در اين پايان نامه، يك مدل شكارچي-شكار با دفاع القايي و بيماري در شكار كه از ديدگاه تكامل زيستي و اكو-اپيدميولوژي ساخته شده است، مورد مطالعه قرار مي‌گيرد. هدف اصلي اين پايان نامه بررسي تأثير بيماري بر پايداري جمعيت در مدل شكارچي-شكار با دفاع القايي است. ابتدا، مثبت بودن و كرانداري يكنواخت جواب‌هاي مدل اشاره شده اثبات مي‌شود. سپس، وجود و پايداري نقاط تعادل، بخصوص نقاط تعادل همزيستي شكار و شكارچي مورد مطالعه قرار مي‌گيرد. در اين مدل حداكثر نه نقطه تعادل وجود دارد. از يك تبديل پارامتري پيچيده براي بررسي ويژگي‌هاي نقاط تعادل همزيستي مدل استفاده مي‌كنيم. همچنين شرايط كافي براي وجود انشعاب هاپف بدست مي‌آيد. براي تحليل انشعاب هاپف و تعيين جهت سيكل‌هاي حدي در مدل شكارچي-شكار با دفاع القايي و بيماري، ضريب اول لياپانوف در مقادير بحراني پارامتر انشعاب محاسبه مي‌شود. در خاتمه شبيه‌سازي‌هاي عددي براي تكميل نتايج تحليلي انجام مي‌شود  

We study a multi-group SAIRS-type epidemic model with vaccination. the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al(2021) to a network of communities. We provide a global stability analysis for the model. We determine the value of the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable if R0<1. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when R0=1. Moreover, if R0>1, the disease-free equilibrium is unstable and a unique equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies. 

  In this thesis, we examine theorems about bifurcation from one-dimensional kernels and generalizations of the Crandell-Rabinowitz theorem. Next, using a bifurcation theorem from one dimensional kernels, the bifurcation in Activator-inhibitor systems are discussed. Also, the effect of nonlocal diffusion on bifurcations and the formation of spatially heterogeneous patterns in the case when the rate of dispersion of the inhibitor is small enough, is study. Unstable steady state solutions and existence of Turing instability for the mentioned nonlocal systems is investigated.

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.Keywords: Bagley–Torvikequation; Chebyshev polynomials; Collocation method; Liouville–Caputo derivative

Encompassing many standard notions such as Dedekind finite and reversible rings we introduce and study a new property for subsets of a ring. We give many examples and characterize some rings such as 2-primal rings with the aid of this notion.In ‎addition‎, We study some properties relatedto zero divisors and reversibility in noncommutative rings .  

  AbstractIn this thesis we study a host-generalist parasitoid model with Holling II functional response where the generalistparasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence ofbifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and adegenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics suchas the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover,there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity ofthe generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation bythe generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacityof the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in theform of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e.,whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populatio   (iii) inboth cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.-parasitoid model; Bogdanov-Takens bifurcation; Hopf bifurcation; Invasion; PersistencKeywords:Hoste; Extinction.

In the thesis, we investigate the well-posedness for abstract Cauchy problems with finite delay. We also study asymptotically equivalence of evolution equations and delay evolution equations.

  In this thesis will first introduce the basic concepts of dynamical systems, differential equa- tions, Philips systems and Poincaré mapping. In chapter two presents a standard form that includes all possible configurations in planar linear systems, and it is shown that the exis- tence of a focus in one zone is sufficient to get three nested limit cycle, independently on the dynamicsoftheanotherlinearzone. Inchapterthree,adiscontinuousplanarpiecewiselinear differential system in R is assumed. if it has a center, real or virtual, then the discontinuous planar piecewise linear differential system having at most two limit cycles.

  Let G be a finite group and H be a subgroup of G. The Chermak-Delgado measureof H with respect to G is defined as mG(H) =|H| |CG(H)|. The set of all subgroupsof G with maximal Chermak-Delgado measure, denoted CD(G), is a sublattice of thelattice of all subgroups of G. In this thesis, we prove that if H   CD(G) then H issubnormal in G and prove if K is a finite group then CD(G K) = CD(G) CD(K).Also, we describe CD(G?Cp) where G has a non-trivial center and p is an odd primeand determine conditions for a wreath product to be a member of its own Chermak-Delgado lattice. Finally, we characterize the structure of finite groups G whoseChermak– Delgado lattice is the interval[G=Z(G)] = {H   L(G)|Z(G) H   G}.

  دراين پايان نامه وجود جواب هاي ضعيف براي يك دستگاه معادلات كيرشهف كسري منفرد و ‎دسته‎ اي از معادلات كيرشهف كسري منفرد با توابع وزن داراي تغيير علامت، با استفاده از روش منيفلد نهاري مورد بررسي قرار مي گيرد. ‌‌‌‌‌‌‌‌‌براي اينكار متناظر با نقاط بحراني نگاشت هاي تاري، منيفلد نهاري به سه زيرمجموعه تقسيم مي شود. به علاوه ثابت مي شود تابعك انرژي روي منيفلد نهاري بازدارنده و از پايين كراندار است و مينيمم هاي موضعي روي منيفلد نهاري نقاط بحراني تابعك انرژي هستند. سپس با نشان دادن وجود دو مينيمم موضعي روي زير مجموعه هاي متناظر با نقاط ماكسيمم و مينيمم نگاشت هاي تاري، وجود دو نقطه بحراني براي تابعك انرژي اثبات خواهد شد.

AbstractIn this thesis, first some definitionsand elementary concepts of analysis and fractional calculus are discussed.Then, by using various fixed point theorems,such as Banach, Krasnoseleskii, Schauder, Leray- Schauder, Leray-Schauder'sNonlinear Alternative and Avery-Peterson, the existence of positive solutionsfor a fractional boundary value problem with boundary conditions involvingReimann-Stieltjes integral is investigated.At the end, the existence anduniqueness of solutions for an integro-differential boundary value problem of Caputotype with nonlocal boundary conditions is studied.  

در اين پايان نامه وجود جواب معادلات ديفرانسيل كسري انتقال-انتشار با استفاده آناليز غير خطي، روش هاي تغييراتي و قضيه گذرگاه كوهستاني بررسي مي شود

اخيرا يكي از موضوعات مورد علاقه رياضي­دانان، مطالعه معادلات ديفرانسيل ضربه­اي به خصوص با عملگرهاي مشتق كسري( مشتق كسري ريمان- ليوويل و كاپاتو) بوده است به منابع [3-4] رجوع شود. با وجود اين وجود مطالعات جزئي روي جواب، يكتايي جواب و پايداري جواب ها براي معادله ديفرانسيل كسري با مشتق هيلفر انجام گرفته است، زيرا كه مشتق كسري هيلفر شكل كلي تري از مشتقات كسري ريمان- ليوويل را ارائه كرده است كه به صورت خلاصه ميتوان آن را يك درونيابي بين مشتق كاپاتو و مشتق ريمان- ليوويل فرض كرد. حال در اين پايان نامه، وجود جواب و پايداري معادلات ديفرانسيل كسري ضربه اي با عملگر مشتق كسري هيلفر مورد مطالعه و بررسي قرار خواهد گرفت.

  In this thesis, an interactive method for solving discrete multi-criteria optimization problems is studied. This methods is based on the pairwise comparison of alternatives to obtain an optimal solution. It is assumed that there are p criteria, m alternatives and a single decision maker who has an implicit increasing quasi-concave utility (value) function that has to be optimized. The main point in this regard is designing a procedure   with lower number of comparison made by the decision make. To this aim, convex cones are used for ranking the alternatives. In the sequel, two methods based on the dual theory and evolutionary algorithms are introduced. These methods obtain an optimal solution by reducing the number of   pairwise comparison and using the decision maker's information.

  AbstractThe concept of a module M being almost N-injective, where N is some module, was introduced by Baba (1989). For a given module M, the class of modules N, for which M is almost N-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module U is almost V-injective, where V is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of V. Let M be a uniform module and V be a finite direct sum of indecomposable modules. Some conditions under which M is almost V-injective are determined, thereby Baba’s result is generalized. A module M that is almost M-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.Keywords: Almost injective module, Direct sum, Finite length module, Uniform module, Indecomposable module, Self-injective module, Indecomposable ring, Von Neumann regular rin

  ‎The ‎objective ‎of ‎this ‎thesis ‎is ‎to ‎obtain ‎sufficient ‎condition‎s for the dynamics of a vector field on the two dimensional center manifold to be topologically equivalent with the versal deformation of two dimensional Bognanov-Takens bifurcation. Next, using the results of this part, the bursting behaviours and the related bifurcations of the neurons in the Pre-Botzinger Complex is investigated.

We obtain ‎ relation between an exact   g -frame and a   g -Riesz basis under some conditions. We alsoprove that the stability of an exact   g -frame for a Hilbert space under perturbation is ‎di‎fferent to that of a   g -frame. These properties of exact   g -frames for Hilbert spaces are not similar to those of exact frames. then we mainly discuss the exact   K-g -frames in theHilbert spaces.  

در اين پايان نامه به بررسي مسئله ي تعادل عملگر تعميم يافته پرداخته شده است ودر ادامه مسئله ي شبه تعادل را مورد بررسي قرار داده وسپس كاربردهاي مسئله ي شبه تعادل در مسئله ي شبه بهينه سازي و مسئله ي نابرابري شبه تغييراتي ارايه شده است.

Firstly we explore some properties of K-frame  in terms of decompositions of operators. Then we  pay more attention to characterize the dual K-Bessel sequences of a given K-frames.                                                                                  

‎In‎ this thesis, we consider the multiplicity and existence of solutions for some kirchhoff-type equations by variational method and critical point theory.Also, we have studied the existence of positive solutions for some partial differential equations.

  ‎In this thesis‎, ‎using variational methods‎, ‎the existence of weak solutions for two margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎For this purpose‎, ‎using certain conditions‎, ‎it is proved that the energy functional satisfies the Palais-Smale condition‎. ‎Then‎, ‎using the Ambrosetti-Rabinowitz condition and the symmetric Mountain Pass Theorem‎, ‎the existance of two nontrivial solutions for the considered equation is proved‎. ‎Also‎, ‎using Lions theorem‎, ‎the concentration of solutions‎, ‎is investigated‎. ‎As an application of the obtained results‎, ‎the existence and concentration of solutions to an elliptic equation with convex-concave nonlinearity is established‎. ‎Finally‎, ‎using the symmetric Mountain Pass Theorem‎, ‎the existence of infinitely many weak solutions to a system of Schrodinger–Kirchhoff equations of fourth-order is investigated

‎A ring with unity is called ni-clean if every element can be expressed as sum of a nilpotent and an idempotent‎. ‎In this thesis‎, ‎we characterize the nil clean matrix rings over fields‎, ‎in fact‎, ‎we prove that for a field $F$ the ring $M_n(F)$ is nil-clean if and only if $F\\cong \\  {Z}_2$‎. ‎As an application‎, ‎we obtain a complete characterization of the finite rank Abelian groups‎ ‎with nil clean endomorphism ring‎. ‎ ‎For a finite commutative ring $R$‎, ‎the nil-clean graph $G_N(R)$ is a simple graph such that the vertex set is the ring $R$ and two ring elements $a$ and $b$ are adjacent if $a+b$ is nil clean in $R$‎. ‎Graph theoretic properties like girth‎, ‎dominating set‎, ‎diameter etc‎. ‎of nil clean graph have been studied.  

 arabolic equations are an important class of partial diferential equations which have many applications in science. Since these equations dont have exact solution, Their numerical solutions have at- tracted lots of researchers. A big class of these equations are Burgers equations. In this thesis some numerical schems based on Splitting methods are derived for this kind of equations. Also some Splitting methods with higher orders for solving a wide range of parabolic equation will be investigated.  

در اين پايان نامه كرانداري وفشردگي عملگرهاي انتگرال ريمان-ليوويل روي فضاهاي موري را بررسي مي كنيم0مشخصه سازي جواب براي معادله انتگرال آبل بدست اورده وباكمك نتايج قضيه نقطه ثابت وجوديكتايي جواب براي مسئله كوشي ثابت خواهد شد  

  ‎This thesis examines the Turing patterns in the Gierer–Meinhardt models‎. ‎The stability of equilibrium points in a system without diffusion is studied‎. ‎Moreover‎, ‎the suficient condition for this system to have Hopf bifurcation is carefully stuided‎. ‎By adding diffusion terms it is shown that‎, ‎influenced by different diffusion coefficients‎, ‎both the stable equilibrium and the bifurcated limit cycle of the Hopf bifurcation are affected by the Turing instability‎. ‎Besides‎, ‎in each case‎, ‎Turings conditions of instability are presented‎.

بررسي اختلال عملگر قاب هاي باناخ در فضاهاي باناخ

‎Let $G$ be an $n$-vertex graph with $m$ edges and with Laplacian spectrum $\\mu_1 \\geq \\mu_2 \\geq‎‎\\cdots\\geq \\mu_{n?1} \\geq \\mu_n = 0$‎. ‎The Laplacian energy is defined as $LE =\\sum_{i=1}^n|\\mu_i-\\frac{2m}{n}|$‎.‎In this thesis‎, ‎all graphs with at most four distinct laplacian eigenvalues are studied‎. ‎Also‎, ‎we use these graphs to obtain some upper and lower bounds for the Laplacian energy of an arbitrary graphs‎.‎Among other results‎, ‎we characterize all such graphs which are bipartite or‎‎have exactly one multiple Laplacian eigenvalue‎.‎Let $\\sigma$ be the‎‎largest positive integer such that $\\mu_\\sigma \\geq \\frac{2m}{n}$‎. ‎The graphs satisfying $\\sigma = n‎ -‎1$ are characterized‎.‎Using this‎, ‎we obtain lower bounds for $LE$ in terms of $n‎, ‎m$‎, ‎and the first Zagreb index‎. ‎In‎‎addition‎, ‎some upper bounds for $LE$ in terms of graph invariants such as $n‎, ‎m$‎,‎maximum degree‎, ‎vertex cover number‎, ‎and spanning tree packing number are presented‎.‎Finally‎, ‎we obtain a relation between Laplacian‎‎energy and Laplacian-energy-like invariant of graphs.

In recent years numerical solution of random ordinary di?erential equations has attracted lots of researchers. Inthisthesisat?rstsomelinearmulti-stepmethodsarederived,andthenundersome assumptions their local error order are established. Then pathwise convergence and B-stabilityofthesemethodsareobtained. Inthefollowing,someimplicitschemesare presented for the pathwise simulation of sti? ordinary di?erential equations, specif- ically an implicit averaged Euler scheme and an implicit averaged midpoint scheme will be considered. Convergence and B-stability proofs of these averaged methods are presented and the numerical schemes are tested for some medical example.

In this thesis first, some concepts and theorems of sobolev space are explianed.Then the existence of nontrivial wenk solutions for a class of quasilinear Schrodinger

  In this thesis, first the definition and elementary concepts of analysis and fractionalcalculus is discussed.Then, by using a various fixed point theorems, such as Banach, Krasnoseleskii, Nonlinearcontractions, Leray-schauder Nonlinear Alternative and Leray-schauder degree,the existence and uniqueness of the solutions for a fractional boundary value probleminvolving Erdelyi-Kober operator is investigated.At the end, by using the conception of measures of noncompactness and the Darbotheorem, the existence of a solution for a >involving Erdelyi-Kober operator is discussed.

در اين پايان نامه نخست مفهوم انتگرال كسري ريمانليويل وانتگرال كسري اردلي كوبر بيان مي گردد در ادامه وجود جواب هاب معادلات ديفرانسيل كسري با استفاده از قضاياي نقطه ثابت را بررسي مي كنيم

  This‎ thesis is devoted to the study ‎of‎ properties and ‎‎‎irre‎d‎ucible‎ representations of ‎Le‎ibniz ‎algebras‎.‎ We use the liezation method to translate some fandamental theorems of Lie ‎‎algebras‎‎ ‎such‎ ‎as‎‎ the Engels theorem, the Levi decomposition, ... to Liebniz ‎algebras.‎ Also, we prove that any irreducible representation of ‎a‎ Leibniz algebra can be obtained from an irreducible representation of the ‎semi-‎simple Lie algebra from the Levi decomposition. As an application, we determine the irreducible representations of sl‎2.‎‎‎

  در سالهاي اخير متيو فضاهاي متريك جزيي وشبه متريك و شبه متريك وزن دار را معرفي كرد . ايشان به ارتباط بين آنها پرداخت وبيان كرد كه تحت چه شرايطي ميتوان فضاي متريك جزيي وشبه متريك را يه يك متر تبديل كرد. يكي از اهداف پايان نامه اين است كه با استفاده از يك نوع خاص از انقباض ضعيف معرفي شده توسط چريچ وبا كمك گرفتن از توابع كنترل ناپيوسته قضييه نقطه ثابت باناخ را به فضاهاي متريك جزيي گسترش  مي دهيم.

In this thesis, we study the fractional calculus and fractional differential equations with Hadamard derivatives, and includes the following parts: in the first chapter, some properties, definitions and theorems of fractional calculus, nonlinear Analysis and fixed point theorems to be introduced that will be used to prove our main results. In the second chapter, the existence and unique   of solutions for a system of Hadamard type fractional differential equations is derived from Leray-Schauders and fixed point theorems guards will be examined. In the next chapter, the existence and uniqueness of solutions ‎using Banachs fixed point theorem for fractional impulsive equations with Hadamard derivatives studied is derived. in the end chapter, existence of solutions for fractional differential equations involving the Hadamard derivatives studied is derivedKeywordsFractional differential, Hadamard fractional derivatives‎, ‎Banachs-fixed point theorem‎, Leray- Schauders theorem‎, Existence of solutions ‎

  In thesis propose nonpolinomial spline and Hermit nonpolynomial spline interpolation and present method to determine optimal value of parametrs which generate minimum error in approximation and used of functions interpolation such the Fouer Series and the Hermite nonpolynomial cubic spline and nonpolynomial cobic spline and interpolated functions for example Runge s Phenomenon Numerical simulations are carried out for the analisis of error in cubic spline and nonpolynomial interpolations.   In this thesis non-polynomial spline for the numerical solution of two-point boundary value problems and singularly perturbed boundary value problems are studied.And it is reduced to sixth order of non-polynomial spline that is used for solving boundary value of second order singularly perturbed.in addition to in both groups of problems, errors and convergence are analyzed.The numerical example are given to illustrate the efficiency of proposed methods.

  ‎In‎ ‎this‎‎ ‎thesis‎,‎‎ ‎first‎‎ ‎we‎‎ ‎investigate‎,‎ ‎the‎‎ ‎existense‎‎ ‎and‎‎ ‎multiplicity‎‎ ‎o‎‎f‎ ‎positive‎‎ ‎solutions‎‎ ‎to‎‎ ‎an‎ ‎N-‎Laplacian‎‎ ‎equation‎‎ ‎in‎‎‎ ‎$ ‎\\mathbb{R}^{N‎‎}‎‎‎ $‎ ‎with‎‎ ‎singular‎‎ ‎and‎‎ ‎exponential‎‎ ‎nonlinearity‎.‎ ‎To‎‎ ‎do‎‎ ‎this‎,‎ ‎we‎‎ ‎use‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎ ‎method‎.‎ ‎First‎‎, ‎we‎‎ ‎prove‎‎ ‎that‎‎ ‎‎‎local‎‎‎ ‎minima‎‎ ‎of‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎‎ ‎i‎n‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎‎ ‎are‎ ‎critical‎‎ ‎‎points‎‎.‎Then‎ ‎we‎‎ ‎divide‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎‎ ‎in‎‎to‎‎ ‎three‎‎ ‎sets‎‎ ‎corresponding‎‎ ‎to‎‎ ‎local‎‎ ‎maxima‎‎, ‎local‎‎ ‎minima‎‎ ‎and‎‎ ‎points‎‎ ‎of‎‎ ‎inflection‎‎ ‎of‎‎ ‎fibering‎‎ ‎‎maps,‎‎‎ ‎and‎‎ ‎we‎‎ ‎‎find‎‎‎ ‎local‎‎ ‎minima‎‎ ‎of‎‎ ‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎ ‎in‎‎ ‎these‎‎‎‎ ‎sets‎.‎ ‎In‎‎ ‎the‎‎ ‎next‎‎ ‎part‎‎‎ ‎of‎‎ ‎this‎‎ ‎thesis‎,‎ ‎we‎‎ ‎study‎‎ ‎the‎‎ ‎existence‎‎ ‎of‎‎ ‎a‎ ‎nontrivial‎‎ ‎solution‎‎ ‎for‎‎ ‎a system‎‎ ‎of‎‎ ‎p‎‎-‎Laplacian‎‎‎ ‎equation‎‎ ‎in‎‎ ‎a‎ ‎bounded‎‎ ‎domain‎‎ ‎and‎‎ ‎under‎‎ ‎the‎‎ ‎Dirichlet‎‎‎ ‎bounda‎ry‎‎ ‎condition‎.‎ ‎For‎‎ ‎this‎‎ ‎problem‎‎‎ ‎we‎‎ ‎prove‎‎ ‎that‎‎‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎ ‎has‎‎ ‎the‎‎ ‎geometry‎‎ ‎of‎‎ ‎mountain‎ ‎pass‎‎. ‎Then‎‎‎‎ ‎using‎‎ ‎the‎‎ ‎S‎addle‎ ‎P‎oint‎‎ ‎Theorem‎‎‎‎ ‎of‎‎ ‎Rabinowitz‎‎‎‎ ‎and‎‎ ‎a‎ ‎generalization‎‎ ‎of‎‎ ‎the‎‎ ‎Landesman‎-‎Lazer‎‎ ‎condition‎,‎ ‎the‎‎ ‎existence‎‎ ‎of‎‎ ‎a‎‎ ‎nontrivial‎‎ ‎solution‎‎ ‎is‎‎ ‎proved‎.‎

  This thesis is devoted to study the number of limit cycles bifurcated from the periodannulus of two polynomial vector fields, under polynomial perturbative of degree n.The analysis is carried out by estimatry the number of zeroes of the correspondingAbelian integrals and averaged function. Chebeyshev criterion is one of the toolsfor deriving sharp upper bound for the number of zeroes of the Abelian integrals.Moreover, the distribution of the bifurcated limit cycles is also considere: