fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

entrainment control in a noisy neural system the open plus closed loop opcl entrainment control put forth by jackson and grosu physica d 85 1995 is applied to an effective neuron system as a way to extract stable limit cycles from a chaotic attractor analogous to the retrieval of memories from a memory searching state additive gaussian white noise representing the natural noise inherent in any real dynamical system is added to the entrainment control mechanism moderate levels of additive noise have little effect on successful entrainment as reflected in phase space plots and lyapunov exponents all three lyapunov exponents are negative which suggests parallels between opcl control and chaotic synchronization al 2

non linear dynamics and chaos control of a physical pendulum with vibrating and rotating support the dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper both analytical and computational results are employed to obtain the characteristics of the system by using lyapunov s direct method the conditions of stability of the relative equilibrium position can be determined melnikov s method is applied to identify the existence of chaotic motion the incremental harmonic balance method is used to find the stable and unstable periodic solutions for the strong non linear system by applying various numerical results such as phase portrait pioncare map time history and power spectrum analysis a variety of the periodic solutions and the phenomena of the chaotic motion can be presented the effects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams further chaotic motion can be verified by using lyapunov exponent and lyapunov dimension the global analysis of basin boundary and fractal structure are observed by the modified interpolated cell mapping method besides non feedback control delayed feedback control adaptive control and variable structure control are used to control the chaos effectively

subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

period doubling bifurcations chaos phase locking and devil s staircase in a bonhoeffer van der pol oscillator abstract the effect of a periodic input current a 1 cos t in the bonhoeffer van der pol oscillator along with a bias a 0 is investigated numerically as the parameter a 1 is varied in the absence of bias by holding the other parameters at constant values typical period doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the feigenbaum scenario when the bias is switched on at the transition to chaos frequency locking is observed in the system the frequency locked intervals exhibit complete devil s staircase similar to the one observed in belousov zhabotinsky reaction

flexible scaling model for use in random field simulation of hydraulic conductivity a new fractal scaling model is proposed for use in aquifer heterogeneity modeling and simulation the new model which is obtained by subordination of fractional brownian motion provides a unifying framework for scaling models of heterogeneity in that it includes previous scaling models as special cases and can also be tuned continuously between these models choosing the subordinator to be a lognormal distribution results in a non gaussian scaling model with a moderate degree of variability that matches data better than previous fractal scaling models thus providing more accurate heterogeneity representations for use in stochastic flow and transport predictions two new stochastic simulation algorithms are constructed from the new models one based on sequential simulation algorithms and the other based on probability field simulation both algorithms can be made conditional on available measurements

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

bifurcation and chaotic motion of a soliton in a long josephson junction oscillator the fluxon dynamics of a long josephson junction ljj is investigated numerically the effect of the surface resistance of the superconductor on the bifurcation phenomenon is examined in detail we show the occurrence of a period doubling route to chaos and the reappearance of low order periodic motions in the ljj by varying the magnitude of the external force the chaotic attractors of the ljj are characterized using maximal lyapunov exponents and a fractal correlation dimension we show the existence of noise induced oscillations in a force free ljj system finally the influence of noise on the periodic and chaotic dynamics is also presented

adaptive control in nonlinear dynamics abstract we extend an adaptive control algorithm recently suggested by huberman and lumer to multi parameter and higher dimensional nonlinear systems this control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour we find that in all cases the recovery time is linearly proportional to the inverse of control stiffness for small stiffness in higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery the control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated we further suggest methods by which periodic motion such as limit cycles can be adaptively controlled and demonstrate the robustness of the procedure in the presence of additive background noise

limit theory for moving averages of random variables with regularly varying tail probabilities let z k infty k infty be iid where the z k s have regularly varying tail probabilities under mild conditions on a real sequence c j j geq 0 the stationary process x n sum infty j 0 c jz n j n geq 1 exists a point process based on x n converges weakly and from this a host of weak limit results for functionals of x n ensue we study sums extremes excedences and first passages as well as behavior of sample covariance functions

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit lagrangian scheme this study develops an explicit two step lagrangian scheme based on the renewal reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media the resulting 3 d anomalous transport simulator provides a flexible platform for modeling transport the first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones the mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step and the subsequent random immobile time follows a hyper exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power law memory function the second step describes well documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow applicability and implementation of the lagrangian solver are further checked against transport observed in various media results show that although the time nonlocal model parameters are predictable for transport with retention in alluvial settings the standard time nonlocal model cannot capture early arrivals retention and early arrivals observed in porous and fractured media can be efficiently modeled by our lagrangian solver allowing anomalous transport to be incorporated into 2 d 3 d models with irregular flow fields extensions of the particle tracking approach are also discussed for transport with parameters conditioned on local aquifer properties as required by transient flow and nonstationary media

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

a variant of wangʼs theorem in this paper we give a new formula of j j for any parameter ideal j in a gorenstein local ring r of positive characteristic in terms of test ideals j j j jd 1 where jd 1 denotes the jd 1 test ideal of r as an application we give a variant of wang s theorem namely we prove that if j is a parameter ideal in a cohen macaulay local ring r m of dimension d 2 with j ms then j m d 1 s 1 resp j m d 1 s 1 1 is integral over j resp if r is not regular moreover we prove that after reduction to characteristic p 0 a similar assertion holds true for cohen macaulay q gorenstein normal local domain essentially of finite type over a field of characteristic zero under some extra assumption

effects of phase space discretization on the long time behavior of dynamical systems abstract we investigate the effects that roundoff errors have on discrete time dynamical systems computer generated orbits terminate in loops whose lengths scale with the machine precision in a universal way we present arguments to explain this fact

bifurcations and chaos of the bonhoeffer van der pol model periodic and chaotic behaviour of the bonhoeffer van der pol model of a nerve membrane driven by a periodic stimulating current a1 cos omega t is investigated results show that there exist ordinary and reversed period doubling cascades and a mode locking state at low driving amplitudes a1 there are period doubling and chaotic states but no impulse solutions when a1 is larger than a0 0 749 there are chaotic reversed period doubling and mode locking states and there also exist impulse trains a mode locking state with period 4 over a very large range of amplitudes is also found at a1 1 7059 the system goes back to a one period state

operator levy motion and multiscaling anomalous diffusion the long term limit motions of individual heavy tailed power law particle jumps that characterize anomalous diffusion may have different scaling rates in different directions operator stable motions y t tg 0 are limits of d dimensional random jumps that are scale invariant according to c h y t y mathrm ct where h is a d ifmmode times else texttimes fi d matrix the eigenvalues of the matrix have real parts 1 ensuremath alpha j with each positive ensuremath alpha j l 2 in each of the j principle directions the random motion has a different fickian or super fickian diffusion dispersion rate proportional to t 1 ensuremath alpha j these motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions subsets of the generalized fractional operator include i a fractional laplacian with a single order ensuremath alpha and a general directional mixing measure m ensuremath theta and ii a fractional laplacian with uniform mixing measure the riesz potential the motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different super fickian rates in the directions parallel and perpendicular to mean flow

effects of fractional mass transfer and chemical reaction on mhd flow in a heterogeneous porous medium abstract this paper presents a study on space fractional anomalous convective diffusion and chemical reaction in the magneto hydrodynamic fluid over an unsteady stretching sheet the fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium a novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level numerical solutions are established by an implicit finite difference scheme the stability and convergence of the method are analyzed results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall there is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter moreover the numerical solutions are compared with exact solutions and good agreement has been observed

numerical solution of space fractional diffusion equation by the method of lines and splines abstract this paper is devoted to the application of the method of lines to solve one dimensional diffusion equation where the classical integer second derivative is replaced by a fractional derivative of the caputo type of order less than 2 as the space derivative a system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the caputo derivative the result is a numerical approach of order o x 2 t m where x and t denote spatial and temporal step sizes and 1 m 5 is an integer which is set by an ode integrator that we used the convergence and numerical stability of the method are considered and numerical tests to investigate the efficiency and feasibility of the scheme are provided

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

quasi socle ideals in gorenstein numerical semigroup rings quasi socle ideals that is the ideals i of the form i q m q in goren stein numerical semigroup rings over fields are explored where q is a parameter ideal and m is the maximal ideal in the base local ring and q 1 is an integer the problems of when i is integral over q and of when the associated graded ring g i l n 0 i n i n 1 of i is cohen macaulay are studied the problems are rather wild examples are given

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

scaling the fractional advective dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries microbial motility is often characterized by run and tumble behavior which consists of bacteria making sequences of runs followed by tumbles random changes in direction as a superset of brownian motion levy motion seems to describe such a motility pattern the eulerian fokker planck equation describing these motions is similar to the classical advection diffusion equation except that the order of highest derivative is fractional email protected 0 2 the lagrangian equation driven by a levy measure with drift is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries the eulerian equation is used to non dimensionalize parameters the amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values salient features of first passage time are studied with respect to scaled parameters

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

moments for tempered fractional advection diffusion equations this paper develops moment formulas for exponentially tempered fractional advection diffusion equations tfades that transition from anomalous to asymptotic diffusion limits over time exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach a fully lagrangian solver cross verified by an implicit eulerian approach is also developed to calculate numerically the complete evolution of moments for the tfades with complex initial and boundary conditions moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion

application of stochastic control techniques to chaotic nonlinear systems a control algorithm based on stochastic control techniques is devised for chaotic nonlinear systems the algorithm uses a state estimator based on the kalman filter and yields performance improvements in at least some regions of state space with respect to that obtainable by use of a controller utilizing only the conditional mean of the system state vector the method is applied to two typical chaotic nonlinear systems the henon heiles system and the lorenz system and their behavior with control is explored numerically

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

links of symbolic powers of prime ideals in this paper we prove the following let r frak m be a cohen macaulay local ring of dimension d 2 suppose that either r is not regular or if r is regular assume that d 3 let t 2 be a positive integer if alpha 1 dots alpha d is a regular sequence contained in frak m t then alpha 1 dots alpha d frak m t subseteq frak m t this result gives an affirmative answer to a conjecture raised by polini and ulrich

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

noise induced escape from attractors the author deals with several aspects of kautz 1987 1988 minimum principle for the noise induced escape from attractors he presents a slightly simpler derivation points out a connection with lyapunov exponents and invariant manifolds and generalises to chaotic maps

noise induced jumps in two coupled duffing oscillators abstract we consider two coupled duffing oscillators with the coexistence of four attractors switching between the coexisting attractors by noises is investigated three types of noises such as gaussian amplitude dependent and dichotomous are used we observed noise induced extrinsic intermittency intermittent dynamics is characterized using power spectrum average and relative residence times on each precoupled attractors and probability distribution of time interval between successive switchings variation of probability distribution of state variables in poincare map is studied as a function of noise strength the effect of various noises is compared the occurrence of crisis induced and type i intermittencies are also presented

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

on advective transport in fractal permeability and velocity fields we consider advective transport in a steady state random velocity field with homogeneous increments such a field is self affine with a power law dyadic semivariogram s proportional to d2 where d is distance and is a hurst coefficient it is characterized by a fractal dimension d e 1 where e is topological dimension as the mean and variance of such a field are undefined we condition them on measurement at some point x0 we then introduce a tracer at another point y0 and invoke elementary theoretical considerations to demonstrate that its conditional mean dispersion is local at all times its conditional mean concentration and variance are given explicitly by well established expressions which however have not been previously recognized as being valid in fractal fields once the conditional mean travel distance s of the tracer becomes large compared to the distance between y0 and x0 the corresponding dispersion and dispersivity tensors grow in proportion to s1 2 where 0 1 this supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods of interpretation from tracer behavior observed in a variety of geologic media under varied flow and transport regimes filtering out modes from the fractal velocity field with correlation scales larger than some s0 allows an asymptotic transport regime to develop when s s0 the corresponding asymptotic dispersivities grow in proportion to s02 when 0 this linear to sublinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained from calibrated numerical models in a variety of media a self affine natural log permeability field gives rise to a self affine velocity field while s is sufficiently small to insure that the variance of the log permeabilities which grows as a power of s remains nominally less than one an analysis of published apparent longitudinal dispersivity data in light of the above theoretical results supports my earlier conclusion that when one juxtaposes data from a large number of generally dissimilar geologic media from a variety of locales one observes a tendency toward self affine behavior with a hurst coefficient 0 25 at any given locale such media may or may not exhibit fractal behavior if they do may or may not be close to 0 25

algorithms for controlling chaotic motion application for the bvp oscillator the bvp oscillator is a biologically and physically important dynamical system exhibiting chaotic motion phaselocking phenomenon and so on in this paper we investigate the controlling of chaos in this system both when periodic and constant currents are present we use the various control algorithms and investigate their effectiveness in particular we study the controlling by the following mechanisms i adaptive control algorithm ii weak periodic perturbations iii ott grebogi yorke method iv weak feedback control and v addition of noise also we show the suppression of chaos by numerical models

synchronization and anti synchronization of chaos in an extended bonhoffer van der pol oscillator using active control abstract in this paper we investigate the complete synchronization and anti synchronization as of double scroll chaotic attractor exhibited by an extended bonhoffer van der pol bvp oscillator using active control technique in both synchronization schemes the oscillators show good transient performance while the as state is further shown to correspond with complete inverse synchronization numerical simulations are also presented to verify the theoretical results

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size

links of prime ideals and their rees algebras abstract in a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in gorenstein rings are equimultiple ideals with reduction number 1 this led to the construction of large families of cohen macaulay rees algebras the first goal of this paper is to extend this result to arbitrary cohen macaulay rings the means of the proof are changed since one cannot depend so heavily on linkage theory we then study the structure of the rees algebra of these links more specifically we describe their canonical module in sufficient detail to be able to characterize self linked prime ideals in the last section multiplicity estimates for classes of such ideals are established