1,721,001 research outputs found
On the determination of the stochasticity threshold of invariant curves
No full textWe consider the problem of determining the stochasticity transition value in nearly-integrable mappings. We perform explicitly a canonical transformation, which conjugates the original mapping to an integrable one, up to a given order in the perturbing parameter. Then we derive a numerical evidence of the existence of an invariant curve associated with the transformed system and, correspondingly, to the original one. In the second part of the paper we implement a numerical method due to M. Hénon [Hénon] for the computation of the rotation number corresponding to a given initial condition. Following an idea of Laskar et al. [1992] and Laskar [1993], we determine with high accuracy the critical breakdown threshold of invariant curves for standard-mapping like systems which allows not only to test Hénon's method but also to compare our analytical results with an accurate numerical one. An application is also made about the accuracy of the leap frog method
Numerical investigation of the break-down threshold for a restricted three-body problem
No full textTransition to chaos in the planar, circular, restricted three body problem is investigated. In particular, a model Hamiltonian suitable for the description of the motion of the asteroid Ceres is introduced. The break-down threshold of invariant surfaces (nearby the position of Ceres in phase space) is computed by means of different numerical methods, which allow to detect the transition from regular to chaotic motion. The experiments show that in the proximity of the location of the asteroid, invariant surfaces cease to exist for a mass ratio (between the perturbing body, Jupiter and the primary, the Sun) between 0.002 and 0.004. We remind that the real value of the above mass ratio is about 0.001 as derived from astronomical observations
Analytical approximation of the dissipative standard map
No full textModem dynamics is increasingly participating in the solution of problems raised by as tronomical observations. This new relationship is being fostered on one side by the im provements in the observations, which in recent years contributed several discoveries of new systems, such as the objects in the Kuiper belt, the pulsar and star companions, to speak only of the most striking ones, and, on the other hand, by the progresses in modem dynamics. The progresses in modem dynamics are due to two factors: the dissemination of fast computers, allowing the numerical studies of very complex systems by a large number of scientists, and the improvement in our understanding of the complex behaviour of Hamiltonian systems. KAM and Nekhorochev theories have shed a light on the subtle and surprizing interplays between regular and chaotic motions; numerical experiments and analytical approximations have shown how these peculiarities are indeed present in astronomically important systems and are instrumental in understanding their formation and evolution
Graphical evolution of the Arnold web: From order to chaos
No full textWe represent graphically the evolution of the set of resonances of a quasi-integrable dynamical system, the so-called Arnold web, whose structure is crucial for the stability properties of the system. The basis of our representation is the use of an original numerical method, whose definition is directly related to the dynamics of orbits, and the careful choice of a model system. We also show the transition from the Nekhoroshev stability regime to the Chirikov diffusive one, which is a generic nontrivial phenomenon occurring in many physical processes, such as slow chaotic transport in the asteroid belt and beam-beam interaction
Analytical approximation of the solution of the dissipative standard map
No full textWe consider a dissipative mapping derived from a modification of the Chirikov standard mapping. For simplicity, we assume that the dissipative strength is of the order of the square of the perturbing parameter of the conservative model. Under this assumption, we derive an analytical approximation of the solution associated to the dissipative mapping. The equations are explicitly solved up to the order 7 in the perturbing parameter.
Having fixed a frequency ω, a comparison of the associated to the dissipative solutions shows that the two curves coincide for low values of the perturbing parameter, while in most cases they diverge as the breakdown threshold of the invariant curve with rotation number ω is approached
On the computation of Lyapunov exponents for discrete time series. Applications to two dimensional symplectic and dissipative mappings
No full textMany techniques have been developed for the measure of the largest Lyapunov exponent of experimental short data series. The main idea, underlying the most common algorithms, is to mimic the method of computation proposed by Benettin and Galgani [1979]. The aim of the present paper is to provide an explanation for the reliability of some algorithms developed for short time series. To this end, we consider two-dimensional mappings as model problems and we compare the results obtained using the Benettin and Galgani method to those obtained using some algorithms for the computation of the largest Lyapunov exponent when dealing with short data series. In particular we focus our attention on conservative systems, which are not widely investigated in the literature. We show that using standard techniques the results obtained for discrete series related to area-preserving mappings are often unreliable, while dissipative systems are easier to analyze. In order to overcome the problem arising with conservative systems, we develop an alternative method, which takes advantage of the existing techniques. In particular, all algorithms provide a good approximation of the largest Lyapunov exponent in the strong chaotic symplectic case and in the dissipative one. However, the application of standard algorithms provides results which are not in agreement with the theoretical expectation for weak chaotic motions, and sometimes also for regular orbits. On the contrary, the method that we propose in this paper seems to work well for the weak chaotic case. Because of the speed of computation, we suggest to use all algorithms to cross-check the results
The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping
No full textEstimate of the transition value of librational invariant curves
No full textWe investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. Henon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter
Diffusion in Hamiltonian quasi-integrable systems
No full textThe characterization of diffusion of orbits in Hamiltonian quasi-
integrable systems is a relevant topic in dynamics. For quasi-integrable Hamiltonian systems a possible model for global diffusion, valid for perturbation larger than
a critical value, was given by Chirikov; while for smaller perturbation the Nekhoroshev theorem leave the possibility of exponentially slow diffusion along a peculiar
the Arnold’s web. We have studied this problem using a numerical approach. The
aim of this chapter is to give the state of the art concerning the detection of slow
Arnold’s diffusion in quasi-integrable Hamiltonian systems
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