Periodic orbits and non-integrability in a cosmological scalar field

Abstract

Abstract. We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing an universe filled with a scalar field which possesses three parameters. The main results are the following.First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level.Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville-Arnol'd, proving that cannot exist any second first integral of class C 1 .It is important to mention that the tools (i.e the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of these periodic orbits for studying the integrability of the Hamiltonian system) used here for proving our results on the cosmological scalar field, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

Averaging generalized scalar field cosmologies II: locally rotationally symmetric Bianchi I and flat Friedmann–Lemaître–Robertson–Walker models

“…Other useful mathematical methods are asymptotic methods and averaging theory [9][10][11][12][13][14][15]. These methods have been applied in cosmology for example in [16][17][18][19][20][21][22][23][24] with interest in early and late-time dynamics. Some works related to Einstein-KG, Maxwell, Yang-Mills and Einstein-Vlasov systems are [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

Averaging generalized scalar field cosmologies I: locally rotationally symmetric Bianchi III and open Friedmann–Lemaître–Robertson–Walker models

“…These results suggest that the asymptotic behavior of time-averaged model is independent of coupling function and geometry. Averaging theory was used in reference [151] to study periodic orbits of Hamiltonian systems describing a universe filled with a scalar field; and in reference [152] to study future asymptotics of LRS Bianchi type III cosmologies with a massive scalar field. In reference [153] a theorem about large-time behaviour of solutions of Spatially Homogeneous (SH) cosmology with oscillatory behaviour (when H is non negative and monotonic decreasing to zero) was 0123456789().…”

On the integrability of a three-dimensional cored galactic Hamiltonian

“…The motivation for the choice of the potential 1 + x 2 + y 2 + z 2 /q comes from the interest of this potential in galactic dynamics, see for instance [2,3,5,6,7,9,12,13,14,15,16]. The parameter q gives the ellipticity of the potential, which ranges in the interval √ 0.6 ≤ q ≤ 1.…”