kolmogorov sinai entropy of many body hamiltonian systems the kolmogorov sinai ks entropy is a central measure of complexity and chaos its calculation for many body systems is an interesting and important challenge in this paper the evaluation is formulated by considering n dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem this approach makes explicit a duality relation that is exactly analogous to one found in a generalized anderson tight binding model and leads to a formally exact expression for the finite time ks entropy within this formalism there is a hierarchy of approximations the final one being a diagonal approximation that only makes use of instantaneous hessians of the potential to find the ks entropy by way of a nontrivial illustration the ks entropy of n identically coupled kicked rotors standard maps is investigated the validity of the various approximations with kicking strength particle number and time are elucidated an analytic formula for the ks entropy within the diagonal approximation is derived and its range of validity is also explored

riemannian theory of hamiltonian chaos and lyapunov exponents this paper deals with the problem of analytically computing the largest lyapunov exponent for many degrees of freedom hamiltonian systems this aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of riemannian geometry a new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of jacobi equation for geodesic spread under general conditions ane effective stability equation is derived an analytic formula for the growth rate of its solutions is worked out and applied to the fermi pasta ulam beta model and to a chain of coupled rotators an excellent agreement is found the theoretical prediction and the values of the lyapunov exponent obtained by numerical simulations for both models

local lyapunov exponents in chaotic systems abstract we define a local lyapunov exponent in chaotic systems so that the lyapunov exponent along a trajectory is the integral over the local exponent thus extending previously used short time lyapunov exponents to a pointwise definition calculations are presented for the dissipative lorenz system the conservative chirikov taylor standard map and a 3d volume preserving flow of interest in magnetohydrodynamics

correlated wishart ensembles and chaotic time series we study the correlated wishart ensembles in the context of time series analysis we are interested in the statistics of eigenlevels viz variables associated with independent eigenmodes in the system the motivation of this work is to study the effect of time series correlations on the wishart ensembles in this connection we derive the level density and the two point function for the correlated wishart ensembles by using the binary correlation method using our analytic results we analyze spectra of autocovariance matrices derived from single variable stationary time series we consider the stochastic time series of gaussian variables with exponentially decaying correlations and time series of chaotic maps viz the arnold map the standard map and the stadium billiard map in both cases correlated time series are encountered and analyzed under the framework of random matrix theory it is shown that the eigenlevel statistics for the chaotic maps follow closely those of correlated wishart ensembles it is indicated that the presence of collective modes in the spectra of autocovariance matrices is related to the integrability of the system

limit sets of automatic sequences abstract we show that for every k automatic sequence there exists a natural number p 0 such that the sequences of the form kpn j n 0 with j 0 p 1 are scaling sequences for f moreover we demonstrate that every limit set is the union of certain basic limit sets

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

automatic sets and delone sets automatic sets are characterized by having a finite number of decimations they are equivalently generated by fixed points of certain substitution systems or by certain finite automata as examples two dimensional versions of the thue morse baum sweet rudin shapiro and paperfolding sequences are presented we give a necessary and sufficient condition for an automatic set to be a delone set in the result is then extended to automatic sets that are defined as fixed points of certain substitutions the morphology of automatic sets is discussed by means of examples

characterizing weak chaos in nonintegrable hamiltonian systems the fundamental role of stickiness and initial conditions abstract weak chaos in high dimensional conservative systems can be characterized through sticky effect induced by invariant structures on chaotic trajectories suitable quantities for this characterization are the higher cummulants of the finite time lyapunov exponents ftles distribution they gather the whole phase space relevant dynamics in one quantity and give information about ordered and random states this is analyzed here for discrete hamiltonian systems with local and global couplings it is also shown that ftles plotted versus initial condition ic and the nonlinear parameter are essential to understand the fundamental role of ics in the dynamics of weakly chaotic hamiltonian systems

navier stokes equations with vorticity in besov spaces of negative regular indices abstract this paper studies the cauchy problem for the three dimensional navier stokes equations and shows that the condition u l 2 2 r 0 t b r 0 r 2 ensures the regularity of the solution on 0 t this improves and extends many previous results

on the stability of periodic orbits of two dimensional mappings we present a closed form stability criterion for the periodic orbits of two dimensional conservative as well as dissipative mappings which are analogous to the poincare maps of dynamical systems our stability criterion has a particularly simple form involving a finite symmetric nearly tridiagonal determinant its derivation is based on an extension of the stability analysis of hill s differential equation to difference equations we apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied standard mapping describing a periodically kicked free rotator as another example we also obtain explicitly and in closed form the intervals of bounded and unbounded solutions of a discrete schrodinger equation for the kronig and penney crystal model

semiclassical chaos in atomic clusters an exploration with ring polymer molecular dynamics the ring polymer molecular dynamics rpmd method was recently shown to provide a powerful framework to address the time evolution of many particle semiclassical systems its underlying hamiltonian formulation is exploited here to define and compute some measures of deterministic chaos namely the lyapunov characteristic exponent applications are presented for weakly bound neon clusters with the aim of assessing the influence of nuclear delocalization on the nonlinear dynamics it is found that the balance between the additional degrees of freedom in the rpmd description and the harmonic nature of the additional interactions leads to more regular dynamics at high temperature but more chaotic dynamics at low temperature intrinsic features of quantum effects on the isomerization dynamics also have a signature on the lyapunov exponent

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

fluctuations of finite time stability exponents in the standard map and the detection of small islands some statistical properties of finite time stability exponents in the standard map can be estimated analytically the mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength in particular the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings the average of the stability matrix inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter the usual accelerator modes and the small islands created through double saddle node bifurcations which come halfway between the positions in interaction strength of the usual accelerator modes are clearly visible in the variance whose time scaling is capable of detecting the presence of islands as small as 0 01 of the phase space we study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions we also discuss the nature of this local approximation in some detail

space time properties of gram schmidt vectors in classical hamiltonian evolution not all tangent space directions play equivalent roles in the local chaotic motions of classical hamiltonian many body systems these directions are numerically represented by basis sets of mutually orthogonal gram schmidt vectors whose statistical properties may depend on the chosen phase space time domain of a trajectory we examine the degree of stability and localization of gram schmidt vector sets simulated with trajectories of a model three atom lennard jones cluster distributions of finite time lyapunov exponent and inverse participation ratio spectra formed from short time histories reveal that ergodicity begins to emerge on different time scales for trajectories spanning different phase space regions in a narrow range of total energy and history length over a range of history lengths the most localized directions were typically the most unstable and corresponded to atomic configurations near potential landscape saddles

a note on the regularity criterion for 3d mhd equations in b 1 space abstract in this note we consider sufficient conditions for the regularity of leray hoph hopf solutions of the 3d incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of b 1 spaces we prove that if u b belongs to the space l 2 0 t b 1 r 3 then the solution u b is regular this extends recent results contained in gala 2011 3

anomalous convergence of lyapunov exponent estimates numerical experiments reveal that estimates of the lyapunov exponent for the logistic map ital x sub ital t 1 ital f ital x sub ital t 4 ital x sub ital t 1 minus ital x sub ital t are anomalously precise they are distributed with a standard deviation that scales as 1 ital n where ital n is the length of the trajectory not as 1 radical ital n the scaling expected from an informal interpretation of the central limit theorem we show that this anomalous convergence follows from the fact that the logistic map is conjugate to a constant slope map the lyapunov estimator is just one example of a chaotic walk we show that whether or not a general chaotic walk exhibits anomalously small variance depends only on the autocorrelation of the chaotic process

a role for a brachistochrone in the broken ergodicity scenario abstract we show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales in turn this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level such a logarithmic dependence has already been shown to hold valid for certain complex systems moreover it represents a key factor to explain the experimentally ubiquitous debye kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field

the calculation of lyapunov spectra abstract linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems the naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations this information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration in this paper we extract information from this matrix through its singular values and the associated orthogonal matrices this given an algebraic foundation to the geometric methods that have been used previously the equations of the orthogonal principal stretching and contraction directions show that asymptotically after a long time they depend only on the location in phase space and not on the orbit as an example we show that these directions sketch out the attractor of the lorenz model

statistics of stretching fields in experimental fluid flows exhibiting chaotic advection stretching fields and their statistical properties are studied experimentally for four distinct two dimensional time periodic confined fluid flows exhibiting chaotic advection a random vortex array for two different reynolds numbers a set of parallel shear layers and a vortex lattice the flows are driven electromagnetically and they are studied by means of precise particle velocimetry we find that for a given flow the probability distributions of log s where s is the local stretching in n cycles can be nearly superimposed for different n when log s is rescaled using the geometrical mean of the stretching distribution the rescaled stretching fields for a given flow at various n are highly correlated spatially when n is large finally the scaled distributions for different flows are similar though there are some differences connected to the degree of spatial symmetry and time reversibility of the flows

statistics of finite time lyapunov exponents in a random time dependent potential the sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite time lyapunov exponent lambda obtained from the elements m ij of the stability matrix m for globally chaotic dynamics lambda tends to a unique value the usual lyapunov exponent lambda infinity for almost all trajectories as t is sent to infinity but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity we compute for a particle moving in a randomly time dependent one dimensional potential how the distribution function p lambda t approaches the limiting distribution p lambda infinity delta lambda lambda infinity our method also applies to the tail of the distribution which determines the growth rates of moments of m ij the results are also applicable to the problem of wave function localization in a disordered one dimensional potential

extreme statistics of complex random and quantum chaotic states complex random states have the statistical properties of the gaussian and circular unitary ensemble eigenstates of random matrix theory even though their components are correlated by the normalization constraint it is nevertheless possible to derive compact analytic formulas for their extreme values statistical properties for all dimensionalities the maximum intensity result slowly approaches the gumbel distribution even though the variables are bounded whereas the minimum intensity result rapidly approaches the weibull distribution since random matrix theory is conjectured to be applicable to chaotic quantum systems we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic the statistical behaviors are consistent with the finite n formulas

characteristic distributions of finite time lyapunov exponents we study the probability densities of finite time or local lyapunov exponents in low dimensional chaotic systems while the multifractal formalism describes how these densities behave in the asymptotic or long time limit there are significant finite size corrections which are coordinate dependent depending on the nature of the dynamical state the distribution of local lyapunov exponents has a characteristic shape for intermittent dynamics and at crises dynamical correlations lead to distributions with stretched exponential tails while for fully developed chaos the probability density has a cusp exact results are presented for the logistic map stackrel ensuremath rightarrow x 4x 1 ensuremath x at intermittency the density is markedly asymmetric while for typical chaos it is known that the central limit theorem obtains and a gaussian density results local analysis provides information on the variation of predictability on dynamical attractors these densities which are used to characterize the nonuniform spatial organization on chaotic attractors are robust to noise and can therefore be measured from experimental data

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

hypersensitivity to perturbations of quantum chaotic wave packet dynamics we reexamine the problem of the loschmidt echo that measures the sensitivity to perturbation of quantum chaotic dynamics the overlap squared m t of two wave packets evolving under slightly different hamiltonian is shown to have the double exponential initial decay ensuremath propto mathrm exp ensuremath mathrm constant ifmmode times else texttimes fi e 2 ensuremath lambda 0 t in the main part of the phase space the coefficient ensuremath lambda 0 is the self averaging lyapunov exponent the average decay overline m ensuremath propto e ensuremath ensuremath lambda 1 t is single exponential with a different coefficient ensuremath lambda 1 the volume of phase space that contributes to overline m vanishes in the classical limit stackrel ensuremath rightarrow ensuremath elzxh 0 for times less than the ehrenfest time ensuremath tau e frac 1 2 ensuremath lambda 0 ensuremath 1 mathrm ln ensuremath elzxh it is only after the ehrenfest time that the average decay is representative for a typical initial condition

statistics of finite time lyapunov exponents in the ulam map the statistical properties of finite time lyapunov exponents at the ulam point of the logistic map are investigated the exact analytical expression for the autocorrelation function of one step lyapunov exponents is obtained allowing the calculation of the variance of exponents computed over time intervals of length n the variance anomalously decays as 1 n 2 the probability density of finite time exponents noticeably deviates from the gaussian shape decaying with exponential tails and presenting 2 n 1 spikes that narrow and accumulate close to the mean value with increasing n the asymptotic expression for this probability distribution function is derived it provides an adequate smooth approximation to describe numerical histograms built for not too small n where the finiteness of bin size trims the sharp peaks

a remark on the blow up criterion of strong solutions to the navier stokes equations abstract in this note we prove that b 1 norm of the vorticity controls the blow up phenomena of strong solutions to the navier stokes equations our criterion is a limiting case in yuan zhang 11 and an improvement of giga s result 3

maximal lyapunov exponent at crises we study the variation of lyapunov exponents of simple dynamical systems near attractor widening and attractor merging crises the largest lyapunov exponent has universal behavior showing abrupt variation as a function of the control parameter as the system passes through the crisis point either in the value itself in the case of an attractor widening crisis or in the slope for an attractor merging crisis the distribution of local lyapunov exponents is very different for the two cases the fluctuations remain constant through a merging crisis but there is a dramatic increase in the fluctuations at a widening crisis

probability distributions of local liapunov exponents for small clusters the probability distribution of the largest local liapunov exponent is evaluated for a classical ar sub 3 cluster at different values of the internal energy ital e for a set of increasing values of the length in which the trajectory is partitioned these distributions can be directly related to the evolution of ergodic behavior particularly to how it exhibits distinctive separable time scales which depend strongly on the energy of the system therefore even though the inequivalence of ergodicity and chaos prohibits a liapunov exponent itself from being a quantitative index of ergodicity we find that 2 the sample distributions used to evaluate liapunov exponents nevertheless can be used for this purpose

an analogue of cobham s theorem for fractals we introduce the notion of k self similarity for compact subsets of n and show that it is a natural analogue of the notion of k automatic subsets of integers we show that various well known fractals such as the triadic cantor set the sierpinski carpet or the menger sponge turn out to be k self similar for some integers k we then prove an analogue of cobham s theorem for compact sets of that are self similar with respect to two multiplicatively independent bases k and l namely we show that x is both a k and an l self similar compact subset of if and only if it is a finite union of closed intervals with rational endpoints

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

automaticity of double sequences generated by one dimensional linear cellular automata abstract we give a complete answer to the question whether a double sequence that is generated by a one dimensional linear cellular automaton and whose states are integers modulo m is k automatic or not

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

intrinsic stickiness and chaos in open integrable billiards tiny border effects rounding border effects at the escape point of open integrable billiards are analyzed via the escape time statistics and emission angles the model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form stickiness chaos and self similar structures for the escape times and emission angles are generated inside backgammon like stripes of initial conditions these stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics tiny rounded borders around 0 1 from the whole billiard size are shown to be sufficient to generate the sticky motion with power law decay esc 1 27 while borders larger than 10 are enough to produce escape times related to the chaotic motion escape exponents in the interval 1 esc 2 are generated due to marginal unstable periodic orbits trapping alternately in time regular and chaotic trajectories

blow up criterion of strong solutions to the navier stokes equations in besov spaces with negative indices a holder type inequality in besov spaces is established and applied to show that every strong solution u t x on 0 t of the navier stokes equations can be continued beyond t t provided that the vorticity t x l22 0 t b r3 l21 0 t b 1 r3 for 0 1

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics