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In this paper, we study the number of limit cycles in the family d H − ε ω = 0 , where H = y 2 2 − ∫ 0 x g ( u ) d u , ω = y f ( x ) d x , with g ( x ) = x ( x 2 − 1 ) ( x 2 − 1 4 ) 2 , and f ( x ) an even polynomial of degree 10. We will consider mainly the bifurcation of limit cycles near the eye-figure loop and the center of d H = 0 . Our investigation focuses on the lower bound of the maximal number of limit cycles for these systems. In particular, we show that the perturbed system can have at least 8 limit cycles when deg ⁡ ( f ( x ) ) = 10 . [ABSTRACT FROM AUTHOR]

In this work we study a class of planar Hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles. Employing Melnikov functions of any order, we show that the class of systems can have at most three limit cycles and the limit is reached. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce and study the notion of homogeneous Tauberian polynomial, aiming at extending the concept of Tauberian operator. Such notion is characterized in terms of the polynomial topology for which we prove a Banach–Alaoglu type theorem. A number of examples show that the behavior of Tauberian polynomials differs from that of Tauberian operators. [Copyright &y& Elsevier]

Abstract: In this paper we study a planar system of the form , , where is a polynomial of degree in , and are polynomials in with degree of and , respectively. Let denote the maximal number of limit cycles having odd multiplicities of this system. We give some lower bounds of . [Copyright &y& Elsevier]

Abstract: In this paper, we study the number of small amplitude limit cycles in arbitrary polynomial systems. It is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating Lyapunov constants and determining the order of the corresponding Hopf bifurcation. It is well known that the difficulty in calculating the Lyapunov constants increases with the increasing of the degree of polynomial systems. So, it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree , which is denoted by . In this paper, by applying the method developed by C. Christopher and N. Lloyd in 1995, and M. Han and J. Li in 2012, we first obtain the lower bounds for , and then prove that if . Finally, we obtain that grows as least as rapidly as for all large (it is proved by M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differential Equations 252 (2012) 3278–3304 that the number of all limit cycles in arbitrary polynomial systems with degree grows as least as rapidly as ). [Copyright &y& Elsevier]

Abstract: For sufficiently small, we consider the planar polynomial differential system , where and are polynomials of degree . By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system. [Copyright &y& Elsevier]

Abstract: In this paper we study the number of limit cycles bifurcating from isochronous surfaces of revolution contained in , when we consider polynomial perturbations of arbitrary degree. The method for studying these limit cycles is based on the averaging theory and on the properties of Chebyshev systems. We present a new result on averaging theory and generalizations of some classical Chebyshev systems which allow us to obtain the main results. [Copyright &y& Elsevier]

Abstract: From a delta series and its compositional inverse , Hsu defined the generalized Stirling number pair . In this paper, we further define from and the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours. [Copyright &y& Elsevier]

Abstract: In this paper, by using q-Volkenborn integral, we construct new generating functions of the new twisted -Bernoulli polynomials and numbers. By applying the Mellin transformation to these generating functions, we obtain integral representations of the new twisted -zeta function and twisted -L-function, which interpolate the twisted -Bernoulli numbers and generalized twisted -Bernoulli numbers at non-positive integers, respectively. Furthermore, relation between twisted -zeta function and twisted -L-function are proved. Some new relations, related to twisted -Bernoulli polynomials and numbers, are given. [Copyright &y& Elsevier]