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a b s t r a c tIn this paper we consider the bifurcation of limit cycles of the systemẋ = y(for ε sufficiently small, where a, b ∈ R − {0}, and P, Q are polynomials of degree n, we obtain that up to first order in ε the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by ε = 0 are (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and n − 1 if a = b. Moreover, there are systems with at least (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and, n − 1 limit cycles if a = b.

In this paper, the numerical solution of one-dimensional linear hyperbolic telegraph equation, which describe wave propagation of electric signals in a cable transmission line, is proposed employing the well known homotopy perturbation method(HPM) and laplace transform(LT). Using Laplace transform scheme, the problem is converted into a partial differential equation without any differentiation respect to time. Homotopy perturbation method is applied in the Laplace transform domain. We use Stehfest's numerical inversion algorithm of Laplace transform to retrieve the time domain solution. The performance was found to be very good. The approximate solutions are in excellent agreement with those by the Chebyshev spectral collocation method (CSCM) and the method uses interpolating scaling functions. To illustrate the method some examples are provided. The results show the simplicity and the efficiency of the method.

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In this paper, by using variational methods, we prove the existence for a class of fractional Hamiltonian systems with Liouville-Weyl fractional derivativeswhere 2 ð1=2, 1, t D 1 and À1 D t are the right and left inverse operators of the corresponding Liouville-Weyl fractional integrals of order respectively, u 2 R, b : R ! R, b 2 L 1 ðRÞ, inf t2R bðtÞ 4 0, , are real parameters and f : R Â R ! R is a function that satisfies some suitable conditions.