With one (Poincare section) parameter and a particular motiom law (that associated to certain determined point of horno-heteroclinic orbits), the usual f'orm of' the llelni kov f'unction seems to be not convenient f'or certain problems. Another favourable form can be obtained by using a supplementary parameter - the arbitrary time constant in the general mentioned motion law. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 285 · 291 Special Issue Dedicated to the Memory of' Prof. Nguyen Van Dao A NOTE ON THE MELNII 0 is intensity of the external forcing ation p cost of period 27T; k > 0 is the linear damping coefficient. Putting x = x 1 , x = x 2 , the equation () can be written as XI = X2 X2, =XI - xr + c(pcost - kx2). () Hence: () ( ) The unperturbed Hamiltonian system (c = 0) possesses two homoclinic orbits - the right C and the left C' - homoclinic to the saddle (origin) 0, respectively encircling two centers J(l,O) and J'(-1,0), intersecting the abscissa ~is Ox= Ox 1 at A(v12,0) an~ A'(-v12, 0). 288 Nguyen Van Dinh The equation of these orbits is H(x.±) = H(x1, x2) = 0 or ± = x2 = ±~ () where + (-) corresponds to upper (lower)-half homoclinic orbits, respectively. The perturbation c(p cost - kx2) destroys C and C' to generate four invariant manifolds: the right and left unstable manifolds Cn and C~ and the right and left stable one Cs and C~. Below, our study is devoted to the right homoclinic orbit C a nd its corresponding stable and unstable invariant manifolds Cs and Cn· A particular (unpertubed) motion law along C is x(t) = x1 (t) = J2sech(t), ±(t) = x2(t) = -J2sech(t)tanh(t) . () x2(0) = -J2sech(O)tanh(O) = 0. () For t = 0, we have x1(0) = J2sech(O) = J2, Thus, the motion (4 .6) is that associated to the point A( J2, 0). The usual form of the Melnikov function is: +oo M(O) = - J J2sech(t - O)tanh(t - e){pcost + kJ2sech(t - O)tanh(t - e) }dt. (4 .8) -oo It gives the estimation of the distance between Cs and Cn on the Poincare section So at the point A( ./2, 0). As in [1], if we vary e, we can study only the variation of the distance between' Cs and Cn on different Poincare sections So at the same point A. The "general" motion law along C is: x(t +a:)= J2sech(t +a:), :i:(t +a:)= -v'2sech(t + o:)tanh(t +a) (4 .9) and the Melnikov function with two .
