On the relevance of chaos for halo stars in the Solar Neighbourhood [GA]


We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun may not be as relevant as previously suggested in erasing phase space signatures of past Galactic accretion events. For this purpose, we analyse Solar Neighbourhood-like volumes extracted from cosmological simulations that naturally account for chaotic orbital behaviour induced by the strongly triaxial and cuspy shape of the resulting dark matter haloes, among other factors. In the approximation of an analytical static triaxial model, our results show that a large fraction of stellar halo particles in such local volumes have chaos onset times (i.e., the timescale at which stars commonly associated with chaotic orbits will exhibit their chaotic behaviour) significantly larger than a Hubble time. Furthermore, particles that do present a chaotic behaviour within a Hubble time do not exhibit significant diffusion in phase space.

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N. Maffione, F. Gomez, P. Cincotta, et. al.
Wed, 5 Aug 15

Comments: 20 pages, 16 figures. Accepted for publication in MNRAS

Anomalous scaling in magnetohydrodynamic turbulence: Effects of anisotropy and compressibility in the kinematic approximation [CL]


The field theoretic renormalization group and the operator product expansion are applied to the model of passive vector (magnetic) field advected by a random turbulent velocity field. The latter is governed by the Navier–Stokes equation for compressible fluid, subject to external random force with the covariance $\propto \delta(t-t’) k^{4-d-y}$, where $d$ is the dimension of space and $y$ is an arbitrary exponent. From physics viewpoints, the model describes magnetohydrodynamic turbulence in the so-called kinematic approximation, where the effects of the magnetic field on the dynamics of the fluid are neglected. The original stochastic problem is reformulated as a multiplicatively renormalizable field theoretic model; the corresponding renormalization group equations possess an infrared attractive fixed point. It is shown that various correlation functions of the magnetic field and its powers demonstrate anomalous scaling behavior in the inertial-convective range already for small values of~$y$. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields (“operators” in the quantum-field terminology), can be systematically calculated as series in $y$. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant.

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N. Antonov and M. Kostenko
Fri, 31 Jul 15

Comments: 14 pages, LaTeX source, two EPS figures. arXiv admin note: text overlap with arXiv:1410.1262

Spin-orbit coupling and chaotic rotation for circumbinary bodies. Application to the small satellites of the Pluto-Charon system [EPA]


Circumbinary bodies are objects that orbit around a more massive binary system. Here we show that, contrarily to the classical two-body problem, circumbinary bodies in planar quasi-circular orbits can present stable non-synchronous rotation. Denoting $n_b$ and $n$ the orbital mean motion of the binary and of the circumbinary body, respectively, there is an entirely new family of spin-orbit resonances at the frequencies $n\pm k\nu/2$, where $\nu = n_b – n$, and $k$ is an integer. In addition, when the natural rotational libration frequency has the same magnitude as $\nu$, the individual resonances overlap and the rotation becomes chaotic. We apply these results to the small satellites in the Pluto-Charon system. We conclude that the rotation of Nix and Styx can be chaotic, and that the rotation of Hydra and Kerberos is stable but not necessarily synchronous.

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A. Correia, A. Leleu and P. Robutel
Tue, 23 Jun 15

Comments: 5 pages, 4 figures, 1 appendix, submitted to Astronomy & Astrophysics

WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations [EPA]


We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer’s law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.

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H. Rein and D. Tamayo
Thu, 4 Jun 15

Comments: Accepted by MNRAS, 13 pages, 4 figures, source code and tutorials available at this http URL

Escape dynamics and fractal basin boundaries in Seyfert galaxies [GA]


The escape dynamics in a simple analytical gravitational model which describes the motion of stars in a Seyfert galaxy is investigated in detail. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the physical $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We also present evidence obtained through numerical simulations that our model can describe the formation and the evolution of the observed spiral structure in Seyfert galaxies. We hope our outcomes to be useful for a further understanding of the escape mechanism in galaxies with active nuclei.

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E. Zotos
Mon, 18 May 15

Comments: Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: substantial text overlap with arXiv:1411.4864, arXiv:1404.4285; text overlap with arXiv:1505.03847

Dynamical Evolution of Multi-Resonant Systems: the Case of GJ876 [EPA]


The GJ876 system was among the earliest multi-planetary detections outside of the Solar System, and has long been known to harbor a resonant pair of giant planets. Subsequent characterization of the system revealed the presence of an additional Neptune mass object on an external orbit, locked in a three body Laplace mean motion resonance with the previously known planets. While this system is currently the only known extrasolar example of a Laplace resonance, it differs from the Galilean satellites in that the orbital motion of the planets is known to be chaotic. In this work, we present a simple perturbative model that illuminates the origins of stochasticity inherent to this system and derive analytic estimates of the Lyapunov time as well as the chaotic diffusion coefficient. We then address the formation of the multi-resonant structure within a protoplanetary disk and show that modest turbulent forcing in addition to dissipative effects is required to reproduce the observed chaotic configuration. Accordingly, this work places important constraints on the typical formation environments of planetary systems and informs the attributes of representative orbital architectures that arise from extended disk-driven evolution.

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K. Batygin, K. Deck and M. Holman
Thu, 2 Apr 15

Comments: 15 pages, 7 figures, accepted to AJ

Symplectic integration for the collisional gravitational $N$-body problem [IMA]


We present a new symplectic numerical integrator designed for collisional gravitational $N$-body problems which makes use of Kepler solvers. The integrator is also reversible and conserves 9 integrals of motion of the $N$-body problem to machine precision. The integrator is second order, but the order can easily be increased by the method of \citeauthor{yos90}. We use fixed time step in all tests studied in this paper to ensure preservation of symplecticity. We study small $N$ collisional problems and perform comparisons with typically used integrators. In particular, we find comparable or better performance when compared to the 4th order Hermite method and much better performance than adaptive time step symplectic integrators introduced previously. The integrator is a promising tool in collisional gravitational dynamics. We plan larger $N$ tests of the method in future work.

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D. Hernandez and E. Bertschinger
Wed, 11 Mar 15

Comments: 11 pages, 7 Figures, to be submitted to MNRAS, comments welcome

The Relative Lyapunov Indicators: Theory and Application to Dynamical Astronomy [CL]


A recently introduced chaos detection method, the Relative Lyapunov Indicator (RLI) is investigated in the cases of symplectic mappings and continuous Hamiltonian systems. It is shown that the RLI is an efficient numerical tool in determining the true nature of individual orbits, and in separating ordered and chaotic regions of the phase space of dynamical systems. A comparison between the RLI and some other variational indicators are presented, as well as the recent applications of the RLI to various problems of dynamical astronomy.

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Z. Sandor and N. Maffione
Thu, 29 Jan 15

Comments: 39 pages, 21 figures. Non proof read version of the paper accepted in Lecture Notes in Physics

Fractal dimension and turbulence in Giant HII Regions [GA]


We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble’s Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.

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H. Caicedo-Ortiz, E. Santiago-Cortes, J. Lopez-Bonilla, et. al.
Wed, 21 Jan 15

Comments: 5 pages, 4 figures

Low-energy capture of asteroids onto KAM tori [EPA]


We present a new method for engineering the artificial capture of asteroids. Based on theories of the chaos-assisted capture of natural satellites of the giant planets, we show how an unbound asteroid that passes close to a regular region of phase space can be easily moved onto the nearby KAM tori and essentially permanently captured with the Earth’s Hill sphere without closing the zero velocity curves. The method has the advantages of a relatively low delta-v requirement and no need for control strategies. An illustration of the method is given for an example asteroid trajectory, demonstrating that it is a viable strategy for the final capture stage of asteroids in the Earth’s neighbourhood.

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P. Verrier and C. McInnes
Thu, 15 Jan 15

Comments: 13 pages, 3 figures, accepted by the Journal of Guidance, Control, and Dynamics

Strange nonchaotic stars [CL]


The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler’s “golden” stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.

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J. Lindner, V. Kohar, B. Kia, et. al.
Fri, 9 Jan 15

Comments: 5 pages, 4 figures, accepted for publication in Physical Review Letters

The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection [CL]


We provide a concise presentation of the Smaller (SALI) and the Generalized Alignment Index (GALI) methods of chaos detection. These are efficient chaos indicators based on the evolution of two or more, initially different, deviation vectors from the studied orbit. After explaining the motivation behind the introduction of these indices, we summon up the behaviors they exhibit for regular and chaotic motion, as well as for stable and unstable periodic orbits, focusing mainly on finite-dimensional conservative systems: autonomous Hamiltonian models and symplectic maps. We emphasize the advantages of these methods in studying the global dynamics of a system, as well as their ability to identify regular motion on low dimensional tori. Finally we discuss several applications of these indices to problems originating from different scientific fields like celestial mechanics, galactic dynamics, accelerator physics and condensed matter physics.

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C. Skokos and T. Manos
Wed, 24 Dec 14

Comments: 51 pages, 23 figures

Transient chaos and fractal structures in planetary feeding zones [EPA]


The circular restricted three body problem is investigated in the context of accretion and scattering processes. In our model a large number of identical non-interacting mass-less planetesimals are considered in planar case orbiting a star-planet system. This description allows us to investigate in dynamical systems approach the gravitational scattering and possible captures of the particles by the forming planetary embryo. Although the problem serves a large variety of complex motion, the results can be easily interpreted because of the low dimensionality of the phase space. We show that initial conditions define isolated regions of the disk, where accretion or escape of the planetesimals occur, these have, in fact, a fractal structure. The fractal geometry of these “basins” implies that the dynamics is very complex. Based on the calculated escape rates and escape times, it is also demonstrated that the planetary accretion rate is exponential for short times and follows a power-law for longer integration. A new numerical calculation of the maximum mass that a planet can reach (described by the expression of the isolation mass) is also derived.

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T. Kovacs and Z. Regaly
Fri, 5 Dec 14

Comments: 6 pages, 4 figures, accepted to ApJ Letters

The effect of Poynting-Robertson drag on the triangular Lagrangian points [EPA]


We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three- body problem, subject to the radial component of Poynting-Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to beta a1 n1 , where beta is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.

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C. Lhotka and A. Celletti
Fri, 5 Dec 14

Comments: 47 pages, 8 figures,

Fidelity and Reversibility in the Three Body Problem [CL]


We present two methods to analyse the global effects of a small perturbation in a non-integrable Hamiltonian system, choosing as a paradigmatic example the restricted planar three body problem and focusing on its Poincar\`e map for the Jacobi invariant. The cumulative effects on the orbit of random or round-off errors leads to a divergence of the perturbed orbit from the exact one. Rather than computing the distance of the perturbed orbit from the reference one after a given number n of time steps, we measure the distance of the reversed orbit (n time steps forwards and backwards) from the initial point. This approach does not require the knowledge of the unperturbed map. The asymptotic equivalence of the Reversibility Error Method (REM) with the forward error is proved for noisy linear maps, and it is shown to characterize the phase space stability of the perturbed map just as the Lyapunov Characteristic Exponent. A second indicator of chaos, the Cumulative Orbital Elements (COE) method is also presented. The loss of memory of the perturbed map is quantified by the Fidelity and its decay rate. It is found that Fidelity behaves in a different way for randomly perturbed regular and for chaotic orbits. This property, already known for one-dimensional maps, is confirmed for the considered planar three body problem suggesting a possible validity for generic hyperbolic systems.

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F. Panichi, L. Ciotti and G. Turchetti
Wed, 3 Dec 14

Comments: 20 pages, 19 figures

Revealing the escape mechanism of three-dimensional orbits in a tidally limited star cluster [GA]


The aim of this work is to explore the escape process of three-dimensional orbits in a star cluster rotating around its parent galaxy in a circular orbit. The gravitational field of the cluster is represented by a smooth, spherically symmetric Plummer potential, while the tidal approximation was used to model the steady tidal field of the galaxy. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. It is of particular interest to locate the escape basins towards the two exit channels and relate them with the corresponding escape times of the orbits. For this purpose, we split our investigation into three cases depending on the initial value of the $z$ coordinate which was used for launching the stars. The most noticeable finding is that the majority of stars initiated very close to the primary $(x,y)$ plane move in chaotic orbits and they remain trapped for vast time intervals, while orbits with relatively high values of $z_0$ on the other hand, form well-defined basins of escape. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the dissolution process and the escape mechanism in open star clusters.

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E. Zotos
Wed, 19 Nov 14

Comments: Published in MNRAS journal. arXiv admin note: text overlap with arXiv:1404.4285

Determining the nature of orbits in a three-dimensional galaxy model hosting a BL Lacertae object [GA]


A three-dimensional dynamical model for a galaxy hosting a BL Lacertae object is constructed. The model consists of a logarithmic potential representing an elliptical host galaxy with a bulge of radius $c_b$ and a dense massive nucleus. Using numerical experiments, we try to distinguish between regular and chaotic motion in both 2D and 3D system. In particular, we investigate how the basic parameters of our model, such as the mass of the nucleus, the internal perturbation and the flattening parameters influence the amount and the degree of chaos. Interesting correlations are presented for both 2D and 3D dynamical models. Our numerical results are explained and supported using elementary theoretical arguments and analytical calculations. Of particular interest, is the local integral of motion which have been found to exist in the vicinity of stable periodic points. The obtained numerical outcomes of the present research, are linked and also compared with several data derived from observations.

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E. Zotos
Tue, 28 Oct 14

Comments: Published in Astronomische Nachrichten (AN) journal

The phase-space of boxy-peanut and X-shaped bulges in galaxies I. Properties of non-periodic orbits [CL]


The investigation of the phase-space properties of structures encountered in a dynamical system is essential for understanding their formation and enhancement. In the present paper we explore the phase space in energy intervals where we have orbits that act as building blocks for boxy-peanut (b/p) and “{\sf X}-shaped” structures in rotating potentials of galactic type. We underline the significance of the rotational tori around the 3D families x1v1 and x1v1$^{\prime}$ that have been bifurcated from the planar x1 family. These tori play a multiple role: (i) They belong to quasi-periodic orbits that reinforce the local density. (ii) They act as obstacles for the diffusion of chaotic orbits and (iii) they attract a large number of chaotic orbits that become sticky to them. There are also bifurcations of unstable families (x1v2, x1v2$^{\prime}$). Their unstable asymptotic curves wind around the x1v1 and x1v1$^{\prime}$ tori generating orbits with hybrid morphologies between that of x1v1 and x1v2. In addition, a new family of multiplicity 2, called x1mul2, is found to be important for the peanut construction. Our work shows also that there are peanut-supporting orbits before the vertical ILR. Non-periodic orbits associated with the x1 family secure this contribution as well as the support of b/p structures at several other energy intervals. Non-linear phenomena associated with complex instability of single and double multiplicity families of periodic orbits show that these structures are not interrupted in regions where such orbits prevail. Depending on the main mechanism behind their formation, boxy bulges exhibit different morphological features. Finally our analysis indicates that “X” features shaped by orbits in the neighbourhood of x1v1 and x1v1$^{\prime}$ periodic orbits are pronounced only in side-on or nearly end-on views of the bar.

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P. Patsis and M. Katsanikas
Tue, 21 Oct 14

Comments: 22 pages, 24 figures, accepted for publication in the MNRAS

The phase-space of boxy-peanut and X-shaped bulges in galaxies II. The relation between face-on and edge-on boxiness [GA]


We study the dynamical mechanisms that reinforce the formation of boxy structures in the \textit{inner} regions, roughly in the middle, of bars observed nearly \textit{face-on}. Outer boxiness, at the ends of the bars, is usually associated with orbits at the inner, radial 4:1 resonance region and can be studied with 2D dynamics. However, in the middle of the bar dominate 3D orbits that give boxy/peanut bulges in the edge-on views of the models. In the present paper we show that 3D quasi-periodic, as well as 3D chaotic orbits sticky to the x1v1 and x1v1$^{\prime}$ tori, especially from the Inner Lindblad Resonance (ILR) region, have boxy projections on the equatorial plane of the bar. The majority of vertically perturbed 2D orbits, initially on the equatorial plane in the ILR resonance region, enhance boxy features in face-on bars. Orbits that build a bar by supporting sharp “{\sf X}” features in their side-on views at energies \textit{beyond} the ILR, may also have a double boxy character. If populated, the extent of the inner boxiness along the major axis is about the same with that of the peanut supporting orbits in the side-on views. At any rate these orbits do not obscure the observation of the boxy orbits of the ILR region in the face-on views, as they contribute more to the surface density at the sides of the bar than to their central parts.

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P. Patsis and M. Katsanikas
Tue, 21 Oct 14

Comments: 12 pages, 13 figures, accepted for publication in the MNRAS

Symplectic map description of Halley's comet dynamics [EPA]


The main features of 1P/Halley chaotic dynamics can be described by a two dimensional symplectic map. Using Mel’nikov integral we semi-analytically determine such a map for 1P/Halley taking into account gravitational interactions from the Sun and the eight planets. We determine the Solar system kick function ie the energy transfer to 1P/Halley along one passage through the Solar system. Our procedure allows to compute for each planet its contribution to the Solar system kick function which appears to be the sum of the Keplerian potential of the planet and of a rotating circular gravitational dipole potential due to the Sun movement around Solar system barycenter. We test the robustness of the symplectic Halley map by directly integrating Newton’s equations over $\sim 2.4\cdot 10^4$ yr around Y2K and by reconstructing the Solar system kick function. Our results show that the Halley map with fixed parameters gives a reliable description of comet dynamics on time scales of $10^4$ yr while on a larger scales the parameters of the map are slowly changing due to slow oscillations of orbital momentum.

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P. Haag, G. Rollin and J. Lages
Wed, 15 Oct 14

Comments: 8 pages, 6 figures

Intermittency in Weak Magnetohydrodynamic Turbulence [CL]


Intermittency is investigated using decaying direct numerical simulations of incompressible weak magnetohydrodynamic turbulence with a strong uniform magnetic field ${\bf b_0}$ and zero cross-helicity. At leading order, this regime is achieved via three-wave resonant interactions with the scattering of two of these waves on the third/slow mode for which $k_{\parallel} = 0$. When the interactions with the slow mode are artificially reduced the system exhibits an energy spectrum with $k_{\perp}^{-3/2}$, whereas the expected exact solution with $k_{\perp}^{-2}$ is recovered with the full nonlinear system. In the latter case, strong intermittency is found when the vector separation of structure functions is taken transverse to ${\bf b_0}$ – at odds with classical weak turbulence where self-similarity is expected. This surprising result, which is being reported here for the first time, may be explained by the influence of slow modes whose regime belongs to strong turbulence. We derive a new log–Poisson law, $\zeta_p = p/8 +1 -(1/4)^{p/2}$, which fits perfectly the data and highlights the dominant role of current sheets.

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R. Meyrand, K. Kiyani and S. Galtier
Tue, 9 Sep 14

Comments: N/A

Escape distribution for an inclined billiard [EPA]


H${\acute{e}}$non [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hill’s problem.

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A. Roy and N. Georgakarakos
Tue, 26 Aug 14

Comments: This is a preprint of a paper published in ‘Regular and Chaotic Dynamics’ in 2012

Galaxies with Supermassive Binary Black Holes: (II) A Model with Cuspy Galactic Density Profiles [GA]


The existence and uniqueness of equilibrium points, including Lagrange Points and Jiang-Yeh Points, of a galactic system with supermassive binary black holes embedded in a centrally cuspy galactic halo are investigated herein. Differing from the previous results of non-cuspy galactic profiles that Jiang-Yeh Points only exist under a particular condition, it is found here that the Lagrange Points, L2, L3, L4 and L5, Jiang-Yeh Points, JY1 and JY2, exist under general conditions. The stability analysis shows that L2, L3, JY1 and JY2 are unstable. However, L4 and L5 are only unstable when the galactic total mass is smaller than a critical mass; otherwise they become neutrally stable centers. These results will be important for further studies on the cores of early-type galaxies.

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I. Jiang and L. Yeh
Mon, 4 Aug 14

Comments: 20 pages, 4 figures, accepted for publication in Astrophysics and Space Science

Continuation and stability deduction of resonant periodic orbits in three dimensional systems [EPA]


In dynamical systems of few degrees of freedom, periodic solutions consist the backbone of the phase space and the determination and computation of their stability is crucial for understanding the global dynamics. In this paper we study the classical three body problem in three dimensions and use its dynamics to assess the long-term evolution of extrasolar systems. We compute periodic orbits, which correspond to exact resonant motion, and determine their linear stability. By computing maps of dynamical stability we show that stable periodic orbits are surrounded in phase space with regular motion even in systems with more than two degrees of freedom, while chaos is apparent close to unstable ones. Therefore, families of stable periodic orbits, indeed, consist backbones of the stability domains in phase space.

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K. Antoniadou, G. Voyatzis and H. Varvoglis
Tue, 29 Jul 14

Comments: Proceedings of the 6th International Conference on Numerical Analysis (NumAn 2014). Published by the Applied Mathematics and Computers Lab, Technical University of Crete (AMCL/TUC), Greece

Lagrangian Coherent Structures from Video Streams of Jupiter [CL]


Jupiter’s fast rotation – one rotation over 10 hours – creates strong jet streams, smearing its clouds into linear bands of dark and light zonal belts that circle the planet on lines of almost constant latitude. Such a high degree of axisymmetry is absent in our own atmosphere. Moreover, Jupiter has the largest and longest-living known atmospheric vortex, the Great Red Spot (GRS). Such vortices abound in nature, but GRS’s size, long-term persistence, and temporal longitudinal oscillations make it unique. Here, we uncover, for the first time, unsteady material structures that form the cores of zonal jets and the boundary of the GRS in Jupiter’s atmosphere. We perform our analysis on a velocity field extracted from a video footage acquired by the NASA Cassini spacecraft.

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A. Hadjighasem and G. Haller
Wed, 16 Jul 14

Comments: to be published in Oberwolfach Reports, in press

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar [GA]


We investigate the dynamics in the close vicinity of and within the critical area in a 2D effective galactic potential with a bar of Zotos. We have calculated Poincar\’e surfaces of section and the basins of escape. In both the Poincar\’e surfaces of section and the basins of escape we find numerical evidence for the existence of a separatrix which hinders orbits from escaping out of the bar region. We present numerical evidence for the similarity between spiral arms of barred spiral galaxies and tidal tails of star clusters.

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A. Ernst and T. Peters
Thu, 3 Jul 14

Comments: 12 pages, 10 figures, accepted by MNRAS

Refined Critical Balance in Strong Alfvénic Turbulence [SSA]


We present numerical evidence that in strong Alfv\’enic turbulence, the critical balance principle—equality of the nonlinear decorrelation and linear propagation times—is scale invariant, in the sense that the probability distribution of the ratio of these times is independent of scale. This result only holds if the local alignment of the Elsasser fields is taken into account in calculating the nonlinear time. At any given scale, the degree of alignment is found to increase with fluctuation amplitude, supporting the idea that the cause of alignment is mutual dynamical shearing of Elsasser fields. The scale-invariance of critical balance (while all other quantities of interest are strongly intermittent, i.e., have scale-dependent distributions) suggests that it is the most robust of the scaling principles used to describe Alfv\’enic turbulence.

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A. Mallet, A. Schekochihin and B. Chandran
Tue, 24 Jun 14

Comments: revtex, 5 pages, 5 figures

Self-organisation and non-linear dynamics in driven magnetohydrodynamic turbulent flows [CL]


Magnetohydrodynamic turbulent flows driven by random mechanical and electromagnetic external forces of zero helicities are investigated by means of direct numerical simulations. It is shown that despite the absence of helicities in the forcing, the system is attracted to self-organized helical states that exhibit laminar behaviour despite the large value of the Reynolds numbers examined. We demonstrate that the correlation time of the external forces is controlling the time spent on these states, i.e. for short correlation times the system remains in the turbulent state while as the correlation time is increased the system spends more and more time in the self-organised states. As a result, time averaged statistics can significantly be affected by the time spent on these states. These results have important theoretical implications for the understanding of the suppression of non-linearities in plasma fusion devises as well as in astrophysical observations.

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V. Dallas and A. Alexakis
Fri, 13 Jun 14

Comments: 7 pages, 6 figures

Torus-fitting method for obtaining action variables in two-dimensional Galactic potentials [GA]


A phase-space distribution function of the steady state in galaxy models that admits regular orbits overall in the phase-space can be represented by a function of three action variables. This type of distribution function in Galactic models is often constructed theoretically for comparison of the Galactic models with observational data as a test of the models. On the other hand, observations give Cartesian phase-space coordinates of stars. Therefore it is necessary to relate action variables and Cartesian coordinates in investigating whether the distribution function constructed in galaxy models can explain observational data. Generating functions are very useful in practice for this purpose, because calculations of relations between action variables and Cartesian coordinates by generating functions do not require a lot of computational time or computer memory in comparison with direct numerical integration calculations of stellar orbits. Here, we propose a new method called a torus-fitting method, by which a generating function is derived numerically for models of the Galactic potential in which almost all orbits are regular. We confirmed the torus-fitting method can be applied to major orbit families (box and loop orbits) in some two-dimensional potentials. Furthermore, the torus-fitting method is still applicable to resonant orbit families, besides major orbit families. Hence the torus-fitting method is useful for analyzing real Galactic systems in which a lot of resonant orbit families might exist.

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H. Ueda, T. Hara, N. Gouda, et. al.
Tue, 13 May 14

Comments: 12 pages, 14 figures

Analytical invariant manifolds near unstable points and the structure of chaos [CL]


It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.

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C. Efthymiopoulos, G. Contopoulos and M. Katsanikas
Thu, 1 May 14

Comments: (in press)

Determining the nature of orbits in disk galaxies with non spherical nuclei [CL]


We investigate the regular or chaotic nature of orbits of stars moving in the meridional plane $(R,z)$ of an axially symmetric galactic model with a flat disk and a central, non spherical and massive nucleus. In particular, we study the influence of the flattening parameter of the central nucleus on the nature of orbits, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits of the main regular families. In an attempt to maximize the accuracy of our results upon distinguishing between regular and chaotic motion, we use both the Fast Lyapunov Indicator (FLI) and the Smaller ALingment Index (SALI) methods to extensive samples of orbits obtained by integrating numerically the equations of motion as well as the variational equations. Moreover, a technique which is based mainly on the field of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used for identifying the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. Varying the value of the flattening parameter, we study three different cases: (i) the case where we have a prolate nucleus (ii) the case where the central nucleus is spherical and (iii) the case where an oblate massive nucleus is present. Furthermore, we present some additional findings regarding the reliability of short time (fast) chaos indicators, as well as a new method to define the threshold between chaos and regularity for both FLI and SALI, by using them simultaneously. Comparison with early related work is also made.

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E. Zotos and N. Caranicolas
Wed, 16 Apr 14

Hyperchaotic Intermittent Convection in a Magnetized Viscous Fluid [CL]


We consider a low-dimensional model of convection in a horizontally magnetized layer of a viscous fluid heated from below. We analyze in detail the stability of hydromagnetic convection for a wide range of two control parameters. Namely, when changing the initially applied temperature difference or magnetic field strength, one can see transitions from regular to irregular long-term behavior of the system, switching between chaotic, periodic, and equilibrium asymptotic solutions. It is worth noting that owing to the induced magnetic field a transition to hyperchaotic dynamics is possible for some parameters of the model. We also reveal new features of the generalized Lorenz model, including both type I and III intermittency.

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W. Macek and M. Strumik
Wed, 26 Mar 14

Chaos in hydrodynamic BL Herculis models [SSA]


We present non-linear, convective, BL Her-type hydrodynamic models that show complex variability characteristic for deterministic chaos. The bifurcation diagram reveals a rich structure, with many phenomena detected for the first time in hydrodynamic models of pulsating stars. The phenomena include not only period doubling cascades en route to chaos (detected in earlier studies) but also periodic windows within chaotic band, type-I and type-III intermittent behaviour, interior crisis bifurcation and others. Such phenomena are known in many textbook chaotic systems, from the simplest discrete logistic map, to more complex systems like Lorenz equations.
We discuss the physical relevance of our models. Although except of period doubling such phenomena were not detected in any BL Her star, chaotic variability was claimed in several higher luminosity siblings of BL Her stars – RV Tau variables, and also in longer-period, luminous irregular pulsators. Our models may help to understand these poorly studied stars. Particularly interesting are periodic windows which are intrinsic property of chaotic systems and are not necessarily caused by resonances between pulsation modes, as sometimes claimed in the literature.

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R. Smolec and P. Moskalik
Fri, 21 Mar 14

Periodic Orbits, Chaos and Manifolds near the Equilibrium Points in the Rotating Plane-Symmetric Potential Field [EPA]


This paper studied the periodic orbits, manifolds and chaos in a rotating plane-symmetric potential field. It is found that the dynamical behaviour near the equilibrium point is totally decided by the structure of the submanifolds and subspaces near the equilibrium point. The non-degenerate equilibrium points are classified into twelve cases. The necessary and sufficient conditions for the linearly stable, non-resonant unstable and resonant equilibrium points are established. Furthermore, it is found that the resonant equilibrium point is a Hopf bifurcation point which leads to the chaotic motion near the resonant equilibrium point; it is found the appearing and disappearing of periodic orbit families near resonant equilibrium points with parametric variation. Besides, it is discovered that the number of periodic orbit families depends on the structure of the submanifolds. In the end, the theory developed here is applied to two particular cases, the rotating homogeneous cube and the circular restricted three-body problem.

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Y. Jiang, H. Baoyin and X. Wang
Tue, 11 Mar 14

Chaotic enhancement of dark matter density in binary systems and galaxies [EPA]


Using symplectic map description we study the capture of galactic dark matter particles (DMP) in two-body and few-body galaxies. This approach allows to model scattering of $10^{16}$ DMP following time evolution of captured particle on about $10^9$ orbital periods. We obtain DMP density distribution inside such galaxies and determine the enhancement factor of their density in galactic center compared to its inter-galactic value as a function of mass ratio of galactic bodies and a ratio of body velocity to velocity of galactic DMP wind. We find that the enhancement factor can be of the order of ten thousands.

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G. Rollin, J. Lages and D. Shepelyansky
Tue, 4 Mar 14

Complexity Phenomena and ROMA of the Magnetospheric Cusp, Hydrodynamic Turbulence, and the Cosmic Web [CEA]


Dynamic Complexity is a phenomenon exhibited by a nonlinearly interacting system within which multitudes of different sizes of large scale coherent structures emerge, resulting in a globally nonlinear stochastic behavior vastly different from that could be surmised from the underlying equations of interaction. The hallmark of such nonlinear, complex phenomena is the appearance of intermittent fluctuating events with the mixing and distributions of correlated structures at all scales. We briefly review here a relatively recent method, ROMA (rank-ordered multifractal analysis), explicitly constructed to analyze the intricate details of the distribution and scaling of such types of intermittent structures. This method is then applied to the analyses of selected examples related to the dynamical plasmas of the cusp region of the magnetosphere, velocity fluctuations of classical hydrodynamic turbulence, and the distribution of the structures of the cosmic gas obtained through large scale, moving mesh simulations. Differences and similarities of the analyzed results among these complex systems will be contrasted and highlighted. The first two examples have direct relevance to the geospace environment and are summaries of previously reported findings. The third example on the cosmic gas, though involving phenomena much larger in spatiotemporal scales, with its highly compressible turbulent behavior and the unique simulation technique employed in generating the data, provides direct motivations of applying such analysis to studies of similar multifractal processes in various extreme environments. These new results are both exciting and intriguing.

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T. Chang, C. Wu, M. Echim, et. al.
Thu, 27 Feb 14

Fractal and Multifractal Analysis of the Rise of Oxygen in Earth's Early Atmosphere [CL]


The rise of oxygen in Earth’s atmosphere that occurred 2.4 to 2.2 billion years ago is known as the Earth’s Great Oxidation, and its impact on the development of life on Earth was profound. The proposed underlying mathematical models are based on physical parameters whose values are currently not well-established owing to uncertainties in geological and biological data. In this paper, a previously developed model of Earth’s atmosphere is modified by adding different strengths of noise to account for the parameters’ uncertainties. The effects of the noise on time variations of oxygen, carbon and methane in Earth’s early atmosphere are investigated by using fractal and multifractal analysis. We show that these time variations cannot properly be described by a single fractal dimension because they exhibit multifractal characteristics. The obtained results also demonstrate that our time series exhibit the multifractality caused by the long-range time correlations.

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S. Kumar, Z. Musielak and M. Cuntz
Fri, 14 Feb 14

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow [CL]


It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to a case of compressible flow of cosmological relevance, governed by the Euler-Poisson equations.

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Wed, 25 Dec 13

Dynamical resonance locking in tidally interacting binary systems [SSA]


We examine the dynamics of resonance locking in detached, tidally interacting binary systems. In a resonance lock, a given stellar or planetary mode is trapped in a highly resonant state for an extended period of time, during which the spin and orbital frequencies vary in concert to maintain the resonance. This phenomenon is qualitatively similar to resonance capture in planetary dynamics. We show that resonance locks can accelerate the course of tidal evolution in eccentric systems and also efficiently couple spin and orbital evolution in circular binaries. Previous analyses of resonance locking have not treated the mode amplitude as a fully dynamical variable, but rather assumed the adiabatic (i.e. Lorentzian) approximation valid only in the limit of relatively strong mode damping. We relax this approximation, analytically derive conditions under which the fixed point associated with resonance locking is stable, and further check these analytic results using numerical integrations of the coupled mode, spin, and orbital evolution equations. These show that resonance locking can sometimes take the form of complex limit cycles or even chaotic trajectories. We provide simple analytic formulae that define the binary and mode parameter regimes in which resonance locks of some kind occur (stable, limit cycle, or chaotic). We briefly discuss the astrophysical implications of our results for white dwarf and neutron star binaries as well as eccentric stellar binaries.

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Thu, 19 Dec 13

Models of magnetic-field evolution and effective viscosity in weakly collisional extragalactic plasmas [CEA]


In weakly collisional extragalactic plasmas such as the intracluster medium, viscous stress and the rate of change of the magnetic-field strength are proportional to the local pressure anisotropy, so subject to constraints imposed by the pressure-anisotropy-driven mirror and firehose instabilities and controlled by the local instantaneous plasma beta. The dynamics of such plasmas is dramatically different from a conventional MHD fluid. The plasma is expected to stay locally in a marginal state with respect to the instabilities, but how it does this is an open question. Two models of magnetic-field evolution are investigated. In the first, marginality is achieved via suppression of the rate of change of the field. In the second, the instabilities give rise to anomalous collisionality, reducing pressure anisotropy to marginal – at the same time decreasing viscosity and so increasing the turbulent rate of strain. Implications of these models are studied in a simplified 0D setting. In the first model, the field grows explosively but on a time scale that scales with initial beta, while in the second, dynamical field strength can be reached in one large-scale turbulence turn-over time regardless of the initial seed. Both models produce very intermittent fields. Both also suffer from strong constraints on their applicability: for typical cluster-core conditions, scale separation between the fluid motions and the microscale fluctuations breaks down at beta~10^4. At larger beta (weaker fields), a fully collisionless plasma dynamo theory is needed in order to justify the growth of the field from a tiny primordial seed. However, the models discussed here are appropriate for studying the structure of the currently observed field as well as large-scale dynamics and thermodynamics of the magnetized ICM or similarly dilute astrophysical plasmas.

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Mon, 16 Dec 13

Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models [GA]


Self-consistent N-body simulations are an efficient tool to study galactic dynamics. However, it can be challenging to use them for the detailed study of individual trajectories (or ensembles of trajectories). Such orbital studies are important to shed light on global phase space properties, which are the underlying cause of observed structures. The potentials needed to describe self-consistent models are in this case time-dependent. For this reason, we aim to investigate the different dynamical properties (such as regular and chaotic motion) of a non-autonomous galactic system, whose time-dependent potential adequately mimics certain realistic trends arising from N-body barred galaxy simulations. We construct a fully time-dependent analytical model, which manages to capture and reproduce several features of an N-body simulation. We model the gravitational potentials of three components (disc, bar and dark matter halo), whose time-dependent parameters are derived from an N-body simulation. We start by studying the dynamical stability of its reduced time-independent 2-degrees of freedom model by charting the different islands of stability associated with certain orbital morphologies and detecting the chaotic and regular regions. We then turn our interest to the full 3-degrees of freedom time-dependent case, where we show a few representative trajectories which experience different typical dynamical behaviours, i.e., an interplay between regular and chaotic motion for different epochs. Finally, we focus on the study of the underlying global dynamical transitions of the time-dependent system in terms of estimating the relative total fraction of (un)stable motion of an ensemble of initial conditions taken from the simulation and evolved with the time-dependent potential. We find that, for such an ensemble, the fraction of regular motion increases with time.

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Fri, 15 Nov 13

Galaxies with Supermassive Binary Black Holes: (I) A Possible Model for the Centers of Core Galaxies [GA]


The dynamics of galactic systems with central binary black holes is studied. The model is a modification from the restricted three body problem, in which a galactic potential is added as an external potential. Considering the case with an equal mass binary black holes, the conditions of existence of equilibrium points, including Lagrange Points and additional new equilibrium points, i.e. Jiang-Yeh Points, are investigated. A critical mass is discovered to be fundamentally important. That is, Jiang-Yeh Points exist if and only if the galactic mass is larger than the critical mass. The stability analysis is performed for all equilibrium points. The results that Jiang-Yeh Points are unstable could lead to the core formation in the centers of galaxies.

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Thu, 7 Nov 13