Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes, (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τm and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The “finite time LCN” is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem, (b) In the case of two fixed black holes M1 and M2 the orbits of photons are separated into three types: orbits falling into M1 (type I), or M2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems.

We discuss specific properties of dynamical systems originating from cosmology and relativity. In particular, we present results of our study of the Bianchi class A cosmological models. We introduce new variables in which the Hamiltonian constraint for all the class A models is solved algebraically. We present results of dimension reduction of the investigated models.

This paper reviews recent advances in several topics of resonant asteroidal dynamics as the role of resonances in the transportation of asteroids and asteroidal debris to the inner and outer solar system; the explanation of the contrast of a depleted 2/1 resonance (Hecuba gap) and a high-populated 3/2 resonance (Hilda group); the overall stochasticity created in the asteroid belt by the short-period perturbations of Jupiter’s orbit, with emphasis in the formation of significant three-period resonances, the chaotic behaviour of the outer asteroid belt, and the depletion of the Hecuba gap.

The present paper reviews our current understanding of the origin and evolution of NEAs, at the light of the results of recent quantitative numerical simulations that have revolutioned the previously accepted scenario.

This report is a continuation of three earlier reviews (Marsden 1996a, 1996b, 1998) that included a summary of our orbital knowledge of the Kuiper Belt. Presented at conferences held in the middle of 1994, 1995 and 1996, respectively, these reviews showed the steadily developing picture of a system dominated by the platinos, librating in the 2:3 mean-motion resonance with Neptune, and the cubewanos, a somewhat more distant population of nonlibrating objects with low orbital eccentricities. The existence of a 3:4 Neptune librator and a 3:5 Neptune librator was also suspected. These librators have now been confirmed, and a possible 4:7 librator and possible second 3:5 librator have also been found. The known and suspected multiple-opposition librators are listed in Table 1. Here it is important to note that the orbital semimajor axes a (in AU), eccentricities e and inclinations i (in degrees with respect to the 2000.0 ecliptic) are mean values that eliminate the large 12-year and 30-year periodicities arising from the indirect perturbations by Jupiter and Satum on sun-centered orbits. The numbers in parentheses are the semimajor axes (in AU) corresponding to the resonances. Following the absolute magnitude H, the entries “Nep.” and “Ura.” show the minimum distances (in AU) from Neptune and Uranus (the latter being of course quite small for the most eccentric 2:3 Neptune librators) within several millennia of the present time.

A set of geocentric variables suitable for the identification of meteoroid streams has been recently proposed and successfully applied to photographic meteor orbits. We describe these variables and the secular invariance of some of them, and discuss their use to improve the search for meteoroid stream parents.

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated.

The twenty most chaotic objects found among first hundred of numbered asteroids are studied. Lyapunov time calculated with and without inner planets indicates that for eleven of those asteroids the strongest chaotic effect results from the resonances with Mars. The filtered semimajor axis displays an abrupt variation only when a close approach to Mars takes place. The study of the behaviour of the critical argument for candidate resonances can reveal which is responsible for the semimajor axis variation. We have determined these resonances for the asteroids in question. For the asteroids chaotic even without the inner planets we have determined the most important resonances with Jupiter, or three-body resonances.

A new semianalytical theory of asteroid motion is presented. The theory is developed on the basis of Kaula’s expansion of the disturbing function including terms up to the second order with respect to the masses of disturbing bodies. The theory is constructed in explicit form that gives the possibility to study separately the influence of different perturbations in the dynamics of minor planets. The mean-motion resonances with major planets as well as mixed three-body resonances can also be taken into account. For the non-resonant case the formulas obtained can be used for deriving the second transformation to calculate the proper elements of an asteroid orbit in closed form with respect to inclinations and eccentricities.

In the present paper we analyse the motion of a massless particle during the capture process in an exterior mean-motion resonance under the effects of an external dissipative force. In particular, we study the orbital evolution from its initial approach to the commensurability up to the final nesting place in the periodic orbit around the equilibrium solution.

Migration of trans–Neptunian objects under their mutual gravitation influence and the influence of the giant planets is investigated. These investigations are based on computer simulation results and on some formulas. We estimated that about 20 % of near–Earth objects with diameter d ≥ 1 km may have come from the Edgeworth–Kuiper belt.

The orbits of 13 Trojan asteroids have been calculated numerically in the model of the outer solar system for a time interval of 100 million years. For these asteroids Milani et al. (1997) determined Lyapunov times less than 100 000 years and introduced the notion “asteroids in stable chaotic motion”. We studied the dynamical behavior of these Trojan asteroids (except the asteroid Thersites which escaped after 26 million years) within 11 time intervals - i.e. subintervals of the whole time - by means of: (1) a numerical frequency analysis (2) the root mean square (r.m.s.) of the orbital elements and (3) the proper elements. For each time interval we compared the root mean squares of the orbital elements (a, e and i) with the corresponding proper element. It turned out that the variations of the proper elements ep in the different time intervals are correlated with the corresponding r.m.s.(e); this is not the case for sin Ip with r.m.s.(i).

This paper concentrates on four key tools for performing star cluster simulations developed during the last decade which are sufficient to handle all the relevant dynamical aspects. First we discuss briefly the Hermite integration scheme which is simple to use and highly efficient for advancing the single particles. The main numerical challenge is in dealing with weakly and strongly perturbed hard binaries. A new treatment of the classical Kustaanheimo-Stiefel two-body regularization has proved to be more accurate for studying binaries than previous algorithms based on divided differences or Hermite integration. This formulation employs a Taylor series expansion combined with the Stumpff functions, still with one force evaluation per step, which gives exact solutions for unperturbed motion and is at least comparable to the polynomial methods for large perturbations. Strong interactions between hard binaries and single stars or other binaries are studied by chain regularization which ensures a non-biased outcome for chaotic motions. A new semi-analytical stability criterion for hierarchical systems has been adopted and the long-term effects on the inner binary are now treated by averaging techniques for cases of interest. These modifications describe consistent changes of the orbital variables due to large Kozai cycles and tidal dissipation. The range of astrophysical processes which can now be considered by N-body simulations include tidal capture, circularization, mass transfer by Roche-lobe overflow as well as physical collisions, where the masses and radii of individual stars are modelled by synthetic stellar evolution.

The gravitational n—body problem is chaotic. Phase trajectories that start very near each other separate rapidly. The rate looks exponential over long times. At any instant, trajectories separated in certain directions move apart rapidly (unstable directions), while those separated in other directions stay about the same (stable directions). Unstable directions lie along eigenvectors that correspond to positive eigenvalues of the matrix of force gradients. The number of positive eigenvalues of that matrix gives the dimensionality of stable regions. This number has been studied numerically in a series of 100—body integrations. It continues to change as long as the integration continues because the matrix changes extremely rapidly. On average, there are about 1.2n unstable directions out of 3n. Issues of dimensionality arise when the tools of ergodic studies are brought to bear on the problem of trajectory separation. A method of estimating the rate of trajectory separation based on matrix descriptions is presented in this note. Severe approximations are required.

The properties of chaos in 2D self-consistent models of barred galaxies are investigated using Kolmogorov-Sinai entropy hKS. These models are constructed with Schwarzschild’s method which combines orbits as elementary building blocks.

Most models are dominated by chaos near the 2/3 of the length of the bar and close to corotation. These locations correspond to regions where star-forming HII regions are observed because gas clouds could shock, shrink and fragment such that star formation could be ignited.

The model the most similar to N-body models shows a peak of hKS between the corners of the rectangular-like x1 orbits and the maximum extension points of the Lagrangian orbits. This emphasizes the role of Lagrangian orbits in the morphology of bars. Most models essentially contain ‘semi-chaotic’ orbits confined inside the corotation.

As a first step towards a comprehensive investigation of stellar motions within globular clusters, we present here the results of a study of stellar orbits in a mildly triaxial globular cluster that follows a circular orbit inside a galaxy. The stellar orbits were classified using the frequency analysis code of Carpintero and Aguilar and, as a check, the Liapunov characteristic exponents were also computed in some cases.

The orbit families were obtained using different start spaces. Chaotic orbits turn out to be very common and while, as could be expected, they are particularly abundant in the outer parts of the cluster, they are still significant in the innermost regions. Their relevance for the structure of the cluster is discussed.

In this paper, we consider the motion of a particle under the gravitational field of a massive straight segment. This model is used as an approximation to the gravitational field of irregular shaped bodies, such as asteroids, comet nuclei and planets’s moons. For tbis potential, we find several families of periodic orbits and bifurcations.

Recent N-body simulations have shown that there is a serious discrepancy between the results of N-body simulations and the results of Fokker-Planck simulations for the evolution of globular and rich open clusters under the influence of the galactic tidal field. In some cases, the lifetime obtained from Fokker-Planck calculations is more than an order of magnitude smaller than those from N-body simulations. In this paper we show that the principal cause for this discrepancy is an oversimplified treatment of the tidal field used in previous Fokker-Planck simulations. We performed new Fokker-Planck calculations using a more appropriate implementation for the boundary condition of the tidal field. The implementation is only possible with anisotropic Fokker-Planck models, while all previous Fokker-Planck calculations rely on the assumption of isotropy. Our new Fokker-Planck results agree well with N-body results. Comparison of the two types of simulations gives a better understanding of the evolution of such clusters.

Tremendous progress has been made recently in modelling the morphology and kinematics of centers of galaxies. Increasingly realistic models are built for central bar, bulge, nucleus and black hole of galaxies, including our own. The newly revived Schwarzschild method has played a central role in these theoretical modellings. Here I will highlight some recent work at Leiden on extending the Schwarzschild method in a few directions. After a brief discussion of (i) an analytical approach to include stochastic orbits (Zhao 1996), and (ii) the “pendulum effect” of loop and boxlet orbits (Zhao, Carollo, de Zeeuw 1999), I will concentrate on the very promising (iii) spectral dynamics method, with which not only can one obtain semi-analytically the actions of individual orbits as previously known, but also many other physical quantities, such as the density in configuration space and the line-of-sight velocity distribution of a superposition of orbits (Copin, Zhao & de Zeeuw 1999). The latter method also represents a drastic reduction of storage space for the orbit library and an increase in accuracy over the grid-based Schwarzschild method.