period annuli of the considered system under small quadratic perturbations are two, respectively. This not only gives at least partially a positive answer to an open conjecture, but also improves the corresponding results in the literature. In addition, we present the configurations of limit cycles of the perturbed system as (2, 0), (1, 1), (1, 0), (0, 2), (0, 1) and (0, 0), where $(i,\, j)$ indicates that the perturbed system has $i$ limit cycles surrounding the positive singularity while it has $j$ limit cycles surrounding the negative one.

Limit Cycles of Perturbed Cubic Isochronous Center via the Second Order Averaging Method

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.