Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits (Q6558853)

The author considers perturbations of a planar Hamiltonian system, where the latter is assumed to have two saddles connected by a heteroclinic orbit, such that the Jacobian matrices at the two saddles have paired real eigenvalues. The aim of the work is to exhibit (real meromorphic) nonintegrability of the perturbed system. This is accomplished using the theory of \textit{J. J. Morales-Ruiz} and \textit{J. P. Ramis} [Methods Appl. Anal. 8, No. 1, 33--95 (2001; Zbl 1140.37352); Ann. Sci. Éc. Norm. Supér. (4) 40, No. 6, 845--884 (2007; Zbl 1144.37023)], namely by showing that if the system has first integrals then the differential Galois group of the system has a commutative connected component. Thus nonintegrability can be established by demonstrating noncommutativity of this component. This means exhibiting a pair of elements which do not commute. Exhibiting any non-identity element of the Galois group is a challenge: although the group is given in principle as linear transformations of a full set of solutions, one needs to know all the algebraic relations among the solutions and all their derivatives to decide if a given linear transformation of the solutions lies in the Galois group. However, as noted, to apply the Ramis-Morales theory one only needs two (noncommuting) elements of the Galois group. The author finds these in the monodromy group, which is a subgroup of the Galois group, when the Melinkov function associated to the system is non-constant, under the additional assumption that the perturbation of the system has a finite Fourier series. \N\NAs the author shows, the result applies in the case of two periodically forced Duffing oscillators:\N\[\N\dot{x_1}=x_2, \text{ } \dot{x_2}=x_1^3+\varepsilon(\beta \cos(\omega t)-\delta x_2),\N\]\N\[\N\dot{x_1}=x_2, \text{ } \dot{x_2}=-x_1^3+\varepsilon(\beta \cos(\omega t)-\delta x_2),\N\]\Nand the two-dimensional periodic forced system\N\[\N\dot{x_1}=x_1-x_2^2, \text{ } \dot{x_2}=-x_2+2x_1x_2+\varepsilon(\beta \cos(\omega t)-\delta x_2),\N\]\Nwhere \(\beta\), \(\omega > 0\) and \(\delta \geq 0\) are constants.