transition to chaos in a shell model of turbulence abstract we study a shell model for the energy cascade in three dimensional turbulence by varying the coefficients of the non linear terms in such a way that the fundamental symmetries of navier stokes are conserved when the control parameter related to the strength of backward energy transfer is small enough the dynamical system has a stable fixed point corresponding to the kolmogorov scaling by using the bi orthogonal decomposition the transition to chaos is shown to follow the ruelle takens scenario for 0 3953 there exists a strange attractor which remains close to the kolmogorov fixed point the intermittency of the chaotic evolution and of the scaling can be described by an intermittent one dimensional map we introduce a modified shell model which has a good scaling behaviour also in the infrared region we study the multifractal properties of this model for large number of shells and for values of slightly above the chaotic transition in this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

complexity in quantum systems we discuss the behavior of a quantum 1 2 spin coupled to a time dependent magnetic field which can be quasiperiodic or random for a quasiperiodic field the time evolution of the system on a coarse grained space has autocorrelation which does not decay and positive topological entropy the information complexity diverges as a stretched exponential although the shannon entropy is zero and there is no divergence of nearby orbits a simple random process is introduced in order to reproduce the main qualitative features of the observed weakly complex behavior

the entropy function for characteristic exponents using a thermodynamic formalism we define an entropy function s a which measures large deviations of the liapunov characteristic exponents of certain hyperbolic dynamical systems the function s a is a legendre transform of a free energy or pressure associated with the dynamical system we show that s a is the noncompact topological entropy of the set of points a with characteristic exponent a that s a a is the hausdorff dimension of a and that a s a is the escape rate from a we explain how to use the formalism for cookie cutters to describe the distribution of scales in the universal period doubled attractor and critical golden circle mapping we relate s a to the renyi entropies prove a conjecture of kantz and grassberger relating the escape rate from hyperbolic repellors and saddles to the characteristic exponents and information dimension and study the fluctuations of escape rates we discuss the escape rate from f x 4 e x 1 x and the behaviour of the associated s a as e 0 finally we discuss how to apply these ideas to experimental time series and non hyperbolic attractors

predictability of velocity and temperature fields in intermittent turbulence we discuss the relation between predictability and the sensitive dependence on initial conditions in turbulent flows the dependence on the reynolds number of the maximum lyapunov exponent of the flow is evaluated within the multifractal model the temporal intermittency of the degree of chaos is found to localize on the degrees of freedom corresponding to high wavenumbers we discuss the consequences for the mechanism for the growth of small perturbations we show that intermittency causes long tails in the distribution of the predictability times both in the velocity field and in a passively advected scalar field though there is no simple relation between these two times numerical evidence of this picture is provided within the framework of a cascade model for velocity and temperature fields in fully developed turbulence

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

estimating the ultimate bound and positively invariant set for the lorenz system and a unified chaotic system to estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general in this paper we attempt to investigate the ultimate bound and positively invariant set for two specific systems the lorenz system and a unified chaotic system we derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the lorenz system for all the positive values of its parameters a b and c and obtain the minimum value of volume for the ellipsoid comparing with the best results in the current literature d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 our new results fill up the gap of the estimate for the cases of 0 a 1 and 0 b 2 x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 furthermore the estimation derived here contains the results given in d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 and x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 as special cases along the same line we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system for its parameter range 0 129 and obtain the minimum value of volume for the ellipsoid the estimate is more accurate than and also extends the result of d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 and x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419

a note on hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system in appl math comput 291 2016 323 339 by amin zarei and saeed tavakoli abstract in the recent paper entitled hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system in appl math comput 291 2016 323 339 by amin zarei and saeed tavakoli they proposed the following new four dimensional 4 d quadratic autonomous hyper chaotic system x 1 a x 2 x 1 x 2 b x 1 x 2 e x 4 x 1 x 3 x 3 c x 3 x 1 x 2 x 1 2 x 4 d x 2 which generates double wing chaotic and hyper chaotic attractors with only one equilibrium point combining theoretical analysis and numerical simulations they investigated some dynamical properties of that system like lyapunov exponent spectrum bifurcation diagram phase portrait hopf bifurcation etc in particular they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems int j bifurc chaos 21 09 2011 2679 2694 by p wang et al however by means of detailed theoretical analysis we show that both the conclusion itself and the derivation of its proof in appl math comput 291 2016 323 339 are erroneous furthermore we point out that the method adopted for studying the ultimate bound of that system is not applicable at all therefore the ultimate bound estimation of that system needs further studying in future work

on the dynamics of navier stokes and euler equations this is a detailed study on certain dynamics of navier stokes and euler equations via a combination of analysis and numerics we focus upon two main aspects a zero viscosity limit of the spectra of linear navier stokes operator b heteroclinics conjecture for euler equation its numerical verification melnikov integral and simulation and control of chaos due to the difficulty of the problem for the full navier stokes and euler equations we also propose and study two simpler models of them

polylogarithms regulators and arakelov motivic complexes we construct an explicit regulator map from bloch s higher chow group complex to the deligne complex of a complex algebraic variety x we define the arakelov motivic complex as the cone of this map shifted by one its last cohomology group is a version of the arakelov chow group defined by h gillet and c soule we relate the grassmannian n logarithms defined as in g5 to the geometry of symmetric space sln c su n for n 2 we recover lobachevsky s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm using the relationship with symmetric spaces we construct the borel regulator on k2n 1 c via the grassmannian n logarithms we study the chow dilogarithm and prove a reciprocity law which strengthens suslin s reciprocity law for milnor s group k 3 on curves our note g5 can serve as an introduction to this paper

estimating generalized lyapunov exponents for products of random matrices we discuss several techniques for the evaluation of the generalized lyapunov exponents which characterize the growth of products of random matrices in the large deviation regime a monte carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general purpose numerical method which is both efficient and easy to implement alternative techniques complementing this method are presented these include the computation of the generalized lyapunov exponents by solving numerically an eigenvalue problem and some asymptotic results corresponding to high order moments of the matrix products taken together the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized lyapunov exponents in a broad range of applications their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows

extinction dynamics of a discrete population in an oasis understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology the first theoretical approach to the problem dates back to the 1950s with the kierstead slobodkin and skellam kiss model namely a continuous reaction diffusion equation for a population growing on a patch of finite size l surrounded by a deadly environment with infinite mortality i e an oasis in a desert the main outcome of the model is that only patches above a critical size allow for population persistence here we introduce an individual based analog of the kiss model to investigate the effects of discreteness and demographic stochasticity in particular we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasistationary distribution of the number of individuals for patch sizes above the critical threshold

a simple construction of grassmannian polylogarithms abstract the classical n logarithm is a multivalued analytic function defined inductively li n z 0 z li n 1 t d log t li 1 z log 1 z in this paper we give a simple explicit construction of the grassmannian n logarithm which is a multivalued analytic function on the quotient of the grassmannian of n dimensional subspaces in c 2 n in generic position to the coordinate hyperplanes by the natural action of the torus c 2 n the classical n logarithm appears at a certain one dimensional boundary stratum we study tate iterated integrals which are homotopy invariant integrals of 1 forms d log f i where f i are rational functions we give a simple explicit formula for the tate iterated integral which describes the grassmannian n logarithm another example is the tate iterated integrals for the multiple polylogarithms on the moduli spaces m 0 n calculated in section 4 4 of goncharov 2005 13 using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms variations of mixed hodge tate structures on x are described by a hopf algebra h x we upgrade tate iterated integrals on a rational complex variety x to elements of h x the coproducts of these elements are very interesting invariants of the iterated integrals in general their calculation is a non trivial problem we show however that working modulo the ideal of h x generated by constant variations there is a simple way to calculate the coproduct it is a pleasure to dedicate this paper to andrey suslin whose works suslin 1984 21 and suslin 1991 22 played an essential role in the development of the story

fast chemical reaction and multiple scale concentration fields in singular vortices two species involved in a simple fast reaction tend to become segregated in patches composed of a single of these reactants these patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs fed by diffusion stirred by advection this boundary and the concentration fields within the patches may tend to present multiple scale characteristics based on this segregated state this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it when concentrations presenting initial self similar fluctuations are advected by a singular vortex first the two sources of singularity i e the self similar initial conditions and the singular vortex are considered isolatedly on the one hand self similar initial conditions are imposed to a diffusion reaction system for one and two dimensional cases on the other hand an imposed singular vortex advects initially on off concentration fields in combination with diffusion and reaction this problem is addressed analytically by characterizing the boundary by a box counting dimension and the concentration fields by a h older exponent and numerically by direct numerical simulations of the advection diffusion reaction equations second the way the two sources hang together shows that depending on the self similar properties of the initial concentration fields the vortex promotes the chemical activity close to its inner smoothed out core or close to the outer region where the boundary starts to spiral for all the considered situations the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws

predictability in systems with many characteristic times the case of turbulence in chaotic dynamical systems an infinitesimal perturbation is exponentially amplified at a time rate given by the inverse of the maximum lyapunov exponent lambda in fully developed turbulence lambda grows as a power of the reynolds number this result could seem in contrast with phenomenological arguments suggesting that as a consequence of physical perturbations the predictability time is roughly given by the characteristic life time of the large scale structures and hence independent of the reynolds number we show that such a situation is present in generic systems with many degrees of freedom since the growth of a non infinitesimal perturbation is determined by cumulative effects of many different characteristic times and is unrelated to the maximum lyapunov exponent our results are illustrated in a chain of coupled maps and in a shell model for the energy cascade in turbulence

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

multiscaling in multifractals multiscaling is shown to be a consequence of multifractality when a lower cutoff is introduced in calculations of correlation functions after a suitable rescaling the correlation function data for different values of seem to fall onto a single curve in the multiscaling regime however we show that there is not a unique functional form at varying but a spread very close to a single curve for each this curve can be computed analytically in terms of the f a spectrum which characterizes the multifractal part of this spectrum can thus be obtained by computing only one moment of the weights at 0

hopf bifurcation analysis of a bam neural network with multiple time delays and diffusion in this paper a bam neural network with multiple time delays and diffusion under homogeneous neumann boundary conditions is investigated by analyzing the corresponding characteristic equations the local stability of the trivial uniform steady state and the existence of hopf bifurcation under two different cases are established respectively by using the normal form theory and the center manifold reduction of partial functional differential equations pfdes explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions numerical simulations are carried out to illustrate the main results

cluster polylogarithms for scattering amplitudes motivated by the cluster structure of two loop scattering amplitudes in yang mills theory we define cluster polylogarithm functions we find that all such functions of weight four are made up of a single simple building block associated with the a2 cluster algebra adding the requirement of locality on generalized stasheff polytopes we find that these a2 building blocks arrange themselves to form a unique function associated with the a3 cluster algebra this a3 function manifests all of the cluster algebraic structure of the two loop n particle mhv amplitudes for all n and we use it to provide an explicit representation for the most complicated part of the n 7 amplitude as an example this article is part of a special issue of journal of physics a mathematical and theoretical devoted to cluster algebras in mathematical physics

hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system based on lorenz system a new four dimensional quadratic autonomous hyper chaotic attractor is presented in this paper it can generate double wing chaotic and hyper chaotic attractors with only one equilibrium point several properties of the new system are investigated using lyapunov exponents spectrum bifurcation diagram and phase portraits using the center manifold and normal form theories the local dynamics the stability and hopf bifurcation at the equilibrium point are analyzed to obtain the ellipsoidal ultimate bound the ultimate bound of the proposed system is theoretically estimated using lagrange multiplier method lagrangian function and local maximizer point by properly choosing p and q matrices an estimation of the ultimate bound region as a function of the lagrange coefficient is obtained using local maximizer point and reduced hessian matrix to demonstrate the evolution of the bifurcation and to show the ultimate bound region numerical simulations are provided

the mixing efficiency of open flows abstract mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary as a measure of the efficiency of stirring the bulk scalar concentration variance is bounded and the bound is shown to be sharp at low peclet number although no specific flow saturating the bound at high peclet number is produced here the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions

vorticity generation by instabilities of chaotic fluid flows abstract we show that at high reynolds number smooth lagrangian chaotic flows are typically linearly unstable and that the perturbed vorticity tends to concentrate on a fractal numerical integration of the relevant linear partial differential equations with reynolds number up to r 10 6 shows that the wavenumber power spectrum of the perturbed vorticity has a powerlaw behavior and that the magnitude of the perturbed vorticity is a multifractal it is then shown that a wavepacket picture whereby vorticity wavepackets are evolved according to ordinary differential equations yields scaling results and appropriately describes the small wavelength features of both the power spectrum and the multifractal dimension spectrum analytical results are derived establishing the validity of the latter approach as well as an equivalence principle for lyapunov partition functions these results are verified by numerical simulations involving a computationally efficient class of chaotic flows

fluctuating hydrodynamics and turbulence in a rotating fluid universal properties we analyze the statistical properties of three dimensional 3d turbulence in a rotating fluid to this end we introduce a generating functional to study the statistical properties of the velocity field mathbit v we obtain the master equation from the navier stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity mathbit ensuremath omega in particular we obtain the differential forms for the analogs of the well known von karman howarth relation for 3d fluid turbulence we examine their behavior in the limit of large rotation our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to mathbit ensuremath omega the hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two dimensional flows to complement these results by using an effective model in the small ensuremath omega limit within a one loop approximation we show that the equal time correlation of the velocity components parallel to mathbit ensuremath omega displays kolmogorov scaling q ensuremath 5 3 wherein as for all other components the equal time correlators scale as q ensuremath 3 in the inertial range where mathbit q is a wave vector in 3d our results are generally testable in experiments and or direct numerical simulations of the navier stokes equation in a rotating frame

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

multiscale mixing efficiencies for steady sources multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales we consider scalars maintained by temporally steady but spatially inhomogeneous sources stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence the mixing efficiencies are rigorously bounded in terms of the p eclet number and specific quantitative features of the source scaling exponents for the bounds at high p eclet number depend on the spectrum of length scales in the source indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena

predictability in the large an extension of the concept of lyapunov exponent we investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales in particular we study the case of three dimensional turbulence at high reynolds numbers by introducing a finite size lyapunov exponent which measures the growth rate of finite size perturbations for sufficiently small perturbations this quantity coincides with the usual lyapunov exponent when the perturbation is still small compared to large scale fluctuations but large compared to fluctuations at the smallest dynamically active scales the finite size lyapunov exponent is inversely proportional to the square of the perturbation size our results are supported by numerical experiments on shell models we find that intermittency corrections do not change the scaling law of predictability we also discuss the relation between the finite size lyapunov exponent and information entropy

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

an algebraic integration for mahler measure there are many examples of several variable polynomials whose mahler measure is expressed in terms of special values of polylogarithms these examples are expected to be related to computations of regulators as observed by deninger d and later by rodriguez villegas r and maillot m while rodriguez villegas made this relationship explicit for the two variable case it is our goal to understand the three variable case and shed some light on the examples with more variables

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

additive polylogarithms and their functional equations let k varepsilon 2 k varepsilon varepsilon 2 the single valued real analytic n polylogarithm mathcal l n mathbb c to mathbb r is fundamental in the study of weight n motivic cohomology over a field k of characteristic 0 in this paper we extend the construction in unver algebra number theory 3 1 34 2009 to define additive n polylogarithms li n k varepsilon 2 to k and prove that they satisfy functional equations analogous to those of mathcal l n under a mild hypothesis we show that these functions descend to an analog of the nth bloch group b n k varepsilon 2 defined by goncharov adv math 114 197 318 1995 we hope that these functions will be useful in the study of weight n motivic cohomology over k e 2

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

anomalous diffusion as modeled by a nonstationary extension of brownian motion if the mean square displacement of a stochastic process is proportional to t ensuremath beta ensuremath beta ensuremath ne 1 then it is said to be anomalous we construct a family of markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean square displacement we label the family as an extended brownian motion and show that they satisfy a langevin equation with time dependent diffusion coefficient if the time derivative of the variance of the process is homogeneous then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the brownian motion for two particles initially separated by a distance x the finite size lyapunov exponent fsle measures the average rate of exponential separation to a distance ax an analytical expression is developed for the fsles of the extended brownian processes and numerical examples presented the explicit construction of these processes illustrates that contrary to what has been stated in the literature a power law mean square displacement is not necessarily related to a breakdown in the classical central limit theorem clt caused by for example correlation fractional brownian motion or correlated continuous time random walk schemes or infinite variance levy motion the classical clt coupled with nonstationary increments can and often does give rise to power law moments such as the mean square displacement

multifractal structure of chaotically advected chemical fields the structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two dimensional flow is investigated we focus our attention on the nonuniformities of the h older exponent of the resulting filamental chemical field they appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold nonuniformities of the h older exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements we study how this affects the scaling exponents of the structure functions displaying anomalous scaling and relate the scaling exponents to the distribution of local lyapunov exponents of the advection dynamics theoretical predictions are compared with numerical experiments

smooth and filamental structures in chaotically advected chemical fields this paper studies the spatial structure of decaying chemical fields generated by a chaotic advection flow and maintained by a spatially smooth chemical source previous work showed that in a regime where diffusion can be neglected large peclet number the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow the scaling exponent gamma q of the qth order structure function depends at leading order linearly on the ratio of the rate of decay of the chemical processes alpha and the average rate of divergence of neighboring fluid parcel trajectories lyapunov exponent h under a homogeneous stretching approximation gamma q q max alpha h 1 which implies that a well defined filamental smooth transition occurs at alpha h this approximation has been improved by using the distribution of finite time lyapunov exponents to characterize the inhomogeneous stretching of the flow however previous work focused more on the behavior of the exponents as q varies and less on the effects of alpha and hence the implications for the filamental smooth transition here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite size rather than finite time lyapunov exponents we clarify the relation between the two distributions we show that the corrected exponents symbol see text depend nonlinearly on alpha with formula see text the magnitude of the correction to the homogeneous stretching approximation formula see text grows as alpha increases reaching a maximum when the leading order transition is reached alpha h the implication of these results is that there is no well defined bulk filamental smooth transition instead it is the case that the chemical field is unambiguously smooth for alpha h max where h max denotes the maximum finite time lyapunov exponent and unambiguously filamental for alpha h with an intermediate character for alpha between these two values theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the cramer function

dynamics of leaking hamiltonian systems in order to understand the dynamics in more detail in particular for visualizing the space filling unstable foliation of closed chaotic hamiltonian systems we propose to leak them up the cutting out of a finite region of their phase space the leak through which escape is possible leads to transient chaotic behavior of nearly all the trajectories the never escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically it is an approximant of the full hamiltonian foliation the better the smaller the leak is the escape rate depends sensitively on the orientation of the leak even if its area is fixed the applications for chaotic advection for chemical reactions superimposed on hydrodynamical flows and in other branches of physics are discussed

discrete time model for chemical or biological decay in chaotic flows reentrance phase transitions we consider a discrete time model of advection reaction and diffusion on a lattice to investigate the steady state spatial structure of chemically decaying substances the time discretization of the dynamics has a considerable impact on these structures additional smooth filamental phase transitions nonexistent in the continuous time description appear we show how these structures and their scaling properties depend on the time step of the discrete dynamics exploiting the analogies of this discrete model with the logistic map some general features are discussed

empirical evidences of a common multifractal signature in economic biological and physical systems abstract recent developments in microcanonical multifractal formalism have lead to a sensible improvement in the numerical techniques for the determination of the multifractal characteristics of real signals with the aid of these techniques we have found empirical evidence of a common multifractal signature in six very different systems ranging from stock market time series to sea surface temperature records these systems are not only found to be multifractal but their singularity spectra are coincident we propose an explanation of this striking coincidence in terms of a cascade process and analyze its consequences

complexes of exact hermitian cubes and the zagier conjecture in this paper we present a new way of constructing a map from the higher bloch group to the higher rational k theory for an algebraic number field the composition of it with the regulator map is expressed in terms of the polylogarithm function to do this we employ exact hermitian cubes and their bott chern forms

bounding biomass in the fisher equation the fisher kolmogorov petrovskii piskunov equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents if the average growth rate is negative then the model has a survival extinction transition the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations the statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities in the limit of zero decorrelation time the velocity field is shown to act as fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity this allows a one dimensional model to predict the biomass productivity and extinction transitions all results are illustrated with a simple growth and stirring model

intermittency in two dimensional turbulence with drag we consider the enstrophy cascade in forced two dimensional turbulence with a linear drag force in the presence of linear drag the energy wave number spectrum drops with a power law faster than in the case without drag and the vorticity field becomes intermittent as shown by the anomalous scaling of the vorticity structure functions using previous theory we compare numerical simulation results with predictions for the power law exponent of the energy wave number spectrum and the scaling exponents of the vorticity structure functions ensuremath zeta 2q we also study both by numerical experiment and theoretical analysis the multifractal structure of the viscous enstrophy dissipation in terms of its r enyi dimension spectrum d q we derive a relation between d q and ensuremath zeta 2q and discuss its relevance to a version of the refined similarity hypothesis in addition we obtain and compare theoretically and numerically derived results for the dependence on separation r of the probability distribution of ensuremath delta stackrel p vec r ensuremath omega the difference between the vorticity at two points separated by a distance r our numerical simulations are done on a 4096 ifmmode times else texttimes fi 4096 grid

reactions in flows with nonhyperbolic dynamics we study the reaction dynamics of active particles that are advected passively by 2d incompressible open flows whose motion is nonhyperbolic this nonhyperbolicity is associated with the presence of persistent vortices near the wake wherein fluid is trapped we show that the fractal equilibrium distribution of the reactants is described by an effective dimension d mathit eff which is a finite resolution approximation to the fractal dimension furthermore d mathit eff depends on the resolution and on the reaction rate 1 ensuremath tau as ensuremath tau is increased the equilibrium distribution goes through a series of transitions where the effective dimension increases abruptly these transitions are determined by the complex structure of cantori surrounding the kolmogorov arnold moser kam islands

stability and hopf bifurcation of a lorenz like system hopf bifurcation is one of the important dynamical behaviors it could often cause some phenomena such as quasiperiodicity and intermittency consequently chaos will happen due to such dynamical behaviors since chaos appears in the lorenz like system to understand the dynamics of such system hopf bifurcation will be explored in this paper first the stability of equilibrium points is presented then hopf bifurcation of the lorenz like system is investigated by applying the normal form theory the conditions guaranteeing the hopf bifurcation are derived further the direction of hopf bifurcation and the stability of bifurcating periodic solutions are also presented finally numerical simulations are given to verify the theoretical analysis it is found that hopf bifurcation could happen when conditions are satisfied the stable bifurcating periodic orbit is displayed chaos will also happen when parameter further increases

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

dedekind zeta motives for totally real number fields let k be a totally real number field for every odd n 3 we construct an element in the category mt k of mixed tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group by a volume calculation we prove that its period is a rational multiple of pi n k mathbb q zeta k 1 n where zeta k 1 n denotes the special value of the dedekind zeta function of k we deduce that the group mathrm ext 1 mathrm mt k mathbb q 0 mathbb q n is generated by the cohomology of a quadric relative to hyperplanes and that zeta k 1 n is a determinant of volumes of geodesic hyperbolic simplices defined over k

analysis of a new chaotic system abstract this paper reports a new three dimensional continuous quadratic autonomous chaotic system modified from the lorenz system in which each equation contains a single quadratic cross product term which is different from the lorenz rossler chen lu systems basic properties of the new system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics