Tag Archives: Dynamical Systems

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

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http://arxiv.org/abs/1512.00707


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.

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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

On the relativistic Lagrange-Laplace secular dynamics for extrasolar systems [CL]

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http://arxiv.org/abs/1510.06523


We study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

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M. Sansottera, L. Grassi and A. Giorgilli
Fri, 23 Oct 15
19/63

Comments: 4 pages, 4 figures, Proceedings IAU Symposium No. S310 (Complex Planetary Systems)

Effective resonant stability of Mercury [CL]

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http://arxiv.org/abs/1510.06543


Mercury is the unique known planet that is situated in a 3:2 spin-orbit resonance nowadays. Observations and models converge to the same conclusion: the planet is presently deeply trapped in the resonance and situated at the Cassini state $1$, or very close to it. We investigate the complete non-linear stability of this equilibrium, with respect to several physical parameters, in the framework of Birkhoff normal form and Nekhoroshev stability theory. We use the same approach adopted for the 1:1 spin-orbit case with a peculiar attention to the role of Mercury’s non negligible eccentricity. The selected parameters are the polar moment of inertia, the Mercury’s inclination and eccentricity and the precession rates of the perihelion and node. Our study produces a bound to both the latitudinal and longitudinal librations (of 0.1 radians) for a long but finite time (greatly exceeding the age of the solar system). This is the so-called effective stability time. Our conclusion is that Mercury, placed inside the 3:2 spin-orbit resonance, occupies a very stable position in the space of these physical parameters, but not the most stable possible one.

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M. Sansottera, C. Lhotka and A. Lemaitre
Fri, 23 Oct 15
56/63

Comments: 9 pages

Conic-Helical Orbits of Planets around Binary Stars do not Exist [CL]

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http://arxiv.org/abs/1510.05345


Oks proposes the existence of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis. We show that planetary orbits initially meeting this description will not continue to do so as the binary pair rotates.

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G. Egan
Tue, 20 Oct 15
92/92

Comments: 3 pages

Multiscale functions, Scale dynamics and Applications to partial differential equations [CL]

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http://arxiv.org/abs/1509.01048


Modeling phenomena from experimental data, always begin with a \emph{choice of hypothesis} on the observed dynamics such as \emph{determinism}, \emph{randomness}, \emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \emph{“With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \emph{multi-scale functions}, \emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale calculus} (see \cite{bohn}). These definitions will be illustrated on the \emph{multi-scale Okamoto’s functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \emph{scale regimes} whereas the equation is \emph{scale invariant}. A typical example of such an equation, is the \emph{Euler-Lagrange equation} and particularly the \emph{Newton’s equation} which will be discussed. Notably, we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton’s equation} and also the \emph{non-linear Schr\”odinger equation} via the \emph{scale Newton’s equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schr\”odinger equation}.

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J. Cresson and F. Pierret
Fri, 4 Sep 15
34/58

Comments: N/A

Finite-Time Singularities in $k=0$ FLRW Cosmologies [CL]

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http://arxiv.org/abs/1507.02241


In this paper, we consider a spatially flat FLRW cosmological model with matter obeying a barotropic equation of state $p = w \mu$, $-1<w\leq1$, and a cosmological constant, $\Lambda$. We use Osgood’s criterion to establish three cases when such models admit finite-time singularities. The first case is for an arbitrary initial condition, with a negative cosmological constant, and phantom energy $w < -1$. We show that except for a very fine-tuned choice of the initial condition $\theta_{0}$, the universe will develop a finite-time singularity. The second case we consider is for a nonnegative cosmological constant, phantom energy, and the expansion scalar being larger than that of the flat-space de Sitter solution, and show that such models only expand forever for $\Lambda = 0$. In all other cases, the universe model develops a finite-time singularity. The final case we consider is for a nonnegative cosmological constant, a matter source with $-1 < w \leq 1$, and an expansion scalar that is asymptotically that of the de Sitter universe. We show that such models will only expand forever when $\Lambda = 0$, otherwise, they will develop a finite-time singularity. This is significant, since the inflationary epoch is a subset of this domain. However, as we show, the inclusion of a bulk viscosity term in the Einstein field equations eliminates this singularity, and the universe expands forever. This could have interesting implications for the role of bulk viscosity in dynamical models of the universe.

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I. Kohli
Thu, 9 Jul 15
7/50

Comments: arXiv admin note: text overlap with arXiv:1505.07770

Shadowing Lemma and Chaotic Orbit Determination [EPA]

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http://arxiv.org/abs/1506.03221


Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. In a simple discrete model, the standard map, we tackle the problem of chaotic orbit determination when observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing orbit is computed by the least squares orbit determination. We test both the convergence of the orbit determination iterative procedure and the behaviour of the uncertainties as a function of the maximum number $n$ of map iterations observed. When the initial conditions belong to a chaotic orbit, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula. Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is added to the initial conditions as parameter to be estimated. The uncertainty of the dynamical parameter decreases like $n^a$ with $a<0$ but not large (of the order of unity). If only the initial conditions are estimated, their uncertainty decreases exponentially with $n$. If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty is polynomial in all parameters, like $n^a$ with $a\simeq 1/2$. The Shadowing Lemma does not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena have significant implications, which remain to be studied, in practical problems of orbit determination involving chaos, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches.

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F. Spoto and A. Milani
Thu, 11 Jun 15
15/55

Comments: N/A

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem [EPA]

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http://arxiv.org/abs/1506.02870


We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamiltonian, we estimate the size of the transformation that maps this Hamiltonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.

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P. Robutel and L. Niederman
Wed, 10 Jun 15
51/53

Comments: N/A

On Singularities in Cosmic Inflation [CL]

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http://arxiv.org/abs/1505.07770


In this paper, we examine a flat FLRW spacetime with a scalar field potential and show by applying Osgood’s criterion to the Einstein field equations that all such models, irrespective of the particular choice of potential develop finite-time singularities. That is, we show that solutions to the field equations rapidly diverge in finite time. This can have important implications for the role of inflation in cosmological models, since one of the implications of this is that within the inflationary epoch, a singularity develops in finite time, which would call into question the role of inflation in the dynamic evolution of our universe. We further point out that a possible reason for this behaviour is that the solutions to the field equations in such inflationary scenarios do not obey global existence and uniqueness properties, which is a typical characteristic of solutions that diverge in finite time.

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I. Kohli
Fri, 29 May 15
25/68

Comments: For submission to: Classical and Quantum Gravity

Capture of Planets Into Mean Motion Resonances and the Origins of Extrasolar Orbital Architectures [EPA]

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http://arxiv.org/abs/1505.01778


The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this work, we employ an integrable Hamiltonian formalism for first order planetary resonances that allows both secondary bodies to have finite masses and eccentricities, and construct a comprehensive theory for resonant capture. Particularly, we derive conditions under which orbital evolution lies within the adiabatic regime, and provide a generalized criterion for guaranteed resonant locking as well as a procedure for calculating capture probabilities when capture is not certain. Subsequently, we utilize the developed analytical model to examine the evolution of Jupiter and Saturn within the protosolar nebula, and investigate the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets. Our calculations show that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e~0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

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K. Batygin
Fri, 8 May 15
7/62

Comments: 22 pages, 15 figures, accepted for publication in MNRAS

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

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http://arxiv.org/abs/1504.00051


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
9/61

Comments: 15 pages, 7 figures, accepted to AJ

Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids [CL]

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http://arxiv.org/abs/1503.06994


We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat FLRW cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the `attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Pad\’e approximants and compare with the slow-roll approximation.

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A. Alho, J. Hell and C. Uggla
Wed, 25 Mar 15
15/38

Comments: 33 pages, 12 figures

The Astrophysics of Resonant Orbits in the Kerr Metric [CL]

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http://arxiv.org/abs/1501.07728


This paper gives a complete characterization of resonant orbits in a Kerr spacetime. A resonant orbit is defined as a geodesic for which the longitudinal and radial orbital frequencies are commensurate. Our analysis is based on expressing the resonance condition in its most symmetric form using Carlson’s integrals. We provide a number of concise formulae for the dependence of resonances on the system parameters. Resonant effects may be observable during the in-spiral of a compact object into a super-massive black hole. When the slowly evolving orbital frequencies pass through a series of low-order resonances, rapid changes in the orbital parameters could produce measurable phase shifts in the emitted gravitational radiation (GW). Resonant orbits may also capture dust leading to electromagnetic emission. The KAM theorem indicates that, low order resonant orbits demarcate the regions where the onset of chaos could occur around a perturbed black-hole. We find that the 1/2 and 2/3 resonances occur at ~4 and 5.4 Schwarzschild radii (Rs) from the event horizon. For compact object in-spirals around super-massive black holes, this region lies within the sensitivity band of space-based GW detectors. For Sgr A*, length scales of ~41 and 55 microarcseconds and timescales of 50 and 79 min respectively should be associated with resonant effects, if Sgr A* is non-spinning. Spin decreases these values by up to ~32% and ~28%. These length-scales are potentially resolvable with VLBI measurements. We find that all low-order resonances are localized to the strong field region r < 50 Rs. This fact guarantees the validity of using approximations based on averaging to model the frequency evolution of a test object in region 50 Rs <r <1000 Rs. The systematic determination of the multipole moments of the central object by observing the orbit of a pulsar, free of chaotic effects, is thus possible.

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J. Brink, M. Geyer and T. Hinderer
Mon, 2 Feb 15
23/49

Comments: 25 Pages, 17 figures

A study of the main resonances outside the geostationary ring [CL]

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http://arxiv.org/abs/1501.06273


We investigate the dynamics of satellites and space debris in external resonances, namely in the region outside the geostationary ring. Precisely, we focus on the 1:2, 1:3, 2:3 resonances, which are located at about 66 931.4 km, 87 705.0 km, 55 250.7 km, respectively. Some of these resonances have been already exploited in space missions, like XMM-Newton and Integral.
Our study is mainly based on a Hamiltonian approach, which allows us to get fast and reliable information on the dynamics in the resonant regions. Significative results are obtained even by considering just the effect of the geopotential in the Hamiltonian formulation. For objects (typically space debris) with high area-to-mass ratio the Hamiltonian includes also the effect of the solar radiation pressure. In addition, we perform a comparison with the numerical integration in Cartesian variables, including the geopotential, the gravitational attraction of Sun and Moon, and the solar radiation pressure.
We implement some simple mathematical tools that allows us to get information on the terms which are dominant in the Fourier series expansion of the Hamiltonian around a given resonance, on the amplitude of the resonant islands and on the location of the equilibrium points. We also compute the Fast Lyapunov Indicators, which provide a cartography of the resonant regions, yielding the main dynamical features associated to the external resonances. We apply these techniques to analyze the 1:2, 1:3, 2:3 resonances; we consider also the case of objects with large area-to-mass ratio and we provide an application to the case studies given by XMM-Newton and Integral.

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A. Celletti and C. Gales
Tue, 27 Jan 15
64/79

Comments: 30 pages, 10 figures

Introduction to the application of the dynamical systems theory in the study of the dynamics of cosmological models of dark energy [CL]

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http://arxiv.org/abs/1501.04851


The theory of the dynamical systems is a very complex subject which has brought several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of the dynamical systems in cosmological settings is less known in spite of the amount of published scientific papers on this subject. In this paper a — mostly pedagogical — introduction to the application in cosmology of the basic tools of the dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these allow to extract essential information on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools is illustrated within the context of the so called $\Lambda$CDM and scalar field models of dark energy. This paper is suitable for teachers, undergraduate and postgraduate students from physics and mathematics disciplines.

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R. Garcia-Salcedo, T. Gonzalez, F. Horta-Rangel, et. al.
Wed, 21 Jan 15
43/52

Comments: 15 pages, 2 figures

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

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http://arxiv.org/abs/1501.03193


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
7/49

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

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

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http://arxiv.org/abs/1412.1630


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
36/56

Comments: 47 pages, 8 figures,

Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology [EPA]

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http://arxiv.org/abs/1411.1860


In this paper we present a new approach to tidal theory. Assuming a Maxwell viscoelastic rheology, we compute the instantaneous deformation of celestial bodies using a differential equation for the gravity field coefficients. This method allows large eccentricities and it is not limited to quasi-periodic perturbations. It can take into account an extended class of perturbations, including chaotic motions and transient events. We apply our model to some already detected eccentric hot Jupiters and super-Earths in planar configurations. We show that when the relaxation time of the deformation is larger than the orbital period, spin-orbit equilibria arise naturally at half-integers of the mean motion, even for gaseous planets. In the case of super-Earths, these equilibria can be maintained for very low values of eccentricity. Our method can also be used to study planets with complex internal structures and other rheologies.

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A. Correia, G. Boue, J. Laskar, et. al.
Mon, 10 Nov 14
11/38

Comments: 16 pages, 13 figures, 2 tables

Earth–Mars Transfers with Ballistic Capture [EPA]

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http://arxiv.org/abs/1410.8856


We construct a new type of transfer from the Earth to Mars, which ends in ballistic capture. This results in a substantial savings in capture $\Delta v$ from that of a classical Hohmann transfer under certain conditions. This is accomplished by first becoming captured at Mars, very distant from the planet, and then from there, following a ballistic capture transfer to a desired altitude within a ballistic capture set. This is achieved by manipulating the stable sets, or sets of initial conditions whose orbits satisfy a simple definition of stability. This transfer type may be of interest for Mars missions because of lower capture $\Delta v$, moderate flight time, and flexibility of launch period from the Earth.

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F. Topputo and E. Belbruno
Mon, 3 Nov 14
31/40

Comments: N/A

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

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http://arxiv.org/abs/1410.4921


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
52/72

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]

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http://arxiv.org/abs/1410.4923


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
62/72

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

Andoyer construction for Hill and Delaunay variables [IMA]

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http://arxiv.org/abs/1409.5880


Andoyer variables are well known for the study of the rigid body dynamics. But these variables were derived by Andoyer through a procedure that can be also used to obtain the Delaunay variables of the Kepler problem in a direct way, without the use of Hamilton-Jacobi theory or non intuitive generating functions.

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J. Laskar
Tue, 23 Sep 14
46/60

Comments: 5 pages, 1 figure

Periodic orbits for 3 and 4 co-orbital bodies [EPA]

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http://arxiv.org/abs/1405.7517


We investigate the natural families of periodic orbits associated with the equilibrium configurations of the the planar restricted $1+n$ body problem for the case $2\leq n \leq 4$ equal mass satellites. Such periodic orbits can be used to model both trojan exoplanetary systems and parking orbits for captured asteroids within the solar system. For $n=2$ there are two families of periodic orbits associated with the equilibria of the system: the well known horseshoe and tadpole orbits. For $n=3$ there are three families that emanate from the equilibrium configurations of the satellites, while for $n=4$ there are six such families as well as numerous additional connecting families. The families of periodic orbits are all of the horseshoe or tadpole type, and several have regions of neutral linear stability.

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P. Verrier and C. McInnes
Fri, 30 May 14
6/74

Comments: 14 pages, 18 figures, accepted by MNRAS

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

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http://arxiv.org/abs/1404.7679


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
32/44

Comments: (in press)

Regularization of the big bang singularity with a time varying equation of state $w > 1$ [CL]

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http://arxiv.org/abs/1403.2122


We study the classical dynamics of the universe undergoing a transition from contraction to expansion through a big bang singularity. The dynamics is described by a system of differential equations for a set of physical quantities, such as the scale factor $a$, the Hubble parameter $H$, the equation of state parameter $w$, and the density parameter $\Omega$. The solutions of the dynamical system have a singularity at the big bang. We study if these solutions can be uniquely extended through the singularity. In particular, we consider the model in which the contracting universe is dominated by a scalar field with a time varying equation of state $w$, which approaches a constant value $w_c$ near the singularity. We prove that, for $w_c > 1$, the singularity is regularizable only for a discrete set of $w_c$ values that satisfy a coprime number condition. Our result implies that the evolution of a bouncing universe through the big bang singularity does not have a continuous classical limit unless the equation of state is extremely fine-tuned.

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B. Xue and E. Belbruno
Tue, 11 Mar 14
58/66

Exploring Vacuum Energy in a Two-Fluid Bianchi Type I Universe [CL]

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http://arxiv.org/abs/1402.1967


We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a two-fluid, non-tilted Bianchi Type I cosmological model. In our model, one of the fluids is a fluid with bulk viscosity, while the other fluid assumes the role of a cosmological constant and represents nonnegative vacuum energy. We begin by completing a detailed fixed-point analysis of the system which gives information about the local sinks, sources and saddles. We then proceed to analyze the global features of the dynamical system by using topological methods such as finding Lyapunov and Chetaev functions, and finding the $\alpha$- and $\omega$-limit sets using the LaSalle invariance principle. The fixed points found were a flat Friedmann-LeMa\^{\i}tre-Robertson-Walker (FLRW) universe with no vacuum energy, a de Sitter universe, a flat FLRW universe with both vacuum and non-vacuum energy, and a Kasner quarter-circle universe. We also show in this paper that the vacuum energy we observe in our present-day universe could actually be a result of the bulk viscosity of the ordinary matter in the universe, and proceed to calculate feasible values of the bulk viscous coefficient based on observations reported in the Planck data. We conclude the paper with some numerical experiments that shed further light on the global dynamics of the system.

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I. Kohli and M. Haslam
Tue, 11 Feb 14
43/55

Instabilities in magnetized rotational flows: A comprehensive short-wavelength approach [CL]

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http://arxiv.org/abs/1401.8276


We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence characterized by an appropriate magnetic Rossby number. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We show in particular that the inductionless versions of MRI that were previously thought to be restricted to comparably steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its field-free profile.

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Mon, 3 Feb 14
45/53

Passive Sorting of Asteroid Material Using Solar Radiation Pressure [CL]

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http://arxiv.org/abs/1401.5750


Understanding dust dynamics in the vicinity of asteroids is key for future science missions and, in the long-term, for asteroid exploitation. This paper analyzes the feasibility of manipulating asteroid material by means of solar radiation pressure. A novel method is proposed for passively sorting material as a function of its grain size or density, where solar radiation pressure is used as a passive in-situ “mass spectrometer”. A simplified analysis shows that in principle this method allows an effective sorting of regolith material. This could have immediate applications for a sample return mission, and for industrial scale in-situ resource utilization to separate and concentrate regolith according to particle size or composition.

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Thu, 23 Jan 14
53/70

Coupled orbital-thermal evolution of Miranda [CL]

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http://arxiv.org/abs/1401.4864


Miranda has a unusually high inclination ($I=4.338^\circ$), and its surface reveals signs of past endogenic activity. Investigations of the dynamical aspects of its orbital evolution suggest probable resonant processes, in particular with Umbriel, as an explanation for the present high inclination of Miranda. The tidal heating induced by gravitational interactions can lead to the rise of eccentricities and, consequently, to the increased dissipation of energy inside the satellite and higher internal temperatures. We study here the possible increase in eccentricities caused by orbital resonances and the resulting endogenic heating on Miranda taking into account its temperature dependent rheology. The coupled orbital-thermal evolution model was run with different rheological models and the thermal parameters starting form a cold thermal state, in radiative equilibrium with the environment. For the nominal parameters of the evolution scenarios studied, the resonances were not sufficient to rise neither the eccentricities nor the internal temperatures significantly. Lowest dissipation function $Q$ of around 100 and final eccentricity of $e\approx0.02$ were obtained during the resonance 3:1 with Umbriel.

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Tue, 21 Jan 14
84/91