Hyperbolic Chaos of Turing Patterns

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.Comment: 4 pages, 4 figure

Absolute Continuity Theorem for Random Dynamical Systems on RdR^d

In this article we provide a proof of the so called absolute continuity theorem for random dynamical systems on $R^d$ which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincar\'e map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.Comment: 46 page

Statistical properties of Lorenz like flows, recent developments and perspectives

We comment on mathematical results about the statistical behavior of Lorenz equations an its attractor, and more generally to the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statisitcal behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated to the existence of physical measures: \emph{in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure}. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial conditions, physical measure, singular-hyperbolicity, expansiveness, robust attractor, robust chaotic flow, positive Lyapunov exponent, large deviations, hitting and recurrence times. Minor typos corrected and precise acknowledgments of financial support added. To appear in Int J of Bif and Chaos in App Sciences and Engineerin

Dephasing representation of quantum fidelity for general pure and mixed states

General semiclassical expression for quantum fidelity (Loschmidt echo) of arbitrary pure and mixed states is derived. It expresses fidelity as an interference sum of dephasing trajectories weighed by the Wigner function of the initial state, and does not require that the initial state be localized in position or momentum. This general dephasing representation is special in that, counterintuitively, all of fidelity decay is due to dephasing and none due to the decay of classical overlaps. Surprising accuracy of the approximation is justified by invoking the shadowing theorem: twice--both for physical perturbations and for numerical errors. It is shown how the general expression reduces to the special forms for position and momentum states and for wave packets localized in position or momentum. The superiority of the general over the specialized forms is explained and supported by numerical tests for wave packets, non-local pure states, and for simple and random mixed states. The tests are done in non-universal regimes in mixed phase space where detailed features of fidelity are important. Although semiclassically motivated, present approach is valid for abstract systems with a finite Hilbert basis provided that the discrete Wigner transform is used. This makes the method applicable, via a phase space approach, e. g., to problems of quantum computation.Comment: 11 pages, 4 figure

On the Numerical Study of the Complexity and Fractal Dimension of CMB Anisotropies

We consider the problem of numerical computation of the Kolmogorov complexity and the fractal dimension of the anisotropy spots of Cosmic Microwave Background (CMB) radiation. Namely, we describe an algorithm of estimation of the complexity of spots given by certain pixel configuration on a grid and represent the results of computations for a series of structures of different complexity. Thus, we demonstrate the calculability of such an abstract descriptor as the Kolmogorov complexity for CMB digitized maps. The correlation of complexity of the anisotropy spots with their fractal dimension is revealed as well. This technique can be especially important while analyzing the data of the forthcoming space experiments.Comment: LATEX, 3 figure

Counting Berg partitions

We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n)

The Aggregation Kinetics of a Simulated Telechelic Polymer

We investigate the aggregation kinetics of a simulated telechelic polymer gel. In the hybrid Molecular Dynamics (MD) / Monte Carlo (MC) algorithm, aggregates of associating end groups form and break according to MC rules, while the position of the polymers in space is dictated by MD. As a result, the aggregate sizes change every time step. In order to describe this aggregation process, we employ master equations. They define changes in the number of aggregates of a certain size in terms of reaction rates. These reaction rates indicate the likelihood that two aggregates combine to form a large one, or that a large aggregate splits into two smaller parts. The reaction rates are obtained from the simulations for a range of temperatures. Our results indicate that the rates are not only temperature dependent, but also a function of the sizes of the aggregates involved in the reaction. Using the measured rates, solutions to the master equations are shown to be stable and in agreement with the aggregate size distribution, as obtained directly from simulation data. Furthermore, we show how temperature induced variations in these rates give rise to the observed changes in the aggregate distribution that characterizes the sol-gel transition.Comment: 9 pages, 10 figure

Lipschitz shadowing implies structural stability

We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. As a corollary, we show that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov.Comment: 11 page

Big Entropy Fluctuations in Statistical Equilibrium: The Macroscopic Kinetics

Large entropy fluctuations in an equilibrium steady state of classical mechanics were studied in extensive numerical experiments on a simple 2--freedom strongly chaotic Hamiltonian model described by the modified Arnold cat map. The rise and fall of a large separated fluctuation was shown to be described by the (regular and stable) "macroscopic" kinetics both fast (ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate" initial conditions by observing (in a long run)spontaneous birth and death of arbitrarily big fluctuations for any initial state of our dynamical model. Statistics of the infinite chain of fluctuations, reminiscent to the Poincar\'e recurrences, was shown to be Poissonian. A simple empirical relation for the mean period between the fluctuations (Poincar\'e "cycle") has been found and confirmed in numerical experiments. A new representation of the entropy via the variance of only a few trajectories ("particles") is proposed which greatly facilitates the computation, being at the same time fairly accurate for big fluctuations. The relation of our results to a long standing debates over statistical "irreversibility" and the "time arrow" is briefly discussed too.Comment: Latex 2.09, 26 pages, 6 figure

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis.Comment: 24 pages, 13 figure