An important problem in a given dynamical system is to determine the existence of a homoclinic orbit. We improve the results of Qin and Xiao [Nonlinearity, 20 (2007), 2305-2317], who present some sufficient conditions for the existence of a homoclinic/heteroclinic orbit for the generalized H@non map. Moreover, an algorithm is presented to locate these homoclinic orbits.
We study the local analytic solutions f of the functional equation f(ψ(zf(z))) = φ(f(z)) for z in some neighborhood of the origin. Whether the solution f vanishes at z = 0 or not plays a critical role for local analytic solutions of this equation. In this paper, we obtain results of analytic solutions not only in the case f(0) = 0 but also for f(0) ≠ 0. When assuming f(0) = 0, for technical reasons, we just get the result for f′(0)≠ 0. Then when assuming f(0) = ω0 ≠ 0, ψ(0) = s # 0, ψ(z) is analytic at z = 0 and ψ(z) is analytic at z = ω0, we give the existence of local analytic solutions f in the case of 0 〈 |sω0| 〈 1 and the case of |sω0| = 1 with the Brjuno condition.