Period Annulus Research Articles

The cyclicity of a class of quadratic reversible system of genus one

In this paper, we investigate the bifurcations of limit cycles in a class of planar quadratic reversible system of genus one x ˙ = y + 4 x 2 , y ˙ = - x 1 - 8 3 y under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two.

On the number of limit cycles in quadratic perturbations of quadratic codimension-four centres

This paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of quadratic codimension-four centres Q4. Gavrilov and Iliev set an upper bound of eight for the number of limit cycles produced from the period annuli around the centre. Based on Gavrilov–Iliev's proof, we prove in this paper that the perturbed system has at most five limit cycles which emerge from the period annuli around the centre. We also show that there exists a perturbed system with three limit cycles produced by the period annuli of Q4.

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n = m = 2 we determine the maximum number of critical periods and their location; to the case n = m = 3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.

The cyclicity of the period annulus of the cubic isochronous center

This paper is concerned with limit cycles which bifurcate from periodic orbits of the cubic isochronous center. It is proved that in this situation, the cyclicity of the period annulus under cubic perturbations is equal to four. Moreover, for each k = 0,1, . . .,4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

Global dynamics of a cubic Hamiltonian system with a periodic annulus linked to equilibria at infinity

In this paper, we investigate the global dynamics of a cubic Hamiltonian system with a periodic annulus linked to equilibria at infinity. We apply the techniques of desingularization and normal sectors to discuss those degenerate equilibria at infinity, show the qualitative properties of the system in various cases, and exhibit the global phase portraits for the system.

On a Maximal Number of Period Annuli

We consider equation x′′ + g(x) = 0, where g(x) is a polynomial, allowing the equation to have multiple period annuli. We detect the maximal number of possible period annuli for polynomials of odd degree and show how the respective optimal polynomials can be constructed.

The cyclicity of the period annulus of a quadratic reversible system with one center of genus one

This paper is concerned with a quadratic reversible and non-Hamiltonian system with one center of genus one. By using the properties of related elliptic integrals and the geometry of some planar curves defined by them, we prove that the cyclicity of the period annulus of the considered system under small quadratic perturbations is two. This verifies Gautier's conjecture about the cyclicity of the related period annulus.

CYCLICITY OF SEVERAL QUADRATIC REVERSIBLE SYSTEMS WITH CENTER OF GENUS ONE

By using the Chebyshev criterion to study the number of zeros of Abelian integrals, developed by M. Grau, F. Ma\(\~n\)osas and J. Villadelprat in [2], we prove that the cyclicity of period annulus of the quadratic reversible systems with center of genus one, classified as (r8), (r13) and (r16) by S. Gautier, L. Gavrilov and I. D. Iliev in [1], under quadratic perturbations is two. These results partially give a positive answer to the conjecture 1 in [1].

The cyclicity of the period annulus of a quadratic reversible system with a hemicycle

The cyclicity of the period annulus of a quadratic reversible and non-Hamiltonian system under quadratic perturbations is studied. The centroid curve method and other mathematical techniques are combined to prove that the related Abelian integral has at most two zeros. This gives a proof of Conjecture 1 in [8] for one case.

The cyclicity and period function of a class of quadratic reversible Lotka–Volterra system of genus one

In this paper, we investigate a class of quadratic reversible Lotka–Volterra system of genus one z˙=iz+z2+bz¯2 with b=3/5. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. Moreover, we prove that the period function of its period trajectories is monotone increasing.

The monotonicity and critical periods of periodic waves of the φ 6 field model

In this paper, we study the relationship between period and energy of periodic traveling wave solutions for the φ6 field model. The various topological phase portraits with periodic annulus are given by using standard phase portrait analytical technique. Some analytic behaviors (convexity, monotonicity and number of critical periods) of the period functions associated with periodic waves are investigated. We prove that the period function has exactly one critical period under certain conditions. Moreover, the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.

Bifurcation of Limit Cycles from a Polynomial Degenerate Center

Abstract Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x2 + y2)/2)m and ẏ = x((x2 + y2)/2)m with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center ẋ = −y, ẏ = x.

A Survey on the Inverse Integrating Factor

The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation z ̇ = i z + z 3 , after a small polynomial perturbation. We first show that, under small perturbations of the form ε P 2 m − 1 ( z , z ̄ ) , where P 2 m − 1 ( z , z ̄ ) is a polynomial of degree 2 m − 1 in which the power of z is odd and the power of z ̄ is even, the only possible distribution of limit cycles is ( u , u ) for all values of u = 0 , 1 , 2 , … , m − 3 . Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of z ̇ = i z + z 3 is m − 3 , for m ≥ 4 . Then we consider a perturbation of the form ε P m ( z , z ̄ ) , where P m ( z , z ̄ ) is a polynomial of degree m in which the power of z is odd and obtain the upper bound m − 5 , for m ≥ 6 . Moreover, we show that the distribution ( u , v ) of limit cycles is possible for 0 ≤ u ≤ m − 5 , 0 ≤ v ≤ m − 5 with u + v ≤ m − 2 and m ≥ 9 .

A note on the period function for certain planar vector fields

Consider a smooth planar autonomous differential equation having a period annulus, 𝒫. We present a new criterion to ensure that the period function has at most one critical period on 𝒫. Our result has a compact form when the differential equation is written as . It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation , where f(z) and are holomorphic functions and h is a C 2 smooth real valuated function.

On the number of critical periods for planar polynomial systems of arbitrary degree

We construct a class of planar systems of arbitrary degree n having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with n. As far as we know, the previous results on this subject gave systems having linear growth.

On the finite cyclicity of open period annuli

Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multi-parameter analytic deformation $X_\lambda$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.

Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes

We study the bifurcations of limit cycles in a class of planarreversible quadratic systems whose critical points are a center, asaddle and two nodes, under small quadratic perturbations. By usingthe properties of related complete elliptic integrals and the geometryof some planar curves defined by them, we prove that at most two limitcycles bifurcate from the period annulus around the center. This boundis exact.

Bifurcation of critical periods from Pleshkan's isochrones

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities 𝒷3. In this paper we prove that if we perturb any of these isochrones inside 𝒷3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers 𝒷2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in 𝒷2. We prove that if we perturb three of them inside 𝒷2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.

Period annuli and multiple solutions for two-point BVD

Abstract In this article we consider equations of the type xʺ+g(x) = 0 and xʺ+ƒ(x)xʹ<sup>2</sup> + g(x) = 0. The Neumann boundary value problem is considered. For polynomials f and g we provide the multiplicity results. These results are based on a thorough analysis of a phase plane. The existence of period annuli is concerned.