By using zeros of elliptic integrals we establish an upper bound for the number of limit cycles that emerge from the period annulus of the Hamiltonian XH in the system X, = XH + e{P,Q), where H -y 7 + x* and P, Q are polynomials in x, y, ( N . \ as a function of the degrees of P and Q. In particular, if (P,Q) = I ^a.ix ',0 I \i = 2 ) with N = 2k + 1 or 2* + 2, this upper bound is Jf c -1.
We consider the Liénard's equation with f(x) polynomial. By using the generalized polar coordinates we establish the maximum possible number of small amplitude limit cycles of such equation in terms of the degree of f(x).
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