This is an introductory article on Chaos giving the definition of Chaos, source and tools of Chaos, routes to chaos, measurement of chaos, chaos through resonance. The Duffing equation has been mentioned as a problem of double resonance leading to chaos and finally some problems relating to chaos in the Solar system have been described.

Applying modern techniques of time series analysis, there are serious indications that the dynamics of the global solar activity is a low dimensional chaos. A simple non-linear dynamo model is qualitatively studied exhibiting a rich dynamical behaviour from steady state via some bifurcation to a chaotic regime.

In a previous paper (Laskar, Nature, 338, 237-238), the chaotic nature of the solar system excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 Myr. In the present an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2(g4−g3)−(s4−s3) related to the motions of perihelions and nodes of the Birth and Mars. An other important secular resonance is identified: (g1−g5)−(s1−s2). Its critical argument stays in libration over 200 Myr with a period of about 10 Myr and amplitude from 85° to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The size of the chaotic regions in the inner solar system are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner solar system can thus be considered as robust against small variations of the initial conditions or of the model. The chaotic regions related to the outer planets frequencies are very thin except for g6 which present variations sufficiently large to be significant over the age of the solar system.

It was proved that the problem of perturbed planar oscillations of a rigid-body in a circular orbit is nonitegrable. Two types of perturbations were considered: solar radiations pressure and the third body torques. In the second part of the paper example of chaotic rotations of a symmetric rigid body in a circular orbit was given. It was shown numerically that the phase space is divided into two separate regions of chaotic and ordered motions.

The role of chaos and instability in evolution of nonlinear, non-stationary stellar system have been discussed. It is possible to distinguish between the only two different cases (i) strongly non-stationary stage when we have the violent relaxation accompanied by the compulsive mixing (ii) weakly non-stationary state or quasilinear case, when the quasidiffusion mixing takes place. In case (i), the chaos and chaotic motion g stars play very important role and in case (ii) the role of any type of instability is important.

We discuss how chaotic dynamics can be converted into regular motion in Bonhoeffer-van der Pol oscillator. Using a control signal proportional to the actual and desired outputs we study the control of fixed points and limit cycles by making time-dependent perturbations of amplitude of external force. We show the round-off induced periodicity in the digital computer simulations of orbits on chaotic attractor. We illustrate the stabilization of unstable periodic or bits by adding periodic pulses of small magnitude.

Instability of orbits in dynamical systems leading to chaos has been reviewed briefly. Stability criteria for some unimodal mapping which provide various periodic regimes during the period doubling bifurcations has been discussed in detail. Stability conditions are reviewed for standard map (or Chirikov-Taylor map), and results obtained for range of values of the non-linear parameter appearing in the map have been studied. Strange attractor has also been discussed.

Vector laypunov functions and stabilities and chaoticities of Functional Differential Equations occurring in Celestial Mechanics and Stellar Dynamics have been discussed. Fuzzy dynamical systems are more realistic while considering problems occurring in Celestial Mechanics and Stellary Dynamics.

We prove a basic result for collisionless galactic models that collective motion not only introduces Landau damping but also intrinsic chaos of typical star dynamics in the phase plane when a small perturbative wave disturbance is present. The Melnikov method is used; the consequences of the chaos and quasilinear diffusion are pointed out.

We have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.

Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

We study a process of stochastization of the motions in an example of the Henon-Heiles model. We propose a new method to study this process a method of the field of directions of motion in the meridional plane. A numerical integration of the equation for the derivative of this field ∂f/∂n to the normal to a trajectory has been made. We denote the points in which ∂f/∂n → ± ∞, and derive the contours of orbit and folds of directions. The growth of ergodity is connected with the increase of a number and an area of the folds. May be, a successive doubling of a number of folds takes place that results in a chaos. In the transition region we found a complex periodical orbit. The interpretation of this fact may be made as an example of cantori. A transition region has very small sizes about 10−4

The dynamical instability of motions is studied numerically for the triplets of stars and galaxies. The models of equal-mass components which are in rest initially, are investigated. The divergence of initially near by trajectories close to exponential one is observed in the configurational as well as the phase space. The characteristic time of such divergenee is less for the configurations which are initially hierarchical. The presence of significant dark matter distributed over the whole volume of system speeds up the rate of stochastisation considerably.

Using mainly Lyapunov Characteristic Indicators, the stochasticity of the orbit of the asteroid 1989 AC has been investigated in the framework of three-body, five-body and seven-body restricted models. From our first results, the orbit of 1989 AC has been found fairly chaotic, through an interplay of the 3/1 resonance with Jupiter and close approaches with the inner planets.

The theoretical aspect of close flybies of space-crafts near planets-gravitational manoeuvres have been studied. It is proved that the space-craft motion with gravitational manoeuvres admits the existence of routing scheme and, consequently shows the existence of quasi-random motion.

Our Universe has probably a space-time with a humpy metric. The space-time has nondegenerate (unequal) geodesics. The Universe is weakly inhomogeneous. The nonstationary inhomogeneities (humpies) of gravitational field change the redshift of any astronomical object - change brightness (Sachs-Wolfe’s effect), and change trajectories of photons. The latter effect is considered here. It is found that a theorem on the null geodesics are instable in the Universe with positive density contrasts. Photons must lose an information of their initial directions of propagation. So such radiation as the Microwave Background Radiation must seem uniform in the Universe.

The evolution of orbital elements of a growing planet during the accumulation process is considered. The planetary orbit undergoes perturbations because of random encounters and collisions with bodies of its accretion zone and also because of gravitational interaction with an already formed massive planet (“Jupiter”). The mass and velocity distributions of the swarm bodies are assumed to be given time-dependent functions. The Fokker-Planck equation describing the behaviour of the distribution function of orbital elements of the growing planet is worked out and solved. The present mean values of the eccentricities and inclinations of orbits of the terrestrial planets can be explained in the case of their accumulation from a single swarm of bodies with mean mass ~ 10−2 M⊕ and with mean eccentricities and inclinations ~ 0.2.

Large scale, or “global” stability of finite radius, razor thin disk of self-gravitating matter in presence of a massive halo surrounding the disk is studied as an eigen-value problem. Effect of the halo mass and size on the allowed global spiral eigenmodes of the disk is discussed.

By the computer simulations we have studied a behaviour of the virial coefficient k(t) for the triples and quintets of galaxies as a function of time. In a number of cases some quasiperiodical oscillations k(t) have been observed; these variations are resulted from an existence of the temporary binary subsystems. Strong negative correlation between k and rmin (rmin is a minimum separation between the bodies) is observed. A presence of a large hidden mass Mn distributed around a triple increases the degree of stochasticity of the process k(t) and decreases the correlation between k and rmin. The process k(t) must not been considered as a random one, as the autocorrelation function does not show any decreasing of the oscillation amplitude with a grow of time.

Theoretical models have been computed for estimating linear polarization from the extended dusty outer layers of the components of close binary stars whose surfaces are distorted by rotation and tidal effects due to the presence of secondary. We have assumed plane-parallel layers of the dusty atmospheres of the components. We have employed a wavelength dependent scattering coefficient, and Rayleigh phase function is used in solving the equation of radiative transfer . It is noticed that polarization increases with decreasing the wavelength and increasing the particle size. Polarization for uniform rotation is larger than that for non-uniform rotation. Polarization for the single stars is always less than that for a binary component.

Subject headings: polarization - radiative transfer - stars: binaries - stars: late-type