Chaotic spiral galaxies

We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic manifolds of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards along the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that the orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure.     

The problem of small divisors in planetary motion

Small divisors caused by certain linear combinations of frequencies appear in all analytical planetary theories. With the exception of the deep resonance between Neptune and Pluto, they can be removed at the expense of introducing secular and mixed secular terms, limiting the domain in which the solution is valid. Because of them classical solutions are known not to converge uniformly; Poincaré referred to them as asymptotic. The KAM theory shows that if one is far enough from exact commensurability and has small enough planetary masses, expansions exist which will converge to quasi-periodic orbits. Solutions showing very small divisors are excluded from this region of convergence. The question of whether they are intrinsic to the problem or are just manifestations of the method of solution is not settled. Problems with a single commensurabily that can be isolated from the rest of the Hamiltonian may have solutions with no small divisors. The problem of two or more commensurabilities remains unsolved.     

Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry, Proc Lond Math Soc ser 2, 27:151–170, 1926; Moser, Commun Pure Appl Math 9:673, 1956 and 11:257, 1958; Moser, Giorgilli, Discret Contin Dyn Syst 7:855, 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., Phys D 29:181, 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice the study of 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.     

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

A numerical study of the orbits of second species of the planar circular RTBP

We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs.     

Satellite encounters

We study numerically the interaction of two small satellites, initially on circular orbits with slightly different radii. We show first that by going to Hill's limit of vanishing masses, one can reduce the problem to a simpler form in which only one dimensionless parameter remains: the reduced impact parameter. We present then a detailed study of the family obtained when this parameter is varied. Each orbit consists of three phases: approach of the two small bodies, interplay, and departure. Fourth-order series are used to represent the asymptotic motion of the two small bodies in the approach and departure phases; these series are matched with a numerical integration of the interplay phase to give an accurate representation of the entire orbit. For each orbit, we compute the net effect of the encounter, essentially characterized by an increase of the separation of the satellite orbits. We compute also the minimal distance of approach of the two satellites. In the limiting cases of large and small impact parameters, the results are compared with the predictions of perturbation theories. Finally we study the “transitions,” which are apparent discontinuities of the family with a sudden change of the direction of departure. We show that they can be explained by the asymptotic approach of the orbit to an unstable periodic solution of Hill's problem. Transitions take place for infinitely many values of the parameter, forming a Cantor-like set.     

Ordered and chaotic Bohmian trajectories

We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.     

The 1/1 resonance in extrasolar systems

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.     

Analytic continuation of periodic orbits from equilibria of two-degree-of-freedom systems in rotating coordinates

A new theory is formulated for the analytic continuation of periodic (and aperiodic) orbits from equilibrium solutions of a two-degree-of-freedom dynamical system in rotating coordinates:% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acbiGab8xDayaacaGaa8xlaiaa-jdacaWFUbGaeqyXduNaa8xpaiaa% -zfadaWgaaWcbaGccaWF4baaleqaaOGaaiilaiqbew8a1zaacaGaey% 4kaSIaaGOmaiaad6gacaWG1bGaeyypa0Jaa8NvamaaBaaaleaakiaa% -LhaaSqabaGccaGGSaGabmiEayaacaGaeyypa0JaamyDaiaacYcace% WG5bGbaiaacqGH9aqpcqaHfpqDaaa!54CD!\[\dot u - 2n\upsilon = V_x ,\dot \upsilon + 2nu = V_y ,\dot x = u,\dot y = \upsilon \]Away from resonance, a family of nonlinear, normal-mode orbits defines an autonomous velocity field u(x, y), u(x, y) represented by convergent algebraic-series expansions in the two position variables. This approach is useful for determining the global structure of solution curves and nonlinear stability of normal modes using Liapunov's direct method. At resonance, the series coefficients are time dependent because stationary modes are incompatible with the equations of motion. By eliminating small divisors, explicit time dependence provides a natural transition from non-resonance to resonance cases within the same theory.     

Evolution of the velocity field in model potentials

We consider the evolution of the structure of the velocity field formed by the loops of the trajectory in a generalized Hénon-Heiles model potential. Box-shaped orbits alternating with periodic trajectories are shown to dominate at low values of the energy integral. Signatures of shell-and tube-shaped structures appear as the energy increases. Tube-shaped orbits are associated with stable periodic trajectories with small resonance ratios. Zones of stochastic orbits appear at values of the energy integral close to its critical value, which corresponds to the opening of the zero-velocity contour.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in Celestial Mechanics and Astrodynamics, e.g., to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic map. The HOTM is only accurate close to the expansion point and therefore the number of revolutions for which the map is accurate tends to be limited. The proper selection of coordinates is of key importance for improving the performance of the HOTM method. In this paper, we investigate the use of different element sets for expressing the high-order map in order to find the coordinates that perform best in terms of accuracy. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.     

Secular Dynamics of Asteroids in the Inner Solar System

In the last three years we have carried out numerical and semi-analytical studies on the secular dynamical mechanisms in the region (semimajor axis a < 2 AU) where the NEA orbits evolve. Our numerical integrations (over a time span of a few Myr) have shown that: (i) the linear secular resonances with both the inner and the outer planets may play an important role in the dynamical evolution of NEAs; (ii) the apsidal secular resonance with Mars could provide an important dynamical transport mechanism by which asteroids in the Mars-crossing region eventually achieve Earth-crossing orbits; (iii) in this region, due to the interaction with the terrestrial planets, the Kozai resonance can occur at small inclinations, with the argument of perihelion ω librating around 0° or 180°, providing a temporary protection mechanism against close approaches to the planets. The location of the linear secular resonances in this zone has also been obtained by an automatic procedure using a semi-numerical method valid for all values of the inclinations and eccentricities of the small bodies, and also in the case of libration of the argument of perihelion. A map of the secular resonances in the (a, i) plane shows — in agreement with the numerical integrations — that all the resonances with the terrestrial and giant planets are present, and also that some of them overlap. Thus the way is now open to fully take into account secular resonances in modelling the dynamical evolution of NEAs. This revised version was published online in July 2006 with corrections to the Cover Date.     

The coalescence of invariant manifolds and the spiral structure of barred galaxies

In a previous paper (Voglis et al., Paper I), we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short-period family of unstable periodic orbits around the Lagrangian points L ₁ or L ₂ create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In this paper, we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a N -body simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of 'stickiness' slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of the order of 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the 'coalescence' of the invariant manifolds. This follows closely the maxima of the observed   m = 2  component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.     

Universal unfolding of symmetric resonances

We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff–Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.     

Order and chaos in spiral galaxies

Spiral galaxies contain both ordered and chaotic orbits. In normal spirals the perturbations are weak (of order 2–10%) and most orbits are ordered. The density wave theory refers mainly to linear perturbations. Nonlinear effects appear in the outer parts of the open spirals (S_b, S_c) and produce the termination of these spirals near the 4/1 resonance. On the other hand in barred spirals the perturbations are relatively strong (of order 100%). Then the outer spirals and the envelope of the bar are composed mainly of chaotic orbits, while the main body of the bar is composed of ordered orbits. The chaotic orbits of the spiral arms of strong barred galaxies are sticky, i.e. they do not escape from the galaxy for at least a Hubble time. The forms of these spirals are delineated by the unstable manifolds of the unstable periodic orbits L_1, L_2 near the ends of the bar and of other unstable periodic orbits inside and outside corotation.     

Motions close to escapes in the rhomboidal four body problem

In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals.     

Numerical study of the three-dimensional transit orbits in the circular restricted three-body problem

Transit orbits are defined as the trajectories that can pass through the neck region of the zero velocity surface in the circular restricted three-body problem (CR3BP). The low-energy transfers in the CR3BP or between two CR3BPs are always through the instrumentality of the transit orbits. In this paper, the distribution of the transit orbits in the six-dimensional phase space is explored by using numerical methods. The necessary and sufficient condition of transition is introduced, which defines the distribution of the transit orbits by using the manifolds of the vertical and horizontal Lyapunov orbits and the transit cones. The relationship between the manifolds of the libration point orbits and the boundary of the transit orbits is discovered. By using this relationship, a fast algorithm for detecting the boundary of the transit orbits is developed. Moreover, this boundary is parametrized by using Fourier series, which makes easy to use the conclusions of this paper in future trajectory optimization and mission design. All the analyses in this paper are based on the Sun?CEarth CR3BP, but the methods introduced here can be extended to any CR3BPs.