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

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

NGC 1300 Dynamics: III. Orbital analysis

We present the orbital analysis of four response models, that succeed in reproducing morphological features of NGC 1300. Two of them assume a planar (2D) geometry with $\Omega_p$=22 and 16 \ksk respectively. The two others assume a cylindrical (thick) disc and rotate with the same pattern speeds as the 2D models. These response models reproduce most successfully main morphological features of NGC 1300 among a large number of models, as became evident in a previous study. Our main result is the discovery of three new dynamical mechanisms that can support structures in a barred-spiral grand design system. These mechanisms are presented in characteristic cases, where these dynamical phenomena take place. They refer firstly to the support of a strong bar, of ansae type, almost solely by chaotic orbits, then to the support of spirals by chaotic orbits that for a certain number of pat tern revolutions follow an n:1 (n=7,8) morphology, and finally to the support of spiral arms by a combination of orbits trapped around L$_{4,5}$ and sticky chaotic orbits with the same Jacobi constant. We have encountered these dynamical phenomena in a large fraction of the cases we studied as we varied the parameters of our general models, without forcing in some way their appearance. This suggests that they could be responsible for the observed morphologies of many barred-spiral galaxies. Comparing our response models among themselves we find that the NGC 130 0 morphology is best described by a thick disc model for the bar region and a 2D disc model for the spirals, with both components rotating with the same pattern speed $\Omega_p$=16 \ksk !. In such a case, the whole structure is included inside the corotation of the system. The bar is supported mainly by regular orbits, while the spirals are supported by chaotic orbits.Comment: 18 pages, 32 figures, accepted for publication in MNRA

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

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

Invariant manifolds and the response of spiral arms in barred galaxies

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points $L_1$ and $L_2$ of a barred galaxy define loci in the configuration space which take the form of a trailing spiral pattern. In the present paper we investigate this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. We also examine the relation of `response' models of barred-spiral galaxies with the theory of the invariant manifolds. Our main results are the following: The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term turned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar's major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-like domain of the configuration space, delimited from below by the boundary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. We construct `spiral response' models on the basis of the theory of the invariant manifolds and examine the connection of the latter to the `response' models (Patsis 2006) used to fit real barred-spiral galaxies, explaining how are the manifolds related to a number of morphological features seen in such models.Comment: 16 Page

Gravitational Wave Signals from Chaotic System: A Point Mass with A Disk

We study gravitational waves from a particle moving around a system of a point mass with a disk in Newtonian gravitational theory. A particle motion in this system can be chaotic when the gravitational contribution from a surface density of a disk is comparable with that from a point mass. In such an orbit, we sometimes find that there appears a phase of the orbit in which particle motion becomes to be nearly regular (the so-called ``stagnant motion'') for a finite time interval between more strongly chaotic phases. To study how these different chaotic behaviours affect on observation of gravitational waves, we investigate a correlation of the particle motion and the waves. We find that such a difference in chaotic motions reflects on the wave forms and energy spectra. The character of the waves in the stagnant motion is quite different from that either in a regular motion or in a more strongly chaotic motion. This suggests that we may make a distinction between different chaotic behaviours of the orbit via the gravitational waves.Comment: Published in Phys.Rev.D76:024018,200

Partial Integrability of 3-d Bohmian Trajectories

In this paper we study the integrability of 3-d Bohmian trajectories of a system of quantum harmonic oscillators. We show that the initial choice of quantum numbers is responsible for the existence (or not) of an integral of motion which confines the trajectories on certain invariant surfaces. We give a few examples of orbits in cases where there is or there is not an integral and make some comments on the impact of partial integrability in Bohmian Mechanics. Finally, we make a connection between our present results for the integrability in the 3-d case and analogous results found in the 2-d and 4-d cases.Comment: 18 pages, 3 figure