Oscillatory behavior of closed isotropic models in second order gravity theory

Homogeneous and isotropic models are studied in the Jordan frame of the second order gravity theory. The late time evolution of the models is analysed with the methods of the dynamical systems. The normal form of the dynamical system has periodic solutions for a large set of initial conditions. This implies that an initially expanding closed isotropic universe may exhibit oscillatory behaviour.Comment: 16 pages, 3 figures. With some minor improvements. To appear in General Relativity and Gravitatio

Unchained polygons and the N-body problem

To the memory of J. Moser, with admiration The simplest solutions of the N-body problem –symmetric relative equilibria – are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus in particular on the relative equilibrium of the equal-mass regular N-gon, assumed horizontal, and study the families of Lyapunov quasi-periodic solutions bifurcating from them in the vertical direction. The proof of the local existence of such solutions relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We then discuss the possibility of continuing the families globally as action minimizers under symmetry constraints by using the fact that, in rotating frames where they become periodic, these solutions are highly symmetric. The paradigmatic examples are the “Eight ” families for an odd number of bodies and the “Hip-Hop ” families for an even number. The first ones generalize Marchal’s P12 family for 3 bodies, which starts with the equilateral triangle and ends with the Eight ([CM, Ma2, CFM, CF, S]); the second ones generalize the Hip-Hop family for 4 bodies, which starts from the square and ends with the Hip-Hop ([CV, CF, TV]). We argue that it is precisely for these two families that global minimization may be used. We also study the relation with the regular N-gon, of the so-called “chain ” choreographies (see [S]): here, only a local minimization property is true (except for N = 3) and moreover the parity plays a deciding role, in particular through the value of the angular momentum. This paper develops [CF]

Communicated by Arieh Iserles.

Abstract We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with O(ɛ j) errors, j = 1, 2, 3(2πɛ denotes the period of the fast oscillations). For second-order systems with large O(ɛ −1) forces, we give the explicit form of the averaged systems with O(ɛ j) errors, j = 1, 2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construc