We study the orbits in a Manko-Novikov type metric (MN) which is a perturbed Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are found in configuration space and on a surface of section for various values of the energy E an
d the z-component of the angular momentum Lz. For relatively large Lz there are two permissible regions of non-plunging motion bounded by two closed curves of zero velocity (CZV), while in the Kerr metric there is only one closed CZV of non-plunging motion. The inner permissible region of the MN metric contains mainly chaotic orbits, but it contains also a large island of stability. We find the positions of the main periodic orbits as functions of Lz and E, and their bifurcations. Around the main periodic orbit of the outer region there are islands of stability that do not appear in the Kerr metric. In a realistic binary system, because of the gravitational radiation, the energy E and the angular momentum Lz of an inspiraling compact object decrease and therefore the orbit of the object is non-geodesic. In fact in an EMRI system the energy E and the angular momentum Lz decrease adiabatically and therefore the motion of the inspiraling object is characterized by the fundamental frequencies which are drifting slowly in time. In the Kerr metric the ratio of the fundamental frequencies changes strictly monotonically in time. However, in the MN metric when an orbit is trapped inside an island the ratio of the fundamental frequencies remains constant for some time. Hence, if such a phenomenon is observed this will indicate that the system is non integrable and therefore the central object is not a Kerr black hole.
We study the dynamics of the scalar field FLRW flat cosmological models within the framework of the Unified Dark Matter (UDM) scenario. In this model we find that the main cosmological functions such as the scale factor of the Universe, the scalar fi
eld, the Hubble flow and the equation of state parameter are defined in terms of hyperbolic functions. These analytical solutions can accommodate an accelerated expansion, equivalent to either the dark energy or the standard $Lambda$ models. Performing a joint likelihood analysis of the recent supernovae type Ia data and the Baryonic Acoustic Oscillations traced by the SDSS galaxies, we place tight constraints on the main cosmological parameters of the UDM cosmological scenario. Finally, we compare the UDM scenario with various dark energy models namely $Lambda$ cosmology, parametric dark energy model and variable Chaplygin gas. We find that the UDM scalar field model provides a large and small scale dynamics which are in fair agreement with the predictions by the above dark energy models although there are some differences especially at high redshifts.
We study the connection between the appearance of a `metastable behavior of weakly chaotic orbits, characterized by a constant rate of increase of the Tsallis q-entropy (Tsallis 1988), and the solutions of the variational equations of motion for the
same orbits. We demonstrate that the variational equations yield transient solutions, lasting for long time intervals, during which the length of deviation vectors of nearby orbits grows in time almost as a power-law. The associated power exponent can be simply related to the entropic exponent for which the q-entropy exhibits a constant rate of increase. This analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits, that we call `Average Power Law Exponent (APLE). We compare the APLE with other established indicators of the literature. In particular, we give examples of application of the APLE in a) a thin separatrix layer of the standard map, b) the stickiness region around an island of stability in the same map, and c) the web of resonances of a 4D symplectic map. In all these cases we identify weakly chaotic orbits exhibiting the `metastable behavior associated with the Tsallis q-entropy.
