# Unraveling the escape dynamics and the nature of the normally hyperbolic invariant manifolds in tidally limited star clusters [GA]

The escape mechanism of orbits in a star cluster rotating around its parent galaxy in a circular orbit is investigated. A three degrees of freedom model is used for describing the dynamical properties of the Hamiltonian system. The gravitational field of the star cluster is represented by a smooth and spherically symmetric Plummer potential. We distinguish between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. The Smaller Alignment Index (SALI) method is used for determining the regular or chaotic nature of the orbits. The basins of escape are located and they are also correlated with the corresponding escape time of the orbits. Areas of bounded regular or chaotic motion and basins of escape were found to coexist in the $(x,z)$ plane. The properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 Lagrange points $L_1$ and $L_2$, are also explored. These manifolds are of paramount importance as they control the flow of stars over the saddle points, while they also trigger the formation of tidal tails observed in star clusters. Bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincar\’e map to the NHIMs are deployed for elucidating the dynamics in the neighbourhood of the saddle points. The extended tidal tails, or tidal arms, formed by stars with low velocity which escape through the Lagrange points are monitored. The numerical results of this work are also compared with previous related work.

E. Zotos and C. Jung
Fri, 24 Feb 17
6/50

Comments: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

# Chaos Control with Ion Propulsion [CL]

The escape dynamics around the triangular Lagrangian point L5 in the real Sun-Earth-Moon-Spacecraft system is investigated. Appearance of the finite time chaotic behaviour suggests that widely used methods and concepts of dynamical system theory can be useful in constructing a desired mission design. Existing chaos control methods are modified in such a way that we are able to protect a test particle from escape. We introduce initial condition maps in order to have a suitable numerical method to describe the motion in high dimensional phase space. Results show that the structure of initial condition maps can be split into two well-defined domains. One of these two parts has a regular contiguous shape and is responsible for long time escape; it is a long-lived island. The other one shows a filamentary fractal structure in initial condition maps. The short time escape is governed by this object. This study focuses on a low-cost method which successfully transfers a reference trajectory between these two regions using an appropriate continuous control force. A comparison of the Earth-Moon transfer is also presented to show the efficiency of our method.

J. Sliz, T. Kovacs and A. Suli
Thu, 23 Feb 17
20/48

Comments: 14 pages, 11 figures, accepted for publication in Astronomische Nachrichten

# The gluon condensation at high energy hadron collisions [CL]

We report that the saturation/CGC model of gluon distribution is unstable under action of the chaotic solution in a nonlinear QCD evolution equation, and it evolves to the distribution with a sharp peak at the critical momentum. We find that this gluon condensation is caused by a new kind of shadowing-antishadowing effects, and it leads to a series of unexpected effects in high energy hadron collisions including astrophysical events. For example, the extremely intense fluctuations in the transverse-momentum and rapidity distributions of the gluon jets present the gluon-jet bursts; a sudden increase of the proton-proton cross sections may fill the GZK suppression; the blocking QCD evolution will restrict the maximum available energy of the hadron-hadron colliders.

W. Zhu and J. Lan
Thu, 9 Feb 17
58/67

Comments: 45 pages, 19 figures, to be published in Nucl. Phys. B

# The structure of invariant tori in a 3D galactic potential [CL]

We study in detail the structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type. We have used the color and rotation method to investigate the properties of the invariant tori in the 4D spaces of section. We compare our results with those of previous works and we describe the morphology of the rotational, as well as of the tube tori in the 4D space. We find sticky chaotic orbits in the immediate neighborhood of sets of invariant tori surrounding 3D stable periodic orbits. Particularly useful for galactic dynamics is the behavior of chaotic orbits trapped for long time between 4D invariant tori. We find that they support during this time the same structure as the quasi-periodic orbits around the stable periodic orbits, contributing however to a local increase of the dispersion of velocities. Finally we find that the tube tori do not appear in the 3D projections of the spaces of section in the axisymmetric Hamiltonian we examined.

M. Katsanikas and P. Patsis
Mon, 9 Jan 17
21/52

Comments: 26 pages, 34 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential [CL]

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis and Zachilas 1994] in order to visualize them. As examples we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

M. Katsanikas, P. Patsis and A. Pinotsis
Mon, 9 Jan 17
26/52

Comments: 12 pages,22 figures, Accepted for publication in the International Journal of Bifurcation and Chaos

# The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation [CL]

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D “confined tori”. Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
42/52

Comments: 10 pages, 14 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Instabilities and stickiness in a 3D rotating galactic potential [CL]

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.

M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
44/52

Comments: 29pages, 25 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Theoretical foundations of the Schrödinger method for LSS formation [CL]

It has been shown that the formation of large scale structures (LSS) in the universe can be described in terms of a Schr$\ddot{o}$dinger-Poisson system. This procedure, known as Schr$\ddot{o}$dinger method, has no theoretical basis, but it is intended as a mere tool to model the N-body dynamics of dark matter halos which form LSS. Furthermore, in this approach the “Planck constant” $\hbar$ in the Schr$\ddot{o}$dinger equation is just a free parameter. In this paper we give a theoretical foundation of the Schr$\ddot{o}$dinger method based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. The order of magnitude of the effective Planck constant is estimated as $\hbar \sim m^{5/3} G^{1/2} (N/<\rho>)^{1/6}$, where $N$ and $m$ are the number and the mass of the dark matter halos, $<\rho_0>$ is their average density, and $G$ the gravitational constant. The relevance of this finding for the study of LSS is discussed.

F. Briscese
Thu, 15 Dec 16
30/59

# Constraints on Bounded Motion and Mutual Escape for the Full 3-Body Problem [EPA]

When gravitational aggregates are spun to fission they can undergo complex dynamical evolution, including escape and reconfiguration. Previous work has shown that a simple analysis of the full 2-body problem provides physically relevant insights for whether a fissioned system can lead to escape of the components and the creation of asteroid pairs. In this paper we extend the analysis to the full 3-body problem, utilizing recent advances in the understanding of fission mechanics of these systems. Specifically, we find that the full 3-body problem can eject a body with as much as 0.31 of the total system mass, significantly larger than the 0.17 mass limit previously calculated for the full 2-body problem. This paper derives rigorous limits on a fissioned 3-body system with regards to whether fissioned system components can physically escape from each other and what other stable relative equilibria they could settle in. We explore this question with a narrow focus on the Spherical Full Three Body Problem studied in detail earlier.

D. Scheeres
Thu, 1 Dec 16
4/75

Comments: Accepted for publication in Celestial Mechanics and Dynamical Astronomy

# On the complexity and the information content of cosmic structures [CEA]

The emergence of cosmic structure is commonly considered one of the most complex phenomena in Nature. However, this complexity has never been defined nor measured in a quantitative and objective way. In this work we propose a method to measure the information content of cosmic structure and to quantify the complexity that emerges from it, based on Information Theory. The emergence of complex evolutionary patterns is studied with a statistical symbolic analysis of the datastream produced by state-of-the-art cosmological simulations of forming galaxy clusters. This powerful approach allows us to measure how many bits of information are necessary to predict the evolution of energy fields in a statistical way, and it offers a simple way to quantify when, where and how the cosmic gas behaves in complex ways. The most complex behaviors are found in the peripheral regions of galaxy clusters, where supersonic flows drive shocks and large energy fluctuations over a few tens of million years. Describing the evolution of magnetic energy requires at least a twice as large amount of bits than for the other energy fields. When radiative cooling and feedback from galaxy formation are considered, the cosmic gas is overall found to double its degree of complexity. In the future, Cosmic Information Theory can significantly increase our understanding of the emergence of cosmic structure as it represents an innovative framework to design and analyze complex simulations of the Universe in a simple, yet powerful way.

F. Vazza
Tue, 29 Nov 16
21/77

Comments: 15 pages, 14 figures. MNRAS accepted, in press

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# Revisiting Evidence of Chaos in X-ray Light Curves: The Case of GRS 1915+105 [HEAP]

Nonlinear time series analysis has been widely used to search for signatures of low-dimensional chaos in light curves emanating from astrophysical bodies. A particularly popular example is the microquasar GRS 1915+105, whose irregular but systematic X-ray variability has been well studied using data acquired by the Rossi X-ray Timing Explorer (RXTE). With a view to building simpler models of X-ray variability, attempts have been made to classify the light curves of GRS 1915+105 as chaotic or stochastic. Contrary to some of the earlier suggestions, after careful analysis, we find no evidence for chaos or determinism in any of the GRS 1915+105 classes. The dearth of long and stationary data sets representing all the different variability classes of GRS 1915+105 make it a poor candidate for analysis using nonlinear time series techniques. We conclude that either very exhaustive data analysis with sufficiently long and stationary light curves should be performed keeping all the pitfalls of nonlinear time series analysis in mind, or alternative schemes of classifying the light curves should be adopted. The generic limitations of the techniques that we point out in the context of GRS 1915+105 affect all similar investigations of light curves from other astrophysical sources.

M. Mannattil, H. Gupta and S. Chakraborty
Tue, 8 Nov 16
8/75

Comments: Accepted in The Astrophysical Journal

# Distributed chaos and Rayleigh-Benard turbulence at very high Ra [CL]

It is shown, by the means of distributed chaos approach and using the experimental data, that at very large Rayleigh number $Ra > 10^{14}$ and Prandtl number $Pr \sim 1$ the Rayleigh-B\'{e}nard turbulence can undergo a transition related to spontaneous breaking of the fundamental Lagrangian relabeling symmetry. Due to the Noether’s theorem helicity plays central role in this process. After the transition the temperature spectrum has a stretched exponential form $E (k) \propto \exp(-k/k_{\beta})^{\beta}$ with $\beta =2/5$ both at the cell midplain and at the near-wall (low boundary) regions. There is a similarity between this phenomenon and the effects of polymer additives.

Wed, 2 Nov 16
13/55

# Strong turbulent convection: distributed chaos and large-scale circulation [CL]

Two types of spontaneous breaking of the space translational symmetry in distributed chaos have been considered for turbulent thermal convection at large values of Rayleigh number. First type is related to boundaries and second type is related to appearance of inertial range of scales. The first type is dominated by vorticity correlation integral: $\int_{V} \langle {\boldsymbol \omega} ({\bf x},t) \cdot {\boldsymbol \omega} ({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ and is characterized by stretched exponential spectrum $\exp-(k/k_{\beta})^{\beta }$ with $\beta =1/2$. The second type is dominated by energy correlation integral: $\int_{V} \langle {\bf u}^2 ({\bf x},t) ~ {\bf u}^2({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ and is characterized by $\beta =3/5$. Good agreement has been established with laboratory experimental data obtained at large values of Rayleigh number $Ra \sim 10^{11}-10^{14}$ (the range relevant to solar photosphere) in upright cylinder cells. Taylor hypothesis transforms the wavenumber spectrum into frequency spectrum $\exp-(f/f_{\beta})^{1/2}$. It is shown that turnover frequency of large-scale circulation (wind): $f_w = f_{\beta}/2$. Results of an experiment in horizontal cylinder are also briefly discussed. The analysis suggests that in this case the large-scale circulation can be considered as a natural (harmonic) part of the distributed chaos.

Wed, 19 Oct 16
27/87

# Vertical stability of circular orbits in relativistic razor-thin disks [CL]

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant by the geodesic flow.

R. Vieira, J. Ramos-Caro and A. Saa
Mon, 10 Oct 16
27/51

Comments: 13 pages, 4 figures; Accepted for publication in Physical Review D

# Effects of magnetic and kinetic helicities on the growth of magnetic fields in laminar and turbulent flows by helical-Fourier decomposition [CL]

We present a numerical and analytical study of incompressible homogeneous conducting fluids using a Fourier-helical representation. We analytically study both small- and large-scale dynamo properties, as well as the inverse cascade of magnetic helicity, in the most general minimal subset of interacting velocity and magnetic fields on a closed Fourier triad. We mainly focus on the dependency of magnetic field growth as a function of the distribution of kinetic and magnetic helicities among the three interacting wavenumbers. By combining direct numerical simulations of the full magnetohydrodynamics (MHD) equations with the Fourier-helical decomposition we numerically confirm that in the kinematic dynamo regime the system develops a large-scale magnetic helicity with opposite sign compared to the small-scale kinetic helicity, a sort of triad-by-triad $\alpha$-effect in Fourier space. Concerning the small-scale perturbations, we predict theoretically and confirm numerically that the largest instability is achived for the magnetic component with the same helicity of the flow, in agreement with the Stretch-Twist-Fold mechanism. Viceversa, in presence of a Lorentz feedback on the velocity, we find that the inverse cascade of magnetic helicity is mostly local if magnetic and kinetic helicities have opposite sign, while it is more non-local and more intense if they have the same sign, as predicted by the analytical approach. Our analytical and numerical results further demonstrate the potential of the helical-Fourier decomposition to elucidate the entangled dynamics of magnetic and kinetic helicities both in fully developed turbulence and in laminar flows.

M. Linkmann, G. Sahoo, M. McKay, et. al.
Thu, 8 Sep 16
1/60

# Modelling resonances and orbital chaos in disk galaxies. Application to a Milky Way spiral model [GA]

Context: Resonances in the stellar orbital motion under perturbations from spiral arms structure play an important role in the evolution of the disks of spiral galaxies. The epicyclic approximation allows the determination of the corresponding resonant radii on the equatorial plane (for nearly circular orbits), but is not suitable in general.
Aims: To expand the study of resonant orbits by analysing stellar motions perturbed by spiral arms with Gaussian-shaped profiles, without any restriction on the stellar orbital configurations, and expand the concept of Lindblad (epicyclic) resonances for orbits with large radial excursions.
Methods: We define a representative plane of initial conditions, which covers the whole phase space of the system. Dynamical maps on representative planes are constructed numerically, in order to characterize the phase-space structure and identify the precise location of the resonances. The study is complemented by the construction of dynamical power spectra, which provide the identification of fundamental oscillatory patterns in the stellar motion.
Results: Our approach allows a precise description of the resonance chains in the whole phase space, giving a broader view of the dynamics of the system when compared to the classical epicyclic approach, even for objects in retrograde motion. The analysis of the solar neighbourhood shows that, depending on the current azimuthal phase of the Sun with respect to the spiral arms, a star with solar kinematic parameters may evolve either inside the stable co-rotation resonance or in a chaotic zone.
Conclusions: Our approach contributes in quantifying the domains of resonant orbits and the degree of chaos in the whole Galactic phase-space structure. It may serve as a starting point to apply these techniques to the investigation of clumps in the distribution of stars in the Galaxy, such as kinematic moving groups.

T. Michtchenko, R. Vieira, D. Barros, et. al.
Thu, 1 Sep 16
55/74

Comments: 17 pages, 14 figures. Submitted to A&A

# Does the Planetary Dynamo Go Cycling On? Re-examining the Evidence for Cycles in Magnetic Reversal Rate [EPA]

The record of reversals of the geomagnetic field has played an integral role in the development of plate tectonic theory. Statistical analyses of the reversal record are aimed at detailing patterns and linking those patterns to core-mantle processes. The geomagnetic polarity timescale is a dynamic record and new paleomagnetic and geochronologic data provide additional detail. In this paper, we examine the periodicity revealed in the reversal record back to 375 Ma using Fourier analysis. Four significant peaks were found in the reversal power spectra within the 16-40-million-year range. Plotting the function constructed from the sum of the frequencies of the proximal peaks yield a transient 26 Myr periodicity, suggesting chaotic motion with a periodic attractor. The possible 16 Myr periodicity, a previously recognized result, may be correlated with “pulsation” of mantle plumes.

A. Melott, A. Pivarunas, J. Meert, et. al.
Mon, 29 Aug 16
20/41

Comments: 4 figures. Submitted to Earth and Planetary Science Letters

# Detecting Dynamical States from Noisy Time Series using Bicoherence [CL]

Deriving meaningful information from observational data is often restricted by many limiting factors, the most important of which is the presence of noise. In this work, we present the use of the bicoherence function to extract information about the underlying nonlinearity from noisy time series. We show that a system evolving in the presence of noise which has its dynamical state concealed from quantifiers like the power spectrum and correlation dimension D2, can be revealed using the bicoherence function. We define an index called main peak bicoherence function as the bicoherence associated with the maximal power spectral peak. We show that this index is extremely useful while dealing with quasi-periodic data as it can distinguish strange non chaos from quasi periodicity even with added noise. We demonstrate this in a real world scenario, by taking the bicoherence of variable stars showing period doubling and strange non-chaotic behavior. Our results indicate that bicoherence analysis can also bypass the method of surrogate analysis using Fourier phase randomization, used to differentiate linear stochastic processes from non linear ones, in conventional methods involving measures like D2.

S. George, G. Ambika and R. Misra
Fri, 19 Aug 16
28/45

Comments: 16 pages, 15 figures, submitted to Nonlinear Dynamics

# Jets or vortices – what flows are generated by an inverse turbulent cascade? [CL]

An inverse cascade – energy transfer to progressively larger scales – is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean flow with zero total momentum was therefore believed to be unidirectional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole was expected. Using direct numerical simulations, we show that in fact neither the box symmetry is respected nor the largest scale is realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasi-linear approximation.

A. Frishman, J. Laurie and G. Falkovich
Wed, 17 Aug 16
6/48

# Influence of a second satellite on the rotational dynamics of an oblate moon [EPA]

The gravitational influence of a second satellite on the rotation of an oblate moon is numerically examined. A simplified model, assuming the axis of rotation perpendicular to the (Keplerian) orbit plane, is derived. The differences between the two models, i.e. in the absence and presence of the second satellite, are investigated via bifurcation diagrams and by evolving compact sets of initial conditions in the phase space. It turns out that the presence of another satellite causes some trajectories, that were regular in its absence, to become chaotic. Moreover, the highly structured picture revealed by the bifurcation diagrams in dependence on the eccentricity of the oblate body’s orbit is destroyed when the gravitational influence is included, and the periodicities and critical curves are destroyed as well. For demonstrative purposes, focus is laid on parameters of the Saturn-Titan-Hyperion system, and on oblate satellites on low-eccentric orbits, i.e. $e\approx 0.005$.

M. Tarnopolski
Tue, 26 Jul 16
39/75

Comments: 19 pages, 9 figures; accepted for publication in Celestial Mechanics and Dynamical Astronomy

# Integrability of motion around galactic razor-thin disks [GA]

We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form $Z(R)\propto[\Sigma(R)]^{-1/3}$, where $R$ is the radial galactocentric coordinate, $Z$ is the z-amplitude (vertical amplitude) of the orbit and $\Sigma$ represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.

R. Vieira and J. Ramos-Caro
Wed, 22 Jun 16
1/50

Comments: Version accepted for publication at Celestial Mechanics and Dynamical Astronomy. Replaces arxiv version arxiv:1206.6501. The final publication is available at Springer via this http URL

# From order to chaos in Earth satellite orbits [EPA]

We consider Earth satellite orbits in the range of semi-major axes where the perturbing effects of Earth’s oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees of freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angles-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances which are of first importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations, and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

I. Gkolias, J. Daquin, F. Gachet, et. al.
Wed, 15 Jun 16
19/54

Comments: 28 pages, 8 figures. Submitted to AJ. Comments are greatly appreciated

# Diffusive chaos in navigation satellites orbits [EPA]

The navigation satellite constellations in medium-Earth orbit exist in a background of third-body secular resonances stemming from the perturbing gravitational effects of the Moon and the Sun. The resulting chaotic motions, emanating from the overlapping of neighboring resonant harmonics, induce especially strong perturbations on the orbital eccentricity, which can be transported to large values, thereby increasing the collision risk to the constellations and possibly leading to a proliferation of space debris. We show here that this transport is of a diffusive nature and we present representative diffusion maps that are useful in obtaining a global comprehension of the dynamical structure of the navigation satellite orbits.

J. Daquin, A. Rosengren and K. Tsiganis
Thu, 2 Jun 16
41/60

Comments: 8 pages, 5 figures, conference Chaos, complexity and transport (Marseille, France)

# Marginal resonances and intermittent behaviour in the motion in the vicinity of a separatrix [CL]

A condition upon which sporadic bursts (intermittent behaviour) of the relative energy become possible is derived for the motion in the chaotic layer around the separatrix of non-linear resonance. This is a condition for the existence of a marginal resonance, i.e. a resonance located at the border of the layer. A separatrix map in Chirikov’s form [Chirikov, B. V., Phys. Reports 52, 263 (1979)] is used to describe the motion. In order to provide a straightforward comparison with numeric integrations, the separatrix map is synchronized to the surface of the section farthest from the saddle point. The condition of intermittency is applied to clear out the nature of the phenomenon of bursts of the eccentricity of chaotic asteroidal trajectories in the 3/1 mean motion commensurability with Jupiter. On the basis of the condition, a new intermittent regime of resonant asteroidal motion is predicted and then identified in numeric simulations.

I. Shevchenko
Tue, 31 May 16
21/70

# On the recurrence and Lyapunov time scales of the motion near the chaos border [CL]

Conditions for the emergence of a statistical relationship between $T_r$, the chaotic transport (recurrence) time, and $T_L$, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than $T_r$, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.

I. Shevchenko
Mon, 30 May 16
32/36

# Supersymmetric Theory of Stochastic ABC Model: A Numerical Study [CL]

In this paper, we investigate numerically the stochastic ABC model, a toy model in the theory of astrophysical kinematic dynamos, within the recently proposed supersymmetric theory of stochastics (STS). STS characterises stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differentials forms over the system’s phase space, X. STS can thereby classify SDEs as chaotic or non-chaotic by identifying the phenomenon of stochastic chaos with the spontaneously broken topological supersymmetry that all SDEs possess. We demonstrate the following three properties of the SEO, deduced previously analytically and from physical arguments: the SEO spectra for zeroth and top degree forms never break topological supersymmetry, all SDEs possesses pseudo-time-reversal symmetry, and each de Rahm cohomology class provides one supersymmetric eigenstate. Our results also suggests that the SEO spectra for forms of complementary degrees, i.e., k and dim X -k, may be isospectral.

I. Ovchinnikov, Y. Sun, T. Ensslin, et. al.
Mon, 2 May 16
39/49

Comments: Revtex 4-1, 9 pages, 3 figures

# Multiple Bifurcations in the Periodic Orbit around Eros [EPA]

We investigate the multiple bifurcations in periodic orbit families in the potential field of a highly irregular-shaped celestial body. Topological cases of periodic orbits and four kinds of basic bifurcations in periodic orbit families are studied. Multiple bifurcations in periodic orbit families consist of four kinds of basic bifurcations. We found both binary period-doubling bifurcations and binary tangent bifurcations in periodic orbit families around asteroid 433 Eros. The periodic orbit family with binary period-doubling bifurcations is nearly circular, with almost zero inclination, and is reversed relative to the body of the asteroid 433 Eros. This implies that there are two stable regions separated by one unstable region for the motion around this asteroid. In addition, we found triple bifurcations which consist of two real saddle bifurcations and one period-doubling bifurcation. A periodic orbit family generated from an equilibrium point of asteroid 433 Eros has five bifurcations, which are one real saddle bifurcation, two tangent bifurcations, and two period-doubling bifurcations.

Y. Ni, Y. Jiang and H. Baoyin
Tue, 26 Apr 16
47/61

# Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system [EPA]

The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points $L_1$ and $L_2$ or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill’s regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and collision times. Our outcomes reveal the high complexity of this planetary system. Furthermore, the numerical analysis suggests a strong dependence of the properties of the considered basins with both the total orbital energy and the initial value of the $z$ coordinate, with a remarkable presence of fractal basin boundaries along all the regimes. Our results are compared with earlier ones regarding the planar version of the Earth-Moon system.

E. Zotos
Wed, 13 Apr 16
7/60

Comments: Published in Astrophysics and Space Science (A&SS) journal. arXiv admin note: text overlap with arXiv:1512.08683, arXiv:1508.05201

# Orbital and escape dynamics in barred galaxies – I. The 2D system [GA]

In this paper we use the two-dimensional (2D) version of a new analytical gravitational model in order to explore the orbital as well as the escape dynamics of the stars in a barred galaxy composed of a spherically symmetric central nucleus, a bar, a flat disk and a dark matter halo component. A thorough numerical investigation is conducted for distinguishing between bounded and escaping motion. Furthermore bounded orbits are further classified into non-escaping regular and trapped chaotic using the Smaller ALingment Index (SALI) method. Our aim is to determine the basins of escape through the two symmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and also to relate them with the corresponding distribution of the escape rates of the orbits. We integrate initial conditions of orbits in several types of planes so as to obtain a more complete view of the overall orbital properties of the dynamical system. We also present evidence that the unstable manifolds which guide the orbits in and out the interior region are directly related with the formation of spiral and ring stellar structures observed in barred galaxies. In particular, we examine how the bar’s semi-major axis determines the resulting morphologies. Our numerical simulations indicate that weak barred structures favour the formation of $R_1$ rings or $R_1’$ pseudo-rings, while strong bars on the other hand, give rise to $R_1R_2$ and open spiral morphologies. Our results are compared with earlier related work. The escape dynamics and the properties of the manifolds of the full three-dimensional (3D) galactic system will be given in an accompanying paper.

C. Jung and E. Zotos
Wed, 13 Apr 16
19/60

Comments: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

# Binary black hole shadows, chaotic scattering and the Cantor set [CL]

We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar–Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably-infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass $M$ separated by $a_1 < a < \sqrt{2} a_1$, where $a_1 = 4M/\sqrt{27}$. To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally-extended spacetime.

J. Shipley and S. Dolan
Mon, 4 Apr 16
21/61

# Chaotic Emission from Electromagnetic Systems Considering Self-Interaction [CL]

The emission of electromagnetic waves from a system described by the H\’enon-Heiles potential is studied in this work. The main aim being to analyze the behavior of the system when the damping term is included explicitly into the equations of motion. Energy losses at the chaotic regime and at the regular regime are compared. The results obtained here are similar to the case of gravitational waves emission, as long we consider only the energy loss. The main difference being that in the present work the energy emitted is explicitly calculated solving the equation of motion without further approximations. It is expected that the present analysis may be useful when studying the analogous problem of dissipation in gravitational systems.

F. Kokubun and V. Zanchin
Thu, 31 Mar 16
28/53

Comments: Typos in Refs. corrected. Other minor changes

# Key Issues Review: Numerical studies of turbulence in stars [SSA]

The numerical simulation of turbulence in stars has led to a rich set of possibilities regarding stellar pulsations, asteroseismology, thermonuclear yields, and formation of neutron stars and black holes. The breaking of symmetry by turbulent flow grows in amplitude as collapse is approached, which insures that the conditions at the onset of collapse are not spherical. This lack of spherical symmetry has important implications for the mechanism of explosion and ejected nucleosynthesis products. Numerical resolution of several different types of three–dimensional (3D) stellar simulations are compared; it is suggested that core collapse simulations may be under-resolved.
New physical effects which appear in 3D are summarized.
Connections between simulations of progenitor explosion and observations of supernova remnants (SNR) are discussed.
Present treatment of boundaries, for mixing regions during He–burning, requires revision.

W. Arnett and C. Meakin
Fri, 18 Mar 16
53/53

Comments: 8 pages, 1 figure, 1 table, submitted to Reports on Progress in Physics

# Chaotic motion and the evolution of morphological components in a time-dependent model of a barred galaxy within a dark matter halo [GA]

Studies of dynamical stability (chaotic versus regular motion) in galactic dynamics often rely on static analytical models of the total gravitational potential. Potentials based upon self-consistent N-body simulations offer more realistic models, fully incorporating the time-dependent nature of the systems. Here we aim at analysing the fractions of chaotic motion within different morphological components of the galaxy. We wish to investigate how the presence of chaotic orbits evolves with time, and how their spatial distribution is associated with morphological features of the galaxy. We employ a time-dependent analytical potential model that was derived from an N-body simulation of a strongly barred galaxy. With this analytical potential we may follow the dynamical evolution of ensembles of orbits. Using the Generalized Alignment Index (GALI) chaos detection method, we study the fraction of chaotic orbits, sampling the dynamics of both the stellar disc and of the dark matter halo. Within the stellar disc, the global trend is for chaotic motion to decrease in time, specially in the region of the bar. We scrutinized the different changes of regime during the evolution (orbits that are permanently chaotic, permanently regular, those that begin regular and end chaotic, and those that begin chaotic and end regular), tracing the types of orbits back to their common origins. Within the dark matter halo, chaotic motion also decreases globally in time. The inner halo (r < 5 kpc) is where most chaotic orbits are found and it is the only region where chaotic orbits outnumber regular orbits, in the early evolution.

Wed, 9 Mar 16
44/71

Comments: 15 pages, 10 figures, accepted for publication in MNRAS

# Electromagnetic radiation of charged particles in stochastic motion [HEAP]

The study of the Brownian motion of a charged particle in electric and magnetic fields fields has many important applications in plasma and heavy ions physics, as well as in astrophysics. In the present paper we consider the electromagnetic radiation properties of a charged non-relativistic particle in the presence of electric and magnetic fields, of an exterior non-electromagnetic potential, and of a friction and stochastic force, respectively. We describe the motion of the charged particle by a Langevin and generalized Langevin type stochastic differential equation. We investigate in detail the cases of the Brownian motion with or without memory in a constant electric field, in the presence of an external harmonic potential, and of a constant magnetic field. In all cases the corresponding Langevin equations are solved numerically, and a full description of the spectrum of the emitted radiation and of the physical properties of the motion is obtained. The Power Spectral Density (PSD) of the emitted power is also obtained for each case, and, for all considered oscillating systems, it shows the presence of peaks, corresponding to certain intervals of the frequency.

T. Harko and G. Mocanu
Tue, 8 Mar 16
8/83

Comments: 24 pages, 22 figures, accepted for publication in EPJC

# Turbulence and distributed chaos with spontaneously broken symmetry [CL]

It is shown that in turbulent flows the distributed chaos with spontaneously broken translational space symmetry (homogeneity) has a stretch exponential spectrum $\exp-(k/k_{\beta})^{\beta }$ with $\beta =1/2$. Good agreement has been established between the theory and the data of direct numerical simulation of a channel flow. An astrophysical application to the large-scale galaxies distribution has been briefly discussed and good agreement with the data of recent Sloan Digital Sky Survey SDSS-III has been established.

Mon, 1 Feb 16
42/42

# Pressure-anisotropy-driven microturbulence and magnetic-field evolution in shearing, collisionless plasma [HEAP]

The nonlinear state of a high-beta collisionless plasma is investigated when an imposed linear shear amplifies or diminishes a uniform magnetic field, driving pressure anisotropies and hence firehose/mirror instabilities. The evolution of the resulting microscale turbulence is considered when the shear is switched off or reversed after one shear time (mimicking local behaviour of a macroscopic flow), so a new macroscale configuration is superimposed on the microscale state left behind by the previous one. There is a threshold value of plasma beta: when $\beta\ll\Omega/S$ (ion cyclotron frequency/shear rate), the emergence of firehose/mirror fluctuations driven unstable by shear and their disappearance when the shear is removed/reversed are quasi-instantaneous compared to the shear time, viz., the decay time of these fluctuations is $\sim\beta/\Omega \ll 1/S$ (this result follows from the free decay of the fluctuations being constrained by the same marginal-stability thresholds as their growth). In contrast, when $\beta\gtrsim\Omega/S$ (“ultra-high” beta), the old microscale state can only be removed on the shear timescale. In this regime, driven firehose fluctuations grow secularly to order-unity amplitudes, compensating for the decay of the mean field and so pinning pressure anisotropy at marginal stability with no appreciable scattering of particles—which is unlike what happens at moderate $\beta$. When the shear reverses, the shearing away of this firehose turbulence compensates for the increase in the mean field and thus prevents growth of the pressure anisotropy, stopping the system from going mirror-unstable. Therefore, at ultra-high beta, the system stays close to the firehose threshold, the mirror instability is largely suppressed, while the mean magnetic energy barely changes at all. Implications for plasma dynamo and thus the origin of cosmic magnetism are discussed.

S. Melville, A. Schekochihin and M. Kunz
Tue, 29 Dec 15
32/54

Comments: MNRAS-style latex, 21 pages, 37 figures, submitted to MNRAS

# Kinematic Dynamo, Supersymmetry Breaking, and Chaos [GA]

The kinematic dynamo (KD) describes the growth of magnetic fields generated by the flow of a conducting medium in the limit of vanishing backaction of the fields onto the flow. The KD is therefore an important model system for understanding astrophysical magnetism. Here, the mathematical correspondence between the KD and a specific stochastic differential equation (SDE) viewed from the perspective of the supersymmetric theory of stochastics (STS) is discussed. The STS is a novel, approximation-free framework to investigate SDEs. The correspondence reported here permits insights from the STS to be applied to the theory of KD and vice versa. It was previously known that the fast KD in the idealistic limit of no magnetic diffusion requires chaotic flows. The KD-STS correspondence shows that this is also true for the diffusive KD. From the STS perspective, the KD possesses a topological supersymmetry and the dynamo effect can be viewed as its spontaneous breakdown. This supersymmetry breaking can be regarded as the stochastic generalization of the concept of dynamical chaos. As this supersymmetry breaking happens in both the diffusive and the non-diffusive case, the necessity of the underlying SDE being chaotic is given in either case. The observed exponentially growing and oscillating KD modes prove physically that dynamical spectra of the STS evolution operator that break the topological supersymmetry exist with both, real and complex ground state eigenvalues. Finally, we comment on the non-existence of dynamos for scalar quantities.

I. Ovchinnikov and T. Ensslin
Tue, 8 Dec 15
25/71

Comments: 10 pages, 1 figure, submitted

# Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance [CL]

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.

A. Marchesiello and G. Pucacco
Thu, 3 Dec 15
65/65

Comments: 36 pages, 10 figures, accepted on International Journal of Bifurcation and Chaos. arXiv admin note: substantial text overlap with arXiv:1401.2855

# Passive scalar mixing at finite correlation times and the Batchelor spectrum [CL]

An elegant model for passive scalar mixing was given by Kraichnan (1968) assuming the velocity to be delta-correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\tau$, using renewing flows. The generalized evolution equation for the 3-D passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\tau \to 0$, and extends it to the next order in $\tau$. It involves third and fourth order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\tau$ (or small Strouhl number), it can be recast using the Landau-Lifshitz approach, to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. Interestingly, to leading order in $\tau$, we show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (1959) form, $E_\theta(k) \propto k^{-1}$, in the viscous-convective limit, independent of $\tau$. When passive scalar fluctuations decay, we show that the decay rate is reduced for finite $\tau$, but the spectrum $E_\theta(k) \propto k^{1/2}$ independent of $\tau$ . We also present results from high resolution ($1024^3$) direct numerical simulations of passive scalar mixing. We find reasonable agreement with predictions of the Batchelor spectrum, during steady state. The scalar spectrum during decay is however shallower than what theory predicts, a feature which remains intriguing.

A. Aiyer, K. Subramanian and P. Bhat
Mon, 30 Nov 15
44/78

Comments: 20 pages, 2 figures, Submitted to JFM

# Order and chaos in a three dimensional galaxy model [GA]

We explore the orbital dynamics of a realistic three dimensional model describing the properties of a disk galaxy with a spherically symmetric central dense nucleus and a triaxial dark matter halo component. Regions of phase space with regular and chaotic motion are identified depending on the parameter values for triaxiality of the dark matter halo and for breaking the rotational symmetry. The four dimensional Poincar\’e map of the three degrees of freedom system is analyzed by a study of its restriction to various two dimensional invariant subsets of its domain.

C. Jung and E. Zotos
Tue, 17 Nov 15
33/87

Comments: Published in Mechanics Research Communications (MRC) journal

# Simple nonlinear models suggest variable star universality [SSA]

Dramatically improved data from observatories like the CoRoT and Kepler spacecraft have recently facilitated nonlinear time series analysis and phenomenological modeling of variable stars, including the search for strange (aka fractal) or chaotic dynamics. We recently argued [Lindner et al., Phys. Rev. Lett. 114 (2015) 054101] that the Kepler data includes “golden” stars, whose luminosities vary quasiperiodically with two frequencies nearly in the golden ratio, and whose secondary frequencies exhibit power-law scaling with exponent near -1.5, suggesting strange nonchaotic dynamics and singular spectra. Here we use a series of phenomenological models to make plausible the connection between golden stars and fractal spectra. We thereby suggest that at least some features of variable star dynamics reflect universal nonlinear phenomena common to even simple systems.

J. Lindner, V. Kohar, B. Kia, et. al.
Mon, 19 Oct 15
17/44

Comments: 9 pages, 9 figures, accepted for publication in Physica D

# Effect of data gaps on correlation dimension computed from light curves of variable stars [IMA]

Observational data, especially astrophysical data, is often limited by gaps in data that arises due to lack of observations for a variety of reasons. Such inadvertent gaps are usually smoothed over using interpolation techniques. However the smoothing techniques can introduce artificial effects, especially when non-linear analysis is undertaken. We investigate how gaps can affect the computed values of correlation dimension of the system, without using any interpolation. For this we introduce gaps artificially in synthetic data derived from standard chaotic systems, like the R{\”o}ssler and Lorenz, with frequency of occurrence and size of missing data drawn from two Gaussian distributions. Then we study the changes in correlation dimension with change in the distributions of position and size of gaps. We find that for a considerable range of mean gap frequency and size, the value of correlation dimension is not significantly affected, indicating that in such specific cases, the calculated values can still be reliable and acceptable. Thus our study introduces a method of checking the reliability of computed correlation dimension values by calculating the distribution of gaps with respect to its size and position. This is illustrated for the data from light curves of three variable stars, R Scuti, U Monocerotis and SU Tauri. We also demonstrate how a cubic spline interpolation can cause a time series of Gaussian noise with missing data to be misinterpreted as being chaotic in origin. This is demonstrated for the non chaotic light curve of variable star SS Cygni, which gives a saturated D$_{2}$ value, when interpolated using a cubic spline. In addition we also find that a careful choice of binning, in addition to reducing noise, can help in shifting the gap distribution to the reliable range for D$_2$ values.

S. George, G. Ambika and R. Misra
Tue, 13 Oct 15
63/64

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# Chaotic features of the last scattering in CMB spectrum [CL]

It is shown, using the data obtained by the Planck space telescope (2009-2013), that the angular CMB Doppler spectrum: $C_l \sim \exp-(l/l_c)$, with $l_c \simeq 300$ in the interval $370 < l < 2500$. The waviness observed along the exponential decay has period (distance between peaks) equal to the same $l_c \simeq 300$. It means that the waviness is generated by the same, presumably chaotic, mechanism that generates the exponential decay. Comparison with deterministic chaos simulations has been briefly discussed.

Fri, 9 Oct 15
7/61

# On geodesic dynamics in deformed black-hole fields [CL]

“Almost all” seems to be known about isolated stationary black holes in asymptotically flat space-times and about the behaviour of {\em test} matter and fields in their backgrounds. The black holes likely present in galactic nuclei and in some X-ray binaries are commonly being represented by the Kerr metric, but actually they are not isolated (they are detected only thanks to a strong interaction with the surroundings), they are not stationary (black-hole sources are rather strongly variable) and they also probably do not live in an asymptotically flat universe. Such “perturbations” may query the classical black-hole theorems (how robust are the latter against them?) and certainly affect particles and fields around, which can have observational consequences. In the present contribution we examine how the geodesic structure of the static and axially symmetric black-hole space-time responds to the presence of an additional matter in the form of a thin disc or ring. We use several different methods to show that geodesic motion may become chaotic, to reveal the strength and type of this irregularity and its dependence on parameters. The relevance of such an analysis for galactic nuclei is briefly commented on.

O. Semerak and P. Sukova
Wed, 30 Sep 15
61/71

# Fractal structures for the Jacobi Hamiltonian of restricted three-body problem [EPA]

We study the dynamical chaos and integrable motion in the planar circular restricted three-body problem and determine the fractal dimension of the spiral strange repeller set of non-escaping orbits at different values of mass ratio of binary bodies and of Jacobi integral of motion. We find that the spiral fractal structure of the Poincar\’e section leads to a spiral density distribution of particles remaining in the system. We also show that the initial exponential drop of survival probability with time is followed by the algebraic decay related to the universal algebraic statistics of Poincar\’e recurrences in generic symplectic maps.

G. Rollin, J. Lages and D. Shepelyansky
Mon, 28 Sep 15
40/67

# Time Series with Tailored Nonlinearities [CL]

It is demonstrated how to generate time series with tailored nonlinearities by inducing well- defined constraints on the Fourier phases. Correlations between the phase information of adjacent phases and (static and dynamic) measures of nonlinearities are established and their origin is explained. By applying a set of simple constraints on the phases of an originally linear and uncor- related Gaussian time series, the observed scaling behavior of the intensity distribution of empirical time series can be reproduced. The power law character of the intensity distributions being typical for e.g. turbulence and financial data can thus be explained in terms of phase correlations.

C. Raeth and I. Laut
Wed, 2 Sep 15
72/87

Comments: 5 pages, 5 figures, Phys. Rev. E, Rapid Communication, accepted

# Practical application of KAM theory to galactic dynamics: II. Application to weakly chaotic orbits in barred galaxies [GA]

Owing to the pioneering work of Contopoulos, a strongly barred galaxy is known to have irregular orbits in the vicinity of the bar. By definition, irregular orbits can not be represented by action-angle tori everywhere in phase space. This thwarts perturbation theory and complicates our understanding of their role in galaxy structure and evolution. This paper provides a qualitative introduction to a new method based on KAM theory for investigating the morphology of regular and irregular orbits based on direct computation of tori described in Paper 1 and applies it to a galaxy disc bar. Using this method, we find that much of the phase space inside of the bar radius becomes chaotic for strong bars, excepting a small region in phase space between the ILR and corotation resonances for orbits of moderate ellipticity. This helps explain the preponderance of moderately eccentric bar-supporting orbits as the bar strength increases. This also suggests that bar strength may be limited by chaos! The chaos results from stochastic layers that form around primary resonances owing to separatrix splitting. Most investigations of orbit regularity are performed using numerical computation of Lyapunov exponents or related indices. We show that Lyapunov exponents poorly diagnose the degree of stochasticity in this problem; the island structure in the stochastic sheaths allow orbit to change morphology while presenting anomalously small Lyapunov exponent values (i.e. weak chaos). For example, a weakly chaotic orbit may appear to change its morphology spontaneously, while appearing regular except during the change itself. The numerical KAM approach sensitively detects these dynamics and provides a model Hamiltonian for further investigation. It may underpredict the number of broken tori for strong perturbations.

M. Weinberg
Wed, 26 Aug 15
2/62

# Helical mode interactions and spectral transfer processes in magnetohydrodynamic turbulence [CL]

Spectral transfer processes in magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of the interacting modes is derived. As expected from the inverse cascade of magnetic helicity in 3D MHD turbulence, mode interactions with like helicities lead to transfer of energy and magnetic helicity to smaller wavenumbers. However, some interactions of modes with unlike helicities also contribute to an inverse energy transfer. As such, an inverse energy cascade for nonhelical magnetic fields is shown to be possible. Furthermore, it is found that high values of the cross-helicity may have an asymmetric effect on forward and reverse transfer of energy, where forward transfer is more quenched in regions of high cross-helicity than reverse transfer. This conforms with recent observations of solar wind turbulence. For specific helical interactions the relation to dynamo action is established.

M. Linkmann, A. Berera, M. McKay, et. al.
Tue, 25 Aug 15
66/69

Comments: 34 pages including references, 4 figures

# Comparing the escape dynamics in tidally limited star cluster models [GA]

The aim of this work is to compare the orbital dynamics in three different models describing the properties of a star cluster rotating around its parent galaxy in a circular orbit. In particular, we use the isochrone and the Hernquist potentials to model the spherically symmetric star cluster and we compare our results with the corresponding ones of a previous work in which the Plummer model was applied for the same purpose. Our analysis takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the tidally limited star cluster. We restrict our investigation into two dimensions and we conduct a thorough numerical analysis distinguishing between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels above the critical escape energy. It is of particular interest to determine the escape basins towards the two exit channels (near the Lagrangian points $L_1$ and $L_2$) and relate them with the corresponding escape times of the orbits.

E. Zotos
Mon, 24 Aug 15
39/43

Comments: Published in MNRAS journal. arXiv admin note: substantial text overlap with arXiv:1411.4864, arXiv:1505.03968, arXiv:1505.03847

# Chaos in Terrestrial Planet Formation [EPA]

Terrestrial planets are thought to be the result of a vast number of gravitational interactions and collisions between smaller bodies. We use numerical simulations to show that practically identical initial conditions result in a wide array of final planetary configurations. This highly chaotic behaviour questions the predictability of different scenarios for the formation and evolution of our solar system and planetary systems in general. However, multiple realisations of the same initial conditions can be used to predict certain global statistics. We present two sets of numerical experiments that quantify this behaviour. Firstly, we demonstrate that simulations with slightly displaced particles are completely divergent after ~500 years, irrespective of initial displacement, particle number, and code accuracy. If a single planetesimal is moved by less than one millimetre, then a different set of planets results — this timescale for chaotic divergence decreases with increasing particle number. Secondly, we show final planetary configurations of initially similar simulations with and without giant planets after evolving them for ~148 Myr. We find that the same simulations including giant planets tend to generate higher mass planets at lower semi-major axes than simulations without gas giants. This prediction can be tested with forthcoming observational programs. By extracting outliers in the observations, we cautiously predict that Kepler-10, Kepler-9, 61 Vir, HD 134060, and HD 51608 may host as yet undetected giant planets.

V. Hoffmann, S. Grimm, B. Moore, et. al.
Thu, 6 Aug 15
48/48

Comments: 17 pages, 15 figures, submitted to MNRAS, simulation outputs available at this https URL