Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n.     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

Limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, $m$ and $m+1$, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree $2m$ and degree $2m+1$, respectively. Both of the bounds can be reached for all $m$.     

Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems

In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5.     

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.     

On the limit cycles for a class of generalized Kukles differential systems

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3},$ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x),$ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x),$ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x),$ $g_{k}(x),$ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1},$ $n_{2},$ $n_{3}$ and $n_{4},$ respectively for each $k=1,2,$ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y,$ $\dot{y}=x$ using the averaging theory of first and second order.     

Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LINARD DIFFERENTIAL SYSTEMS

Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y + εh_l~1(x) + ε~2h_l~2(x),y=-x- ε(f_n~1(x)y~(2p+1) + g_m~1(x)) + ∈~2(f_n~2(x)y~(2p+1) + g_m~2(x)),which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials h_l~1 and h_l~2 have degree l;f_n~1and f_n~2 have degree n;and g_m~1,g_m~2 have degree m.p ∈ N and[·]denotes the integer part function.     

Bifurcation of limit cycles from a 4-dimensional center in 1:n resonance

We study the bifurcation of limit cycles from the periodic orbits of a linear differential system in R⁴ in resonance 1:n perturbed inside a class of piecewise linear differential systems, which appear in a natural way in control theory. Our main result shows that at most 1 limit cycle can bifurcate using expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed.     

On the Non-Existence of Periodic Orbits for a Class of Two-Dimensional Kolmogorov Systems

In this paper we introduce an explicit expression of first integral, then we prove the nonexistence of periodic orbits, then consequently the non-existence of limit cycles of two-dimensional Kolmogorov system, where R(x, y), S (x, y), P (x, y), Q(x, y),M (x, y), N (x, y) are homogeneous polynomials of degrees m, a, n, n, b, b, respectively. We introduce concrete example exhibiting the applicability of our result.     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method

By using the averaging method, we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane. More specifically, we analyze the global shape of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits with linear center , . The perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms.     

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.     

Pointwise Approximation By Monotone Sequences of Polynomials

In a recent paper [2], Gal and Szabados obtained, for $f \in C_{\left[ { - 1,1} \right]}$ , sequences {Pn} and{Qn} satisfying Qn(x) ≦ Qn+1(x) ≦ f(x) ≦ Pn+1 ≦ Pn(x)such that $$||P_n (x) - Q_n (x)||\underline { \leqslant 8} \sum\limits_{k = [n/2] - 1}^\infty {k^{ - 1} E_k (f),{\text{ }}n\underline{ \geqslant 4} } $$ under the condition $$\sum\limits_{k = 1}^\infty {k^{ - 1} K_k (f) < \infty } $$ . Xie and Zhou in [4] showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous function, making no condition, in a quite delicate constructive way just by perturbation by constants of a subsequence of the best approximation polynomials. By considering that the pointwise estimate for such type of approximation might be potentially useful in some algebraic approximation cases, one should be interested to establish Jackson type rate. However, this problem is not easy. This paper will present an affirmative answer.     

Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

Numerical computation of periodic orbits that bifurcate from stationary solutions of ordinary differential equations

The computation of periodic orbits of autonomous ordinary differential equations is considered. A new method especially suited for computing limit cycles that bifurcate from stationary solutions is established. This novel approach is very easy to implement, the method requiring the programming of the right-hand side only. Another topic of this paper is the demonstration of how a direct method should be applied in the computation of Hopf bifurcation points. These methods solve boundary-value problems by means of standard software only. The methods are tested on four examples arising in different application areas including chemistry and biology.     

On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Summary For the nonlinear system , which has a family { _h } of closed orbits, we consider perturbations of the type , whereP andQ are arbitrary polynomials. The abelian integralsA(h) corresponding to this family { _h } are investigated. By deriving differential equations forA(h) and proving monotonicity for quotients of abelian integrals, we obtain results on the number of zeros of abelian integrals and, hence, on the number of closed orbits _h which persist as limit cycles of the perturbed system (^*). In particular, a uniqueness theorem for limit cycles of (^*) with quadratic polynomialsP, Q is proved. Moreover, whenP, Q are of arbitrary degree, a lower bound for the possible number of limit cycles of (^*) is derived.     

Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.