fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

levy flights in quenched random force fields l evy flights characterized by the microscopic step index f are for fl2 the case of rare events considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f ensuremath 2 in the short range case and the critical fall off exponent 2f ensuremath 2 in the long range case by means of a dynamic renormalization group analysis based on the momentum shell integration method we determine flows fixed point and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient unlike the case of ordinary brownian motion in a quenched force field characterized by a single critical dimension or fall off exponent d 2 two critical dimensions appear in the l evy case a critical dimension or fall off exponent d f below which the diffusion coefficient exhibits anomalous scaling behavior i e algebraic spatial behavior and long time tails and a critical dimension or fall off exponent d 2f ensuremath 2 below which the force correlations characterized by a nontrivial fixed point become relevant as a general result we find in all cases that the dynamic exponent z characterizing the mean square displacement locks onto the l evy index f independent of dimension and independent of the presence of weak quenched disorder

anomalous rotational relaxation a fractional fokker planck equation approach in this study we have analytically obtained the relaxation function in terms of rotational correlation functions based on brownian motion for complex disordered systems in a stochastic framework we found out that the rotational relaxation function has a fractional form for complex disordered systems which indicates that relaxation has nonexponential character and obeys the kohlrausch william watts law following the mittag leffler decay

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

arc length based maximal lyapunov functions and domains of attraction estimation for polynomial nonlinear systems in the phase space of a dynamical system containing an asymptotically stable equilibrium point the arc length function alf is defined as sum of length differential elements of phase trajectories starting from state points and ending at the equilibrium point it is shown that receding from the origin and verging on the domain of attraction doa boundary causes alf to be increased drastically according to the latter issue and regarding to other properties it is shown that alf is a maximal lyapunov function to this end a numerical method to approximate the alf as a polynomial function is proposed to ensure that the approximated function has positive value and negative derivative inside the desired region the homotopy continuation method is used thus the approximated alf presents an ensured lyapunov behavior to estimate doa the efficacy of the proposed method is demonstrated by several simulation examples

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

design of state feedback control for polynomial systems with quadratic performance criterion and control input constraints abstract this paper presents a novel convex optimization approach to design state feedback control for polynomial systems design criteria are comprised of a quadratic cost function and bounded magnitudes of control inputs specifically we formulate a control synthesis of closed loop systems operated in a given bounded domain characterized by a semi algebraic set we consider an extended class of rational lyapunov functions and derive an upper bound of the cost function together with a state feedback control law by exploiting bounds on the control input magnitudes the controller design condition can be cast as a parameter dependent linear matrix inequality pdlmi which is convex optimization and can be efficiently solved by sum of squares sos technique in addition we derive a sufficient condition to compute a lower bound of the cost function when choosing polynomial structure of the solution candidate the lower bound can also be written as pdlmi numerical examples are provided to illustrate the effectiveness of the proposed design

stability analysis of implicit polynomial systems this technical note develops a robust stability analysis method for a class of implicit polynomial systems subject to uncertain constant parameters a polynomial with respect to the state and uncertain parameters lyapunov function is computed to assess the robust local stability of the system and to provide a robust estimate of the domain of attraction the results are given in terms of linear matrix inequalities that are affinely dependent on the system state and uncertain parameters numerical examples illustrate the effectiveness of the developed approach

a difference method for solving parabolic equations of order 2n in this paper a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined the resulting system is solved by efficient implicit iterations in some numerical examples maple modules are designed for the purpose of testing and using the method

stochastic processes with stable distributions in random environments the asymptotic behavior in random environments of random flights with stable distribution laws is analyzed by the field theoretic renormalization group random force fields with isotropic divergenceless curl free and unconstrained pair correlation functions with both finite and infinite correlation length are considered stability of the effective distribution laws in the scaling limit is determined and the scaling dimension of time is calculated in the ensuremath varepsilon mathit d mathit c d expansion where mathit d mathit c is the critical dimension of the model textcopyright 1996 the american physical society

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

high order p stable multistep methods abstract recently numerical integration of the special ivp y f t y whose solutions are of periodic type have been considered of great interest p stability is the appropriate requirement for numerical methods integrating periodic problems as established by lambert and watson 1976 a p stable linear multistep method has order no greater than two in this paper two step multi stage linear methods are proposed for numerical integration of periodic problems and p stable methods of this type obtained in a simple form up to order eight are shown finally those methods are successfully tested

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

explicit two step methods with minimal phase lag for the numerical integration of special second order initial value problems and their application to the one dimensional schro dinger equation abstract two step sixth order methods with phase lag of order eight ten and twelve are developed for the numerical integration of the special second order initial value sproblem an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by chawla rao and neta this journal 1986 1987

an explicit four step phase fitted method for the numerical integration of second order initial value problems abstract an explicit four step method with phase lag of infinite order is developed for the numerical integration of second order initial value problems extensive numerical testing indicates that this new method can be generally more efficient than other four step methods

computing output feedback controllers to enlarge the domain of attraction in polynomial systems the problem of computing controllers to enlarge the domain of attraction da of equilibrium points of polynomial systems is considered in order to deal with such a problem a technique for computing static nonlinear output feedback controllers which maximize the largest estimate of the da leda induced by a given polynomial lyapunov function is proposed the main contribution of the note is to show that a lower bound of the maximum achievable leda and a corresponding controller can be obtained through linear matrix inequality optimizations moreover a necessary condition for tightness of this lower bound is presented which is also a sufficient condition to establish the tightness of the lower bound of the leda for a given controller

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

inertial effects in anomalous dielectric relaxation of symmetrical top molecules the linear dielectric response of an assembly of noninteracting symmetrical top molecules each of which is free to rotate in space is evaluated in the context of fractional dynamics the infinite hierarchy of differentialrecurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional klein kramers equation on solving this hierarchy in terms of matrix continued fractions as in the normal rotational diffusion the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters for the limiting case of spherical top molecules the solution is obtained in terms of an ordinary continued fraction it is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies vt 1 where t is the debye relaxation time and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies in the far infrared region so that gordon s sum rule for integral dipolar absorption is satisfied

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

rational lyapunov functions for estimating and controlling the robust domain of attraction this paper addresses the estimation and control of the robust domain of attraction rda of equilibrium points through rational lyapunov functions lfs and sum of squares sos techniques specifically continuous time uncertain polynomial systems are considered where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set the estimation problem consists of computing the largest estimate of the rda lerda provided by a given rational lf the control problem consists of computing a polynomial static output controller of given degree for maximizing such a lerda in particular the paper shows that the computation of the best lower bound of the lerda for chosen degrees of the sos polynomials which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities bmis can be reformulated as a quasi convex optimization problem under some conditions moreover the paper provides a necessary and sufficient condition for establishing tightness of this lower bound lastly the paper discusses the search for optimal rational lfs using the proposed strategy

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

a new class of long tailed pausing time densities for the ctrw we present some asymptotic results for the family of pausing time densities having the asymptotic t property t t ln1 t t 1 in particular we show that for this class of pausing time densities the mean squared displacement r2 t is asymptotically proportional to ln t t and the asymptotic distribution of the displacement has a negative exponential form

estimating the domain of attraction for uncertain polynomial systems estimating the domain of attraction da of equilibrium points is a problem of fundamental importance in systems engineering several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the da leda for a given lyapunov function lf however the problem of estimating the robust da rda that is the da guaranteed for all possible uncertainties in an uncertain system it is still an unsolved problem in this paper lmi methods are proposed for estimating the rda in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope specifically the issue of computing the robust leda rleda that is the intersection of all ledas is considered for common and parameter dependent lfs providing constant and parameter dependent lower bounds the computation of approximations with simple shape of the rleda in the case of parameter dependent lfs is also discussed

a numerical approach to nonlinear two point boundary value problems for odes in this paper we propose a numerical approach to solve some problems connected with the implementation of the newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two point bvps for odes with mixed linear boundary conditions by using the finite difference method

an optimal algorithm for certain boundary value problem abstract the o h 4 finite difference scheme for the second derivative u x leads to a coherent pentadiagonal matrix which is factorized into two tridiagonal matrices this factorization is used to derive an optimal algorithm for solving a linear system of equations with the pentadiagonal matrix as an application a nonlinear system of ordinary differential equations is approximated by an o h 4 convergent finite difference scheme this scheme is solved by the implicit iterative method applying the algorithm at each iteration a mathematica module designed for the purpose of testing and using the method is attached

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

stable equilibrium based on levy statistics stochastic collision models approach we investigate equilibrium properties of two very different stochastic collision models i the rayleigh particle and ii the driven maxwell gas for both models the equilibrium velocity distribution is a levy distribution the maxwell distribution being a special case we show how these models are related to fractional kinetic equations our work demonstrates that a stable power law equilibrium which is independent of details of the underlying models is a natural generalization of maxwell s velocity distribution

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

inner approximating reachable sets for polynomial systems with time varying uncertainties in this paper we propose a convex programming based method to address a long standing problem of inner approximating backward reachable sets of state constrained polynomial systems subject to time varying uncertainties the backward reachable set is a set of states from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties it is equal to the zero sublevel set of the unique lipschitz viscosity solution to a hamilton jacobi partial differential equation hje we show that inner approximations of the backward reachable set can be formed by zero sublevel sets of its viscosity supersolutions consequently we reduce the inner approximation problem to a problem of synthesizing polynomial viscosity supersolutions to this hje such a polynomial solution in our method is synthesized by solving a single semidefinite program we also prove that polynomial solutions to the formulated semidefinite program exist and can produce a convergent sequence of inner approximations to the interior of the backward reachable set in measure under appropriate assumptions this is the main contribution of this paper several illustrative examples demonstrate the merits of our approach

a higher order finite difference method for solving a system of integro differential equations abstract a finite difference method and gauss method of numerical integration are used to approximate the integro differential system of eq 1 with the global error z h o h4 the resulting system of algebraic equations is solved by efficient implicit iterations a mathematica module is designed for the purpose of testing and using the method an example of an integral differential equation with the green s function as the kernel is solved

safe nonlinear control design for input constrained polynomial systems using sum of squares programming we study the feedback stabilisation problem for input affine polynomial systems subject to polytopic input constraints first we characterise a subset of the state space which we refer to as the

replacing the finite difference methods for nonlinear two point boundary value problems by successive application of the linear shooting method abstract this paper studies numerical solution of nonlinear two point boundary value problems for second order ordinary differential equations first it establishes a connection between the finite difference method and the quasi linearization method we prove that using finite differences to discretize the sequence of linear differential equations arising from quasi linearization newton method on operator level leads to the usual iteration formula of the newton finite difference method from the provided derivation it can easily be inferred that such a relation holds also for the picard and the constant slope methods based on this result we propose a way of replacing the newton picard and constant slope finite difference methods by respective successive application of the linear shooting method this approach has a number of advantages it removes the necessity of solving systems of algebraic equations hence working with matrices altogether compared to the usual finite difference method with general solver it reduces the number of computational operations from o n 3 where n is the number of mesh points to only o n

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

a higher accuracy finite difference method for an elliptic equation of order four the elliptic equation 1 with boundary value conditions either 2 or 3 is solved by an o h4 convergent finite difference method combined with an implicit iterative method the method is tested by examples of boundary value problems with use of the enclosed mathematica module solve bhpeq the module solve bhpeq gives options either it implements the o h4 convergent nine mesh points scheme or the o h2 convergent standard five points scheme when the finite difference approximation of eq 1 is considered

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

continuous time random walks and traveling fronts we present a geometric approach to the problem of propagating fronts into an unstable state valid for an arbitrary continuous time random walk with a fisher kolmogorov petrovski piskunov growth reaction rate we derive an integral hamilton jacobi type equation for the action functional determining the position of reaction front and its speed our method does not rely on the explicit derivation of a differential equation for the density of particles in particular we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non markovian random processes

aging properties of an anomalously diffusing particule we report new results about the two time dynamics of an anomalously diffusing classical particle as described by the generalized langevin equation with a frequency dependent noise and the associated friction the noise is defined by its spectral density proportional to 1 at low frequencies with 0 1 subdiffusion or 1 2 superdiffusion using laplace analysis we derive analytic expressions in terms of mittag leffler functions for the correlation functions of the velocity and of the displacement while the velocity thermalizes at large times slowly in contrast to the standard brownian motion case 1 the displacement never attains equilibrium it ages we thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion we provide a closed form analytic expression for the fluctuation dissipation ratio characterizing aging

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

global asymptotic stabilisation of rational dynamical systems based on solving bmi abstractin this paper the global asymptotic stabiliser design of rational systems is studied in detail to develop the idea the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law this in turn is followed by the satisfaction of the linear growth condition i e lipschitz at zero based on a linear matrix inequality solution the system in the new coordinate is globally asymptotically stabilised and then leading to the global asymptotic stabilisation of the primary system the polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem to confirm the capability of this method three examples are highlighted

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

o h 4 locally overconvergent semidiscrete scheme for the equation u t u xx f t x u abstract in the equation ut uxx f t x u the second derivative uxx is approximated by the finite difference operator lh which has o h2 local truncation error at the points h and 1 h next to the ends of the interval 0 1 and o h4 at the interior mesh points 2h 3h 1 2h it is proved that the obtained semidiscrete scheme is o h4 globally convergent the semidiscrete scheme is solved by the pure implicit finite difference scheme combined with the method of iteration and the gauss elimination method for tridiagonal matrices the nonstationary liouville s equation ut uxx e u has been solved by the proposed algorithm

levy scaling the diffusion entropy analysis applied to dna sequences we address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis dea n scafetta p hamilton and p grigolini fractals 9 193 2001 this method is based on the evaluation of the shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations rather than on the measurement of the variance of this diffusion process as done with the traditional methods we compare the dea to the traditional methods of scaling detection and prove that the dea is the only method that always yields the correct scaling value if the scaling condition applies furthermore dea detects the real scaling of a time series without requiring any form of detrending we show that the joint use of dea and variance method allows to assess whether a time series is characterized by levy or gauss statistics we apply the dea to the study of dna sequences and prove that their large time scales are characterized by levy statistics regardless of whether they are coding or noncoding sequences we show that the dea is a reliable technique and at the same time we use it to confirm the validity of the dynamic approach to the dna sequences proposed in earlier work

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

anomalous diffusion in the nonasymptotic regime we analyze some properties of the one dimensional levy flights assuming that the one step transition rates depend on the flight length x as p alpha x equivalent to x alpha 2 for flights on a finite 2m 1 site lattice we can define an effective size dependent diffusion coefficient d alpha m equivalent to m 1 alpha 1 1 alpha if alpha are well defined provided that alpha r 3 these moments exhibit a power law singularity if alpha 1 and r 2 3 the short and intermediate time properties of the generalized mean square displacement are then studied numerically this work suggests the conditions under which the asymptotic analytical formulas obtained in the literature by the use of the generalized central limit theorem could be applied to finite time experiments these formulas should work much better if alpha is close to zero than in the alpha 1 neighborhood

anomalous diffusion stable processes and generalized functions the evolution equations in real space and time corresponding to a class of anomalous diffusion processes arc examined as special cases evolution equations corresponding to stable processes are derived using the theory of generalized functions recovering some known results differently interpreted and an evolution law for stable processes of order unity

growing surfaces with anomalous diffusion results for the fractal kardar parisi zhang equation in this paper i study a model for a growing surface in the presence of anomalous diffusion also known as the fractal kardar parisi zhang equation fkpz this equation includes a fractional laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a levy flight it is shown that for a specific choice of parameters of the fkpz equation the equation can be solved exactly in one dimension so that all the critical exponents which describe the surface that grows under fkpz can be derived for that case afterwards the self consistent expansion sce is used to predict the critical exponents for the fkpz model for any choice of the parameters and any spatial dimension it is then verified that the results obtained using sce recover the exact result in one dimension at the end a simple picture for the behavior of the fractal kpz equation is suggested and the upper critical dimension of this model is discussed

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

region of attraction estimation using invariant sets and rational lyapunov functions this work addresses the problem of estimating the region of attraction ra of equilibrium points of nonlinear dynamical systems the estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a lyapunov function moreover we present conditions for the existence of lyapunov functions linked to the positively invariant set formulation we propose connections to fundamental results on estimates of the ra are presented and support the search of lyapunov functions of a rational nature we then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the ra at each iteration

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function