We study the capture of galactic dark matter particles (DMP) in two-body and few-body systems with a symplectic map description. This approach allows modeling the scattering of $10^{16}$ DMPs after following the time evolution of the captured particl
e on about $10^9$ orbital periods of the binary system. We obtain the DMP density distribution inside such systems and determine the enhancement factor of their density in a center vicinity compared to its galactic value as a function of the mass ratio of the bodies and the ratio of the body velocity to the velocity of the galactic DMP wind. We find that the enhancement factor can be on the order of tens of thousands.
We use the methods of quantum chaos and Random Matrix Theory for analysis of statistical fluctuations of PageRank probabilities in directed networks. In this approach the effective energy levels are given by a logarithm of PageRank probability at a g
iven node. After the standard energy level unfolding procedure we establish that the nearest spacing distribution of PageRank probabilities is described by the Poisson law typical for integrable quantum systems. Our studies are done for the Twitter network and three networks of Wikipedia editions in English, French and German. We argue that due to absence of level repulsion the PageRank order of nearby nodes can be easily interchanged. The obtained Poisson law implies that the nearby PageRank probabilities fluctuate as random independent variables.
We study the capture of galactic dark matter particles in the Solar System produced by rotation of Jupiter. It is shown that the capture cross section is much larger than the area of Jupiter orbit being inversely diverging at small particle energy. W
e show that the dynamics of captured particles is chaotic and is well described by a simple symplectic dark map. This dark map description allows to simulate the scattering and dynamics of $10^{14}$ dark matter particles during the life time of the Solar System and to determine dark matter density profile as a function of distance from the Sun. The mass of captured dark matter in the radius of Neptune orbit is estimated to be $2 cdot 10^{15} g$. The radial density of captured dark matter is found to be approximately constant behind Jupiter orbit being similar to the density profile found in galaxies.
We build up a directed network tracing links from a given integer to its divisors and analyze the properties of the Google matrix of this network. The PageRank vector of this matrix is computed numerically and it is shown that its probability is inve
rsely proportional to the PageRank index thus being similar to the Zipf law and the dependence established for the World Wide Web. The spectrum of the Google matrix of integers is characterized by a large gap and a relatively small number of nonzero eigenvalues. A simple semi-analytical expression for the PageRank of integers is derived that allows to find this vector for matrices of billion size. This network provides a new PageRank order of integers.
We analyze the statistical properties of Poincare recurrences of Homo sapiens, mammalian and other DNA sequences taken from Ensembl Genome data base with up to fifteen billions base pairs. We show that the probability of Poincare recurrences decays i
n an algebraic way with the Poincare exponent $beta approx 4$ even if oscillatory dependence is well pronounced. The correlations between recurrences decay with an exponent $ u approx 0.6$ that leads to an anomalous super-diffusive walk. However, for Homo sapiens sequences, with the largest available statistics, the diffusion coefficient converges to a finite value on distances larger than million base pairs. We argue that the approach based on Poncare recurrences determines new proximity features between different species and shed a new light on their evolution history.
