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Analytical Construction of Homoclinic Orbits of Two- And Three-Dimensional Dynamical Systems

“…The value a :$2)23 is obtained by not di$cult calculation. The initial value corresponding to the decaying solution was numerically estimated by means of selection as aLSK :$2)206 Emaci [7].…”

“…One obtains the following algebraic equations: [7] (line 3), and a corresponding checking solution with a numerically estimated exact initial value aLSK :$2)206 obtained by computer [7] (line 1).…”

“…Finally, one has b "1)453 which is the last step of the solution of the boundary-value problem (15). Figure 1 presents a comparison of the analytical solution of the form QPA (19) (line 2), a corresponding analytical solution of the form of PA obtained in reference [7] (line 3), and a corresponding checking solution with a numerically estimated exact initial value aLSK :$2)206 obtained by the computer [7] (line 1).…”

“…The existence of localized waves in Hamiltonian systems consisting of weakly coupled non-linear oscillators was proved in reference [6]. The HT orbits of the non-linear Schrodinger equation were analytically approximated by constructing Pade' approximants (PA) in reference [7].…”

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Padé approximant

“…The Padé approximants (PA) were successfully used for the homoclinic orbitals of two-dimensional axisymmetric breathers by Vakakis. [10] The quasi-Padé approximants (QPA) were used to construct the homo-clinic orbitals in a nonlinear Schrodinger equation system by Mikhlin. [11] The analytic homoclinic orbitals in the nonautonomous Duffing equation and in the Van der Pol-Duffing equation with weakly coupled nonlinear oscillators were computed in Ref.…”