Ulam method for the Chirikov standard map

We introduce a generalized Ulam method and apply it to symplectic dynamical maps with a divided phase space. Our extensive numerical studies based on the Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. Our numerical data show that the exponent of this decay is approximately equal to the exponent of Poincar\'e recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.Comment: 13 pages, 13 figures, high resolution figures available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor corrections in text and fig. 12 and revised discussio

On the representation of maps by Lie transforms

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Scrh\"oder--Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension

Poincar\'e recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincar\'e recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincar\'e exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincar\'e recurrences.Comment: 11 pages, 14 figures, high resolution figures and video mpeg files available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod

Universal diffusion near the golden chaos border

We study local diffusion rate $D$ in Chirikov standard map near the critical golden curve. Numerical simulations confirm the predicted exponent $\alpha=5$ for the power law decay of $D$ as approaching the golden curve via principal resonances with period $q_n$ ($D \sim 1/q^{\alpha}_n$). The universal self-similar structure of diffusion between principal resonances is demonstrated and it is shown that resonances of other type play also an important role.Comment: 4 pages Latex, revtex, 3 uuencoded postscript figure

Exploring Transition from Stability to Chaos through Random Matrices

This study explores the application of random matrices to track chaotic dynamics within the Chirikov standard map. Our findings highlight the potential of matrices exhibiting Wishart-like characteristics, combined with statistical insights from their eigenvalue density, as a promising avenue for chaos monitoring. Inspired by a technique originally designed for detecting phase transitions in spin systems, we successfully adapt and apply it to identify analogous transformative patterns in the context of the Chirikov standard map. Leveraging the precision previously demonstrated in localizing critical points within magnetic systems in our prior research, our method accurately pinpoints the Chirikov resonance-overlap criterion for the chaos boundary at $K\approx 2.43$, reinforcing its effectiveness.Comment: 10 pages, 6 figure

Fractal Weyl law for quantum fractal eigenstates

The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.Comment: 4 pages, 5 figs, minor modifications, research at http://www.quantware.ups-tlse.fr