chaos control in economical model by time delayed feedback method abstract a two dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of pyragas time delayed feedback control the method avoids fancy data processing used in the ott grebogi yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy behrens feichtinger model we show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system s economic properties

control of unstable steady states by extended time delayed feedback time delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states we present an application of extended time delay autosynchronization introduced by socolar et al phys rev e 50 3245 1994 to an unstable focus this system represents a generic model of an unstable steady state which can be found for instance in hopf bifurcation in addition to the original controller design we investigate effects of control loop latency and a bandpass filter on the domain of control furthermore we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase we present an analysis of the domain of control and support our results by numerical calculations

exact mean field solutions of the asymmetric random average process we consider the asymmetric random average process arap with continuous mass variables and parallel discrete time dynamics studied recently by krug garcia and rajesh majumdar both j statist phys 99 2000 the model is defined by an arbitrary state independent fraction density function r with support on the unit interval we examine the exactness of mean field steady state mass distributions in dependence of and identify as a conjecture based on high order calculations the class of density functions yielding product measure solutions additionally the exact form of the associated mass distributions p m is derived using these results we show examplary the exactness of the mean field ansatz for monomial fraction densities r n 1 rn 2 with n 2 for verification we calculate the mass distributions p m explicitly and prove directly that product measure holds furthermore we show that even in cases where the steady state is not given by a product measure very accurate approximants can be found in the class

analytical investigation of modulated time delayed feedback control the influence of time dependent control parameters on time delayed feedback schemes for control of chaos is investigated by analytical means for the logistic map the linear stability of the period two orbit subjected to a modulated time delayed feedback loop is calculated we find enhanced control performance due to phase lags between the periodic orbit and the controller

analytical properties and optimization of time delayed feedback control time delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems if the equations governing the system dynamics are known the success of the method can be predicted by a linear stability analysis of the desired orbit unfortunately the usual procedures for evaluating the floquet exponents of such systems are rather intricate we show that the main stability properties of the system controlled by time delayed feedback can be simply derived from a leading floquet exponent defining the system behavior under proportional feedback control optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay differential equations the method is valid for low dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the rossler system and the duffing oscillator

adaptive localized replay an efficient integration scheme for accurate simulation of coarsening dynamical systems coarsening dynamical systems cds of ordinary differential equations odes have as a defining feature the presence of discontinuities in the governing equations due to the removal of vanishing elements from the system simple commonly used integration strategies for odes neglect these discontinuities resulting in large errors in contrast available methods to resolve discontinuities exhibit total running times that scale like o n 2 as the number of elements n the resulting need to choose between either speed or accuracy frustrates attempts to gather well converged statistical data from large ensembles of elements a common goal in studies of coarsening in this work we introduce a numerical scheme called adaptive localized replay alr which overcomes this dilemma provided the underlying system has sparse neighbor dependence by analyzing the system after each timestep and then selectively re simulating only those small parts of the system that were affected by discontinuities we attain high order accuracy while maintaining running times of o n using a reference implementation of our algorithm applied to a sample problem from the field of faceted surface evolution we obtain convergence and runtime results for the alr method which compares favorably with existing methods

transient and self similar dynamics in thin film coarsening abstract we study coarsening in a simplified model of one dimensional thin films of viscous fluids on hydrophobic substrates lubrication theory shows that such films are unstable and dewet to form droplets that then aggregate over long timescales the masses and positions of the droplets can be described by a coarsening dynamical system cds consisting of odes and deletion rules we develop discrete and continuous mean field models that reproduce the well known n t o t 2 5 long time statistical power law for the number of drops a lifshitz slyozov wagner type lsw continuous model predicts the self similar distribution of drop masses matching with histograms produced by cds simulations and the discrete mean field model we also describe the distribution of drops following homogeneous versus heterogeneous dewetting and use these as initial conditions for the cds simulations that yield characteristic staircasing transients transients can also include recurring spike formation behavior in the mass distribution for idealized initial conditions we show that the transient dynamics can span the full coarsening process bypassing the power law regime entirely

models of stress fluctuations in granular media we investigate in detail two models describing how stresses propagate and fluctuate in granular media the first one is a scalar model where only the vertical component of the stress tensor is considered in the continuum limit this model is equivalent to a diffusion equation where the r ole of time is played by the vertical coordinate plus a randomly varying convection term we calculate the response and correlation function of this model and discuss several properties in particular related to the stress distribution function we then turn to the tensorial model where the basic starting point is a wave equation which in the absence of disorder leads to a ray like propagation of stress in the presence of disorder the rays acquire a diffusive width and the angle of propagation is shifted a striking feature is that the response function becomes negative which suggests that the contact network is mechanically unstable to very weak perturbations the stress correlation function reveals characteristic features related to the ray like propagation which are absent in the scalar description our analytical calculations are confirmed and extended by a numerical analysis of the stochastic wave equation

granular contact force density of states and entropy in a modified edwards ensemble a method has been found to analyze edwards granular contact force probability functional for a special case as a result the granular contact force probability density functions are obtained from first principles for this case the results are in excellent agreement with the experimental and simulation data the derivation assumes edwards flat measure a density of states dos that is uniform within the metastable regions of phase space the enabling assumption supported by physical arguments and empirical evidence is that correlating information is not significantly recursive through loops in the packing maximizing a state counting entropy results in a transport equation that can be solved numerically for the present this has been done using the mean structure approximation projecting the dos across all angular coordinates to more clearly identify its predominant nonuniformities these features are 1 the grain factor psi related to grain stability and strong correlation between the contact forces on the same grain and 2 the structure factor upsilon related to newton s third law and strong correlation between neighboring grains

operator splitting methods for the 2d convective cahn hilliard equation abstract in this work we present operator splitting methods for the two dimensional nonlinear fourth order convective cahn hilliard equation with specified initial condition and periodic boundary conditions the full problem is split into hyperbolic nonlinear diffusion and linear fourth order problems we prove that the semi discrete approximate solution obtained from the operator splitting method converges to the weak solution numerical methods are then constructed to solve each sub equations sequentially the hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method while the one dimensional hyperbolic conservation laws are solved using front tracking algorithm the front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step the nonlinear diffusion problem is solved by a linearized implicit finite volume method which is unconditionally stable the linear fourth order equation is solved using a pseudo spectral method which is based on an exact solution finally some numerical experiments are carried out to test the performance of the proposed numerical methods

delayed feedback control of the lorenz system an analytical treatment at a subcritical hopf bifurcation we develop an analytical approach for the delayed feedback control of the lorenz system close to a subcritical hopf bifurcation the periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique we utilize a modification based on an unstable delayed feedback controller the analytical approach employs the center manifold theory and the near identity transformation we derive the characteristic equation for the floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain the analytical results are supported by numerical analysis of the original system of nonlinear differential difference equations

simple models for granular force networks abstract a remarkable feature of static granular matter is the distribution of force along intricate networks even regular inter particle contact networks produce wildly inhomogeneous force networks where certain chains of particles carry forces far larger than the mean although the full information of each inter particle contact requires analysis of the stress tensor potentially including friction and elasticity there is a rich tradition of approximating these interactions with simple scalar models that focus on the geometry of the system in this paper we briefly review past theoretical approaches to understanding the geometry of force networks we then investigate the structure of experimentally derived granular force networks using a simple algorithm to obtain corresponding graphs we compare our observations with the results of geometric models including random bond percolation which show similar spatial distributions without enforcing vector force balance our findings suggest that some aspects of the mean geometry of granular force networks may be captured by these simple descriptions

influence of stable floquet exponents on time delayed feedback control the performance of time delayed feedback control is studied by linear stability analysis analytical approximations for the resulting eigenvalue spectrum are proposed our investigations demonstrate that eigenbranches that develop from the stable lyapunov exponents of the free system also have a strong influence on the control properties either by hybridization or by a crossing of branches which interchanges the role of the leading eigenvalue our findings are confirmed by numerical analysis of two particular examples the toda and the r ossler models more important is the verification by actual electronic circuit experiments here the observed reduction of control domains can be attributed to these additional eigenvalue branches the investigations lead to a thorough analytical understanding of the stability properties in time delayed feedback systems

optimal periodic orbits of chaotic systems invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor

limitation of generalized delayed feedback control abstract it had been proved that the delayed feedback control dfc of chaos has a limitation as follows if the linear variational equation about the target hyperbolic unstable periodic orbit upo has an odd number of real characteristic multipliers greater than unity the upo can never be stabilized by the dfc with any value of feedback gain we show that this limitation also holds for generalized dfcs based on the feedback of the difference between the current state and delayed states which include the extended time delay auto synchronization etdas and an exponential dfc

persistence of chaos in a time delayed feedback controlled duffing system this paper concerns global phase structures of a time delayed feedback controlled two well duffing system the remains of a global stretch and fold structure along an unstable manifold which develops from an unstable fixed point in function space reveals that the global chaotic dynamics is inherited from the original system by the controlled system the remains of the original chaotic dynamics causes a highly complicated domain of attraction for target orbits and a long chaotic transient before convergence

time delayed feedback control of unstable periodic orbits near a subcritical hopf bifurcation we show that the pyragas delayed feedback control can stabilize an unstable periodic orbit upo that arises from a generic subcritical hopf bifurcation of a stable equilibrium in an n dimensional dynamical system this extends results of fiedler et al b fiedler v flunkert m georgi p hovel e scholl refuting the odd number limitation of time delayed feedback control phys rev lett 98 11 2007 114101 who demonstrated that such a feedback control can stabilize the upo associated with a two dimensional subcritical hopf normal form the pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization as well as knowledge of the period of the targeted upo we apply feedback in the directions tangent to the two dimensional center manifold we parameterize the feedback gain by a modulus and a phase angle and give explicit formulae for choosing these two parameters given the period of the upo in a neighborhood of the bifurcation point we show first heuristically and then rigorously by a center manifold reduction for delay differential equations that the stabilization mechanism involves a highly degenerate hopf bifurcation problem that is induced by the time delayed feedback when the feedback gain modulus reaches a threshold for stabilization both of the genericity assumptions associated with a two dimensional hopf bifurcation are violated the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied and the real part of the cubic coefficient of the normal form vanishes our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two parameter plane in each case the pyragas type feedback successfully stabilizes the branch of small amplitude upos in a neighborhood of the original bifurcation point provided that the feedback phase angle satisfies a certain restriction

delayed feedback control of forced self sustained oscillations we consider a weakly nonlinear van der pol oscillator subjected to a periodic force and delayed feedback control without control the oscillator can be synchronized by the periodic force only in a certain domain of parameters however outside of this domain the system possesses unstable periodic orbits that can be stabilized by delayed feedback perturbation the feedback perturbation vanishes if the stabilization is successful and thus the domain of synchronization can be extended with only small control force we take advantage of the fact that the system is close to a hopf bifurcation and derive a simplified averaged equation which we are able to treat analytically even in the presence of the delayed feedback as a result we obtain simple analytical expressions defining the domain of synchronization of the controlled system as well as an optimal value of the control gain the analytical theory is supported by numerical simulations of the original delay differential equations

dynamics of domain walls governed by the convective cahn hilliard equation the convective cahn hilliard cch equation ut uxx u u 3 xx d 2 u 2 x 0 has been suggested recently for the description of several physical phenomena including spinodal decomposition of driven phase separating systems in an external field instability of steps moving on a crystal surface and faceting of growing thermodynamically unstable surfaces in this paper the dynamics of domain walls kinks governed by the convective cahn hilliard equation is studied by means of asymptotic and numerical methods a special attention is paid to the dynamics of kink pairs and triplets that play crucial role in the coarsening ostwald ripening process for the driving parameter d d 0 2 3 and large distance between the kinks l 1 analytical formulas are found that describe the motion of the kink pairs and triplets the analytical formulas are in excellent agreement with the results of direct numerical simulations of cch equation they explain the logarithmically slow coarsening observed for the kink dynamics governed by cch equation when the distances between the kinks become large unlike the fast coarsening for moderate distances between the kinks at intermediate stages 2005 elsevier b v all rights reserved

on distribution of contact forces in random granular packings abstract in an earlier paper ngan phys rev e 68 2003 011301 proposed a statistical mechanics based theory to predict the mathematical form of the probability force distribution of contact forces in stressed granular packings however the wrong choice of the mean force constraint in that work leads to unsatisfactory prediction in the 3d situation although the 2d prediction is very accurate the present work shows that a different mean strain constraint chosen for the force distribution can yield the correct prediction when compared to computer simulation results in both 2d and 3d furthermore new simulation results are used to verify the interpretation that the shape of the force distribution measured by a parameter in the predicted force distribution is a structure sensitive term

some basic remarks on eigenmode expansions of time delay dynamics abstract we describe in elementary terms how eigenmode expansions can be used to deal with differential difference equations as particular applications we present the full analytical solution of linear stochastic time delay systems and the weakly nonlinear analysis of nonlinear differential difference equations in the limit of large time delay our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the lambert w function and on the explicit formula for the eigenfunctions of the adjoint equation

on a higher order convective cahn hilliard type equation a higher order convective cahn hilliard type equation that describes the faceting of a growing surface is considered with periodic boundary conditions by using a galerkin approach the existence

memory controlled diffusion memory effects require for their incorporation into random walk models an extension of the conventional equations the linear fokker planck equation for the probability density p stackrel p vec r t is generalized by including nonlinear and nonlocal spatial temporal memory effects the realization of the memory kernel is restricted due the conservation of the basic quantity p a general criteria is given for the existence of stationary solutions in case the memory kernel depends on p polynomially transport may be prevented owing to the delay effects a finite amount of particles remains localized and the further transport is terminated for diffusion with nonlinear memory effects we find an exact solution in the long time limit although the mean square displacement exhibits diffusive behavior higher order cumulants offer differences to diffusion and they depend on the memory strength

asymptotic behavior of a stationary silo with absorbing walls we study the nearest neighbors one dimensional uniform q model of force fluctuations in bead packs 1 a stochastic model to simulate the stress of granular media in two dimensional silos the vertical coordinate plays the role of time and the horizontal coordinate the role of space the process is a discrete time markov process with state space 1 n at each layer time the weight supported by each grain is a random variable of mean one its own weight plus the sum of random fractions of the weights supported by the nearest neighboring grains at the previous layer the fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in 0 1 independent of everything the remaining weight is given to the left neighbor in the boundaries a uniform fraction of the weight leans on the wall of the silo this corresponds to absorbing boundary conditions for this model we show that there exists a unique invariant measure the mean weight at site i under the invariant measure is i n 1 i we prove that its variance is frac 1 2 i n 1 i 2 o n3 and the covariances between grains i j are of order o n3 moreover as n the law under the invariant measure of the weights divided by n2 around site integer part of rn r 0 1 converges to a product of gamma distributions with parameters 2 and 2 r 1 r 1 sum of two exponentials of mean r 1 r 2 liu et al 2 proved that for a silo with infinitely many weightless grains any product of gamma distributions with parameters 2 and 2 with 0 are invariant our result shows that as the silo grows the model selects exactly one of these gamma s at each macroscopic place

mean field theory for coarsening faceted surfaces a mean field theory is developed for the scale invariant length distributions observed during the coarsening of generic one dimensional faceted surfaces this theory closely follows the lifshitz slyozov wagner theory of ostwald ripening in two phase systems but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling work on particle coalescence and induces an unexpected coupling between the convolution and the rate of facet loss as a generic framework the theory concisely illustrates how the universal processes of facet removal and neighbor merger are moderated by the system specific mean field velocity describing average rates of length change for a simple example facet dynamics associated with the directional solidification of a binary alloy agreement between the predicted scaling state and that observed after direct numerical simulation of 40 000 000 facets is reasonable given the limiting assumption of noncorrelation between neighbors relaxing this assumption is a clear path forward toward improved quantitative agreement with data in the future

control of unstable steady states by time delayed feedback methods we show that time delayed feedback methods which have successfully been used to control unstable periodic orbits provide a tool to stabilize unstable steady states we present an analytical investigation of the feedback scheme using the lambert function and discuss effects of both a low pass filter included in the control loop and nonzero latency times associated with the generation and injection of the feedback signal

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

anisotropy in granular media classical elasticity and directed force chain network a general approach is presented for understanding the stress response function in anisotropic granular layers in two dimensions the formalism accommodates both classical anisotropic elasticity theory and linear theories of anisotropic directed force chain networks perhaps surprisingly two peak response functions can occur even for classical anisotropic elastic materials such as triangular networks of springs with different stiffnesses in such cases the peak widths grow linearly with the height of the layer contrary to the diffusive spreading found in stress only hyperbolic models in principle directed force chain networks can exhibit the two peak diffusively spreading response function of hyperbolic models but all models in a particular class studied here are found to be in the elliptic regime

coarsening dynamics of a one dimensional driven cahn hilliard system we study the one dimensional cahn hilliard equation with an additional driving term representing say the effect of gravity we find that the driving field e has an asymmetric effect on the solution for a single stationary domain wall or kink the direction of the field determining whether the analytic solutions found by leung j stat phys 61 345 1990 are unique the dynamics of a kink antikink pair bubble is then studied the behavior of a bubble is dependent on the relative sizes of a characteristic length scale mathit e mathrm ensuremath 1 where e is the driving field and the separation l of the interfaces for el ensuremath gg 1 the velocities of the interfaces are negligible while in the opposite limit a traveling wave solution is found with a velocity v ensuremath propto e l for this latter case el ensuremath ll 1 a set of reduced equations describing the evolution of the domain lengths is obtained for a system with a large number of interfaces and implies a characteristic length scale growing as et 1 2 numerical results for the domain size distribution and structure factor confirm this behavior and show that the system exhibits dynamical scaling from very early times textcopyright 1996 the american physical society

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

comparison of time delayed feedback schemes for spatiotemporal control of chaos in a reaction diffusion system with global coupling time delayed feedback control for stabilizing time periodic spatial patterns is investigated in a generic reaction diffusion system with global coupling we focus on the case of low dimensional chaos where unstable patterns admit only a single unstable mode spatial degrees of freedom are taken into account to define different control schemes the efficiency of these schemes is discussed where control forces are motivated by physical requirements as well as by the possibility of obtaining analytically exact results we find that control schemes that contain the full feedback of the inhibitor variable may finally destroy the control performance thus schemes that omit the inhibitor might be more efficient our numerical findings are explained in terms of floquet spectra and compared with analytical solutions of particular coupling schemes

the control of higher dimensional chaos comparative results for the chaotic satellite attitude control problem abstract using simulations this paper reports the success and failure of three experiments each using a different technique to control the same higher dimensional chaotic system the three techniques are a simple delayed feedback control method originally suggested by pyragas the otani jones technique and a higher dimensional variation of the ogy method the three methods are applied to a six dimensional system which describes the attitude dynamics of a satellite rigid body subjected to deterministic external perturbations which induce chaotic motion when no control is effected the attitude of the satellite is controlled by three orthogonal pairs of thrusters aligned with the principal axes and the system can be described by a modified set of euler equations the three control methods are compared in terms of the prior calculations required the real time computational requirements and the effectiveness of the method in stabilizing the system the results show that the method of pyragas compares very favourably with the other two techniques requiring no prior calculation and having a very low real time computational overhead pyragas method also provides the most satisfactory control solution the problems in presenting a theoretical justification of pyragas method are briefly discussed and the connection between the method and a recent technique for the location of unstable fixed points in chaotic systems is highlighted

faceting and coarsening dynamics in the complex swift hohenberg equation the complex swift hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers optical parametric oscillators and photorefractive oscillators we show that with real coefficients this equation exhibits two classes of localized states localized in amplitude only or localized in both amplitude and phase the latter are associated with phase winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them the localized states take the form of defects connecting phase winding states with equal and opposite phase lag and can be stable over a wide range of parameters the formation of these defects leads to faceting of states with initially spatially uniform phase depending on parameters these facets may either coarsen indefinitely as described by a cahn hilliard equation or the coarsening ceases leading to a frozen faceted structure

coarsening dynamics of the convective cahn hilliard equation abstract we characterize the coarsening dynamics associated with a convective cahn hilliard equation cch in one space dimension first we derive a sharp interface theory through a matched asymptotic analysis two types of phase boundaries kink and anti kink arise due to the presence of convection and their motions are governed to leading order by a nearest neighbors interaction coarsening dynamical system cds theoretical predictions on cds include the characteristic length l m for coarsening exhibits the temporal power law scaling t1 2 provided l m is appropriately small with respect to the peclet length scale l p binary coalescence of phase boundaries is impossible ternary coalescence only occurs through the kink ternary interaction two kinks meet an anti kink resulting in a kink direct numerical simulations performed on both cds and cch confirm each of these predictions a linear stability analysis of cds identifies a pinching mechanism as the dominant instability which in turn leads to kink ternaries we propose a self similar period doubling pinch ansatz as a model for the coarsening process from which an analytical coarsening law for the characteristic length scale l m emerges it predicts both the scaling constant c of the t1 2 regime i e l m ct 1 2 as well as the crossover to logarithmically slow coarsening as l m crosses l p our analytical coarsening law stands in good qualitative agreement with large scale numerical simulations that have been performed on cch

delayed feedback control of dynamical systems at a subcritical hopf bifurcation we consider the delayed feedback control of a torsion free unstable periodic orbit originated in a dynamical system at a subcritical hopf bifurcation close to the bifurcation point the problem is treated analytically using the method of averaging we discuss the necessity of employing an unstable degree of freedom in the feedback loop as well as a nonlinear coupling between the controlled system and controller to demonstrate our analytical approach the specific example of a nonlinear electronic circuit is taken as a model of a subcritical hopf bifurcation