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.

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.

The dynamical system considered in this paper is an inelastic beam whose supports are subjected to a harmonic excitation. This paper first explores whether the system has chaotic motion. The appearance of the irregular time history, strange attractor on the Poincaré map as well as period-doubling bifurcation phenomenon strongly indicates that chaos indeed exist in this system. After finding the chaos phenomenon, this paper continues to investigate the relationship between the decay time of the autocorrelation function and the largest Lyapunov exponent. The Poincaré mapping points are chosen to be the sampled function of the discrete autocorrelation function. It's found that a power model of regression analysis can fit with good accuracy the data points, which are composed of the mapping times for the autocorrelation to decay into the square of the mean of the Poincaré points and the corresponding largest Lyapunov exponent.

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.