The non-integrability of the Zipoy-Voorhees metric

The low frequency gravitational wave detectors like eLISA/NGO will give us the opportunity to test whether the supermassive compact objects lying at the centers of galaxies are indeed Kerr black holes. A way to do such a test is to compare the gravitational wave signals with templates of perturbed black hole spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV) spacetime (known also as the $\gamma$ spacetime) can be included in the bumpy black hole family, because it can be considered as a perturbation of the Schwarzschild spacetime background. Several authors have suggested that the ZV metric corresponds to an integrable system. Contrary to this integrability conjecture, in the present article it is shown by numerical examples that in general ZV belongs to the family of non-integrable systems.Comment: 10 pages, 13 figure

Orbits in the H2O molecule

We study the forms of the orbits in a symmetric configuration of a realistic model of the H2O molecule with particular emphasis on the periodic orbits. We use an appropriate Poincar\'e surface of section (PSS) and study the distribution of the orbits on this PSS for various energies. We find both ordered and chaotic orbits. The proportion of ordered orbits is almost 100% for small energies, but decreases abruptly beyond a critical energy. When the energy exceeds the escape energy there are still non-escaping orbits around stable periodic orbits. We study in detail the forms of the various periodic orbits, and their connections, by providing appropriate stability and bifurcation diagrams.Comment: 21 pages, 14 figures, accepted for publication in CHAO

Periodic Orbits and Escapes in Dynamical Systems

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central "bumpy" black hole. When the energy reaches its escape value a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.Comment: 28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical Astronomy

How to observe a non-Kerr spacetime

We present a generic criterion which can be used in gravitational-wave data analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background spacetime from one into a non-Kerr background spacetime. The criterion exploits the fact that when an integrable system, such as the system that describes geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space which initially corresponded to resonances disintegrate so as to form the so called Birkhoff chains on a surface of section, according to the Poincar\'{e}-Birkhoff theorem. The KAM curves of these islands in such a chain share the same ratio of frequencies, even though the frequencies themselves vary from one KAM curve to another inside an island. On the other hand, the KAM curves, which do not lie in a Birkhoff chain, do not share this characteristic property. Such a temporal constancy of the ratio of frequencies during the evolution of the gravitational-wave signal will signal a non-Kerr spacetime which could then be further explored.Comment: 4 pages, 2 figure

Asymptotic Orbits in Barred Spiral Galaxies

We study the formation of the spiral structure of barred spiral galaxies, using an $N$-body model. The evolution of this $N$-body model in the adiabatic approximation maintains a strong spiral pattern for more than 10 bar rotations. We find that this longevity of the spiral arms is mainly due to the phenomenon of stickiness of chaotic orbits close to the unstable asymptotic manifolds originated from the main unstable periodic orbits, both inside and outside corotation. The stickiness along the manifolds corresponding to different energy levels supports parts of the spiral structure. The loci of the disc velocity minima (where the particles spend most of their time, in the configuration space) reveal the density maxima and therefore the main morphological structures of the system. We study the relation of these loci with those of the apocentres and pericentres at different energy levels. The diffusion of the sticky chaotic orbits outwards is slow and depends on the initial conditions and the corresponding Jacobi constant.Comment: 17 pages, 24 figure

Analytical description of the structure of chaos

We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the H\'{e}non map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(\xi, \eta)$ in which the product $\xi\eta=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $\Phi$ to the plane $(x,y)$, giving "Moser invariant curves". We find that the series $\Phi$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $\kappa$ of the H\'{e}non map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c \leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41\leq c< c_{max} \simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a "structure of chaos". Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $\Phi$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the H\'{e}non parameter $\kappa$, i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure

Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians

We consider normal forms in `magnetic bottle' type Hamiltonians of the form $H=\frac{1}{2}(\rho^2_\rho+\omega^2_1\rho^2) +\frac{1}{2}p^2_z+hot$ (second frequency $\omega_2$ equal to zero in the lowest order). Our main results are: i) a novel method to construct the normal form in cases of resonance, and ii) a study of the asymptotic behavior of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order $r$, the series remainder in both constructions decreases with increasing $r$ down to a minimum, and then it increases with $r$. The computed minimum remainder turns to be exponentially small in $\frac{1}{\Delta E}$, where $\Delta E$ is the mirror oscillation energy, while the optimal order scales as an inverse power of $\Delta E$. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behavior. In the resonant case, our novel method allows to compute a `quasi-integral' (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non resonant normal form yields theorerical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.Comment: 20 pages, 7 figure

Resurvey of order and chaos in spinning compact binaries

This paper is mainly devoted to applying the invariant, fast, Lyapunov indicator to clarify some doubt regarding the apparently conflicting results of chaos in spinning compact binaries at the second-order post-Newtonian approximation of general relativity from previous literatures. It is shown with a number of examples that no single physical parameter or initial condition can be described as responsible for causing chaos, but a complicated combination of all parameters and initial conditions is responsible. In other words, a universal rule for the dependence of chaos on each parameter or initial condition cannot be found in general. Chaos does not depend only on the mass ratio, and the maximal spins do not necessarily bring the strongest effect of chaos. Additionally, chaos does not always become drastic when the initial spin vectors are nearly perpendicular to the orbital plane, and the alignment of spins cannot trigger chaos by itself.Comment: 16 pages, 7 figure