Class Of Cubic Systems Research Articles

The Number of Limit Cycles for a Class of Cubic Systems with Multiple Parameters

In this paper, we study the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations. By using the Melnikov function theory we obtain that five limit cycles can be bifurcated from a period annulus. We also study the Hopf bifurcation at the center surrounded by the annulus.

Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves

Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves

The study on cyclicity of a class of cubic systems

<p style='text-indent:20px;'>In this paper, we consider a class of cubic systems with polynomial perturbation of the degree at most <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and estimate the upper bound of the number of isolated zeros of its Abelian integral. Furthermore, we obtain the distributions of limit cycles bifurcated from a <inline-formula><tex-math id="M2">\begin{document}$ Z_4 $\end{document}</tex-math></inline-formula>-equivariant system with <inline-formula><tex-math id="M3">\begin{document}$ 5 $\end{document}</tex-math></inline-formula> centers.</p>

A number of limit cycle of sextic polynomial differential systems via the averaging theory

The main purpose of this paper is to study the number of limit cycles of sextic polynomial differential systems (SPDS) via the averaging theory which is an extension to the study of cubic polynomial vector fields in (Nonlinear Analysis 66 (2007), 1707--1721), where we provide an accurate upper bound of the maximum number of limit cycles that SPDS can have bifurcating from the period annulus surrounding the origin of a class of cubic system.

First Integrals and Phase Portraits of Planar Polynomial Differential Cubic Systems with the Maximum Number of Invariant Straight Lines

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.

Solution of center-focus problem for a class of cubic systems

For a class of cubic systems, the authors give a representation of the nth order Liapunov constant through a chain of pseudo-divisions. As an application, the center problem and the isochronous center problem of a particular system are considered. They show that the system has a center at the origin if and only if the first seven Liapunov constants vanish, and cannot have an isochronous center at the origin.

Double Bifurcation of Nilpotent Focus

In this paper, an interesting bifurcation phenomenon is investigated — a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 ⊃ (1 ∪ 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z2 equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.

Renormalisation group and isochronous oscillations

We show how the condition of isochronicity can be studied for two dimensional systems in the renormalization group (RG) context. We find a necessary condition for the isochronicity of the Cherkas and another class of cubic systems. Our conditions are satisfied by all the cases studied recently by Bardet et al \cite{bard} and Ghose Choudhury and Guha

Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center

In this paper we deal with a cubic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have 5 limit cycles by using bifurcation theory.

The center conditions and bifurcation of limit cycles at the infinity for a cubic polynomial system

Abstract In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system.

The generalized center problem of resonant infinity for a polynomial differential system

This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for p : − q resonant infinity.

ON THIRD-ORDER NILPOTENT CRITICAL POINTS: INTEGRAL FACTOR METHOD

For third-order nilpotent critical points of a planar dynamical system, the problem of characterizing its center and focus is completely solved in this article by using the integral factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic systems under small perturbations, it is proved that there exist eight small-amplitude limit cycles created from a nilpotent critical point.

NEW STUDY ON THE CENTER PROBLEM AND BIFURCATIONS OF LIMIT CYCLES FOR THE LYAPUNOV SYSTEM (I)

The center problem and bifurcations of limit cycles for the planar dynamical system with double zero eigenvalues are studied. The concepts of focal values and Lyapunov constants as well as their computational method are given. As an example, for a class of cubic system, the corresponding results are obtained.

Uniqueness of limit cycles for a class of cubic system with an invariant straight line

A class of cubic system with an invariant straight line as given in an equation is studied. It is proved that this system has at most, one limit cycle surrounding a weak focal O ( 0 , 0 ) , and has at most, one limit cycle surrounding weak focal N ( 1 n , 0 ) .

The Limit Cycles for a Class of Cubic Systems II

The Limit Cycles for a Class of Cubic Systems II

Acoustic waves in (1 1 0) layered structures

We study the acoustic waves of layered structures formed by slabs of cubic crystals grown along the [1 1 0] direction. We consider structures formed by AlN, GaN and InN in the zinc-blende structure because of the possible applications of these materials, although the results obtained here are valid for a wider class of cubic systems. The anisotropy of the materials has been taken into account and the different propagation directions including symmetry directions and general directions have been considered. The dispersion relations for these propagation directions have been obtained. The coupling between the spatial components of the elastic displacements is different for symmetry and general propagation directions. The spatial localization of the modes in the structures is presented also.

Limit cycle containing nine critical points in its interior for a class of cubic systems

Discuss a class of real planar cubic systems with a critical point O(0,0) of nine orders and obtain the conditions for its limit cycle surrounding the origin, and prove that when small pertubations of coefficients are made, the critical point O(0,0) of nine orders is split into nine real simple critical points and the limit cycle surrounding the origin becomes the limit cycle containing nine critical points in its interior.

A Class of Cubic Systems with Two Centers or Two Foci

In this paper, the necessary and sufficient conditions for a class of cubic differential systems to possess two centers are given. Some conditions for the systems to have two weak foci are also derived by using the computer algebra system Maple.

Centers and Foci for a Class of Cubic Systems

Definition 2. The equilibrium O(0, 0) of system (1) is called a focus of order n if vk = 0, k = 1, . . . , n− 1, and vn 6= 0. The existence of cubic systems with foci of orders 6, 7, and 8 was proved in [1], [2], and [3], respectively. It was shown in [4] that there exist cubic systems with 11 small-amplitude limit cycles. Therefore, there exist cubic systems for which O(0, 0) is a focus of order 11. In the present paper, we obtain necessary and sufficient conditions for the point O(0, 0) to be a center of system (1) and prove the existence of a system of the form (1) for which O(0, 0) is a focus of order 12. The software package Mathematica has been used in the computations. For system (1), we have a 1 : −1 resonance [5]. In (1), we set X = x+Cy and Y = y. Then we obtain the system

Conditions for origin to be a center and the bifurcation of limit cycles in a class of cubic systems

Conditions for origin to be a center and the bifurcation of limit cycles in a class of cubic systems