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arXiv1995

Lindstedt series, ultraviolet divergences and Moser's theorem

Moser's invariant tori for a class of nonanalytic quasi integrable even hamiltonian systems are shown to be analytic in the perturbation parameter. We do so by exhibiting a summation rule for the divergent series (``Lindstedt series") that formally define them. We find additional cancellations taking place in the formal series, besides the ones already known and necessary in the analytic case (i.e. to prove convergence of Lindtsedt algorithm for Kolmogorov's invariant tori). The method is interpreted in terms of a non renormalizable quantum field theory, considerably more singular than the one we pointed out in the analytic case.

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arXiv1995

Random band matrix approach to chaotic scattering: the average $S$-matrix and its pole distribution

Random band matrices relevant for open chaotic systems are introduced and studied. The scattering model based on such matrices may serve for the description of preequilibrium chaotic scattering. In the limit of a large number of open channels we calculate the average $S$-matrix and $S$-matrix's pole distribution which are found to reduce to those of the full matrix (GOE) case under proper renormalization of the energy scale and strength of coupling to the continuum.

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arXiv1996

Burgers Velocity Fields and the Electromagnetic Forcing in Schroedinger's Interpolating Dynamics

We explore a connection of the deterministically forced Burgers equation for local velocity fields with probabilistic solutions of the Schrödinger boundary data problem. An issue of deducing the most likely interpolating dynamics from the given initial and terminal probability density data is here investigated to give account of the perturbation by external electromagnetic fields. A suitable extension of the Hopf-Cole logarithmic transformation allows to deal with nonconservative drift fields and implies the validity of appropriately generalised heat equations, which completely determine the dynamics.

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arXiv1996

Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures

The thermodynamic approach to non-equilibrium dynamics describes the state of macroscopic systems by means of a collection of intensities or intensive variables. The latter are by definition the differentials of the entropy with respect to the set of extensive constraints. The environment is directly involved in controlling the intensities. The isolation paradigm is negated. The general principles substanciating the approach are restated and expanded to multi-component systems. The procedure is applied to the prediction of transport processes (viscosity and thermal conductivity) of mixtures of atomic gases. Theoretical results are compared with published experimental data.

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arXiv1996

Asymptotic formulae for the Lyapunov spectrum of fully-developed shell model turbulence

We study the scaling behavior of the Lyapunov spectra of a chaotic shell model for 3D turbulence. First, we quantify localization of the Lyapunov vectors in the wavenumber space by using the numerical results. Using dimensional arguments of Kolmogorov-type, we then deduce explicitly the asymptotic scaling behavior of the Lyapunov spectra. This in turn is confirmed by numerical results. This shell model may be regarded as a rare example of high-dimensional chaotic systems for which an analytic expression is known for the Lyapunov spectrum. Implications for the Navier-Stokes turbulence is given. In particular we conjecture that the distribution of Lyapunov exponents is {\it not} singular at null exponent.

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arXiv1996

Backscattering-Induced Crossover in Deterministic Diffusion

We study diffusion in a one-dimensional periodic array of scatterers modeled by a simple map. The chaotic scattering process for this map can be changed by a control parameter and exhibits the dynamics of a crisis in chaotic scattering. We show that the strong backscattering associated with the crisis mechanism induces a crossover which leads to different asymptotic laws for the parameter-dependent diffusion coefficient. These laws are obtained from exact diffusion coefficient results and are supported by simple random walk models. We argue that the main physical feature of the crossover should be present in many other dynamical systems with non-equilibrium transport.

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arXiv1996

Classical Kolmogorov scaling is inconsistent with local coupling

We consider cascade models of turbulence which are obtained by restricting the Navier-Stokes equation to local interactions. By combining the results of the method of extended self-similarity and a novel subgrid model, we investigate the inertial range fluctuations of the cascade. Significant corrections to the classical scaling exponents are found. The dynamics of our local Navier-Stokes models is described accurately by a simple set of Langevin equations proposed earlier as a model of turbulence [Phys. Rev. E {\bf 50}, 285 (1994)]. This allows for a prediction of the intermittency exponents without adjustable parameters. Excellent agreement with numerical simulations is found.

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arXiv1996

Spatio-temporal chaos and patterns formation in nonequilibrium media: phenomenological model of electronic turbulence

The processes in nonequilibrium dissipative media caused by coherent structure formation and lead to the complicated dynamics are of interest for nonlinear physics. Here we consider a model of the flow of interacting electronics patterns. Under electronic pattern we understand a small volume of active media, containing electron oscillators. The superradiation phenomenon takes place due to the evolving radiation instability within each small volume of the flow. It is numerically shown that the dynamics of radiation is changed from spatio-temporal complex oscillations to arising of spatial disorder, i. e. chaotic spatial pattern formation.

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arXiv1996

An Invitation to Algorithmic Information Theory

I'll outline the latest version of my limits of math course. The purpose of this course is to illustrate the proofs of the key information-theoretic incompleteness theorems of algorithmic information theory by means of algorithms written in a specially designed version of LISP. The course is now written in HTML with Java applets, and is available at http://www.research.ibm.com/people/c/chaitin/lm . The LISP now used is much friendlier than before, and because its interpreter is a Java applet it will run in the Netscape browser as you browse my limits of math Web site.

chao-dyn↗
arXiv1996

Do zeta functions for intermittent maps have branch points?

We present numerical evidence that the dynamical zeta function and the Fredholm determinant of intermittent maps with a neutral fix point have branch point singularities at z=1 We consider the power series expansion of zeta function and the Fredholm determinant around z=0 with the fix point pruned. This power series is computed up to order 20, requiring 10^5periodic orbits. We also discuss the relation between correlation decay and the nature of the branch point. We conclude by demonstrating how zeros of zeta functions with thermodynamic weights that are close to the branch point can be efficiently computed by a resummed cycle expansion. The idea is quite similar to that of Padeé approximants, but the ansatz is a generalized series expansion around the branch point instead of a rational function.

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arXiv1996

Classical Billiards in a Magnetic Field and a Potential

We study billiards in plane domains, with a perpendicular magnetic field and a potential. We give some results on periodic orbits, KAM tori and adiabatic invariants. We also prove the existence of bound states in a related scattering problem.

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arXiv1997

Fluctuation patterns and conditional reversibility in nonequilibrium systems

Fluctuations of observables as functions of time, or "fluctuation patterns", are studied in a chaotic microscopically reversible system that has irreversibly reached a nonequilibrium stationary state. Supposing that during a certain, long enough, time interval the average entropy creation rate has a value $s$ and that during another time interval of the same length it has value $-s$ then we show that the relative probabilities of fluctuation patterns in the first time interval are the same as those of the reversed patterns in the second time interval. The system is ``conditionally reversible'' or irreversibility in a reversible system is "driven" by the entropy creation: while a very rare fluctuation happens to change the sign of the entropy creation rate it also happens that the time reversed fluctuations of all other observables acquire the same relative probability of the corresponding fluctuations in presence of normal entropy creation. A mathematical proof is sketched.

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arXiv2004

Singularities of Transition Processes in Dynamical Systems: Qualitative Theory of Critical Delays

The paper gives a systematic analysis of singularities of transition processes in dynamical systems. General dynamical systems with dependence on parameter are studied. A system of relaxation times is constructed. Each relaxation time depends on three variables: initial conditions, parameters $k$ of the system and accuracy $ε$ of the relaxation. The singularities of relaxation times as functions of $(x_0,k)$ under fixed $ε$ are studied. The classification of different bifurcations (explosions) of limit sets is performed. The relations between the singularities of relaxation times and bifurcations of limit sets are studied. The peculiarities of dynamics which entail singularities of transition processes without bifurcations are described as well. The analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems.

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arXiv1997

A KAM type theorem for systems with round-off errors

Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth periodic on the angle perturbations of the twist map, bounding the discretized map from above and from below. Using the well known Moser's theorem we prove the existence of invariant curves for these smooth approximations. As a result we are able to prove that any trajectory of the discretized twist map is eventually periodic. We discuss also some questions, concerning the application of the intersection property in Moser's theorem and the generalization of our results for the twist map in Lobachevski plane.

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arXiv1997

Scale-dependent functions in statistical hydrodynamics: a functional analysis point of view

Most of dynamic systems which exhibit chaotic behavior are also known to posses self-similarity and manifest strong fluctuations of all possible scales.The meaning of this terms is not always same. In present note we make an attempt to formulate the problem in the framework of functional analysis. The statistical hydrodynamics is taken as a vivid physical example. The links to wavelet analysis, multi-fractal formalism and Nottale's scale relativity are presented.

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arXiv1997

Search for Low-Dimensional Chaos in Observational Data

We introduce the reader to the "global flow reconstruction method". The purpose of the method is to see if a given temporal sequence has been generated by a low dimensional dynamics, and to determine that dimension and other properties of the dynamics. The method is shown to work very well on the well known Roessler system. An application of the method to the observational data of the light curve of the irregular variable star R Scuti is presented. This light curve is shown to have been generated by a 4 dimensional dynamics. From the functional form of the flow we find that the Lyapunov dimension is close to 3.1.

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