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 On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation Jihua Yang Keywords:piecewise smooth Hamilton system; limit cycle; generalized homoclinic loop; Melnikov function Abstract: This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give the upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than the case with one straight line of separation and more limit cycles can be bifurcated.