Chaos in cosmological Hamiltonians

This paper summarises a numerical investigation which aimed to identify and characterise regular and chaotic behaviour in time-dependent Hamiltonians H(r,p,t) = p^2/2 + U(r,t), with U=R(t)V(r) or U=V[R(t)r], where V(r) is a polynomial in x, y, and/or z, and R = const * t^p is a time-dependent scale factor. When p is not too negative, one can distinguish between regular and chaotic behaviour by determining whether an orbit segment exhibits a sensitive dependence on initial conditions. However, chaotic segments in these potentials differ from chaotic segments in time-independent potentials in that a small initial perturbation will usually exhibit a sub- or super-exponential growth in time. Although not periodic, regular segments typically exhibit simpler shapes, topologies, and Fourier spectra than do chaotic segments. This distinction between regular and chaotic behaviour is not absolute since a single orbit segment can seemingly change from regular to chaotic and visa versa. All these observed phenomena can be understood in terms of a simple theoretical model.Comment: 16 pages LaTeX, including 5 figures, no macros require

Phase Space Transport in Noisy Hamiltonian Systems

This paper analyses the effect of low amplitude friction and noise in accelerating phase space transport in time-independent Hamiltonian systems that exhibit global stochasticity. Numerical experiments reveal that even very weak non-Hamiltonian perturbations can dramatically increase the rate at which an ensemble of orbits penetrates obstructions like cantori or Arnold webs, thus accelerating the approach towards an invariant measure, i.e., a near-microcanonical population of the accessible phase space region. An investigation of first passage times through cantori leads to three conclusions, namely: (i) that, at least for white noise, the detailed form of the perturbation is unimportant, (ii) that the presence or absence of friction is largely irrelevant, and (iii) that, overall, the amplitude of the response to weak noise scales logarithmically in the amplitude of the noise.Comment: 13 pages, 3 Postscript figures, latex, no macors. Annals of the New York Academy of Sciences, in pres

Numerical tests of dynamical friction in gravitational inhomogeneous systems

In this paper, I test by numerical simulations the results of Del Popolo & Gambera (1998),dealing with the extension of Chandrasekhar and von Neumann's analysis of the statistics of the gravitational field to systems in which particles (e.g., stars, galaxies) are inhomogeneously distributed. The paper is an extension of that of Ahmad & Cohen (1974), in which the authors tested some results of the stochastic theory of dynamical friction developed by Chandrasekhar & von Neumann (1943) in the case of homogeneous gravitational systems. It is also a continuation of the work developed in Del Popolo (1996a,b), which extended the results of Ahmad & Cohen (1973), (dealing with the study of the probability distribution of the stochastic force in homogeneous gravitational systems) to inhomogeneous gravitational systems. Similarly to what was done by Ahmad & Cohen (1974) in the case of homogeneous systems, I test, by means of the evolution of an inhomogeneous system of particles, that the theoretical rate of force fluctuation d F/dt describes correctly the experimental one, I find that the stochastic force distribution obtained for the evolved system is in good agreement with the Del Popolo & Gambera (1998) theory. Moreover, in an inhomogeneous background the friction force is actually enhanced relative to the homogeneous case.Comment: 12 pages; 2 encapsulated figures. Aatronomy and Astrophysics, in prin

Radial orbit instability as a dissipation-induced phenomenon

This paper is devoted to Radial Orbit Instability in the context of self-gravitating dynamical systems. We present this instability in the new frame of Dissipation-Induced Instability theory. This allows us to obtain a rather simple proof based on energetics arguments and to clarify the associated physical mechanism.Comment: 15 pages. Published in Monthly Notices of the RAS by the Royal Astronomical Society and Blackwell Publishing. Corrected for page style, typos, and added reference

LP-VIcode: a program to compute a suite of variational chaos indicators

An important point in analysing the dynamics of a given stellar or planetary system is the reliable identification of the chaotic or regular behaviour of its orbits. We introduce here the program LP-VIcode, a fully operational code which efficiently computes a suite of ten variational chaos indicators for dynamical systems in any number of dimensions. The user may choose to simultaneously compute any number of chaos indicators among the following: the Lyapunov Exponents, the Mean Exponential Growth factor of Nearby Orbits, the Slope Estimation of the largest Lyapunov Characteristic Exponent, the Smaller ALignment Index, the Generalized ALignment Index, the Fast Lyapunov Indicator, the Othogonal Fast Lyapunov Indicator, the dynamical Spectra of Stretching Numbers, the Spectral Distance, and the Relative Lyapunov Indicator. They are combined in an efficient way, allowing the sharing of differential equations whenever this is possible, and the individual stopping of their computation when any of them saturates.Comment: 26 pages, 9 black-and-white figures. Accepted for publication in Astronomy and Computing (Elsevier