error analysis and comparative study of numerical methods for the parabolic equation applied to tunnel propagation modeling parabolic equation pe methods have been widely applied to the modeling of wireless propagation in tunnel environments however the relevant literature does not include concrete guidelines for the choice of the parameters of these methods and the tradeoffs involved this paper provides a comprehensive analysis of the two sources of error that arise when pe methods are employed for the modeling of radio wave propagation scenarios the well known numerical dispersion error stemming from the finite difference solvers for pe and the approximation error stemming from the use of pe for the solution of wave propagation problems that are subject to maxwell s equations the analysis is performed for four methods three of which have been already used in pe based propagation studies namely the crank nicolson cn scheme the alternative direction implicit adi method and its locally one dimensional lod adi version the fourth method is the mitchell fairweather mf adi scheme that has been recently shown to be a promising alternative technique for tunnel propagation modeling the proposed method leads to robust criteria for the choice of spatial discretization in realistic propagation scenarios as shown via numerical examples

the structured controllability radii of higher order systems abstract in this paper we shall deal with the problem of calculation of the controllability radius of higher order dynamical systems of the form ex k t a k 1 x k 1 t a 1 x 1 t a 0 x t bu t where the leading coefficient matrix e need not assumed to be non singular by using multi valued linear operators we are able to derive computable formulas of the controllability radius under the assumption that the system s coefficient matrices are subjected to structured affine perturbations some examples are provided to illustrate the obtained results

the structured controllability radius of linear delay systems in this article we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x t a 0 x t a 1 x t h 1 a k x t h k bu t by using multi valued linear operators we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

recursive two way parabolic equation approach for modeling terrain effects in tropospheric propagation the fourier split step method is a one way marching type algorithm to efficiently solve the parabolic equation for modeling electromagnetic propagation in troposphere the main drawback of this method is that it characterizes only forward propagating waves and neglects backward propagating waves which become important especially in the presence of irregular surfaces although ground reflecting boundaries are inherently incorporated into the split step algorithm irregular surfaces such as sharp edges introduce a formidable challenge in this paper a recursive two way split step algorithm is presented to model both forward and backward propagation in the presence of multiple knife edges the algorithm starts marching in the forward direction until the wave reaches a knife edge the wave arriving at the knife edge is partially reflected by imposing the boundary conditions at the edge and is propagated in the backward direction by reversing the paraxial direction in the parabolic equation in other words the wave is split into two components and the components travel in their corresponding directions the reflected wave is added to the forward wave in each range step to obtain the total wave the wave splitting is performed each time a wave is incident on one of the knife edges this procedure is repeated until convergence is achieved inside the entire domain

analysis of the effects of irregular terrain on radio wave propagation based on a three dimensional parabolic equation the effects from lateral variations of irregular terrain on the propagation of radio waves are considered by the representation of the vector fields in terms of two scalar hertz potentials in spherical coordinates the combination of three dimensional parabolic equations for these potentials with an impedance boundary condition for the ground followed by a transformation of variables will define a boundary condition problem characterized by equations displaying coefficients that depend on the terrain height function and its partial derivatives the problem solution through the crank nicolson scheme will lead to a sparse system of linear equations which will be solved by a direct method the resulting numerical model will be applied to test cases to show similar features to those from examples described in the literature taken as references for validation purposes

radius of approximate controllability of linear retarded systems under structured perturbations abstract in this paper we shall deal with the problem of calculation of the radius of approximate controllability in the banach state space k n l 2 h k 0 k n for linear retarded systems of the form x t a 0 x t a 1 x t h 1 a k x t h k b u t by using multi valued linear operators we are able to derive computable formulas for this radius when the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

robust stochastic resonance signal detection and adaptation in impulsive noise stochastic resonance sr occurs when noise improves a system performance measure such as a spectral signal to noise ratio or a cross correlation measure all sr studies have assumed that the forcing noise has finite variance most have further assumed that the noise is gaussian we show that sr still occurs for the more general case of impulsive or infinite variance noise the sr effect fades as the noise grows more impulsive we study this fading effect on the family of symmetric ensuremath alpha stable bell curves that includes the gaussian bell curve as a special case these bell curves have thicker tails as the parameter ensuremath alpha falls from 2 the gaussian case to 1 the cauchy case to even lower values thicker tails create more frequent and more violent noise impulses the main feedback and feedforward models in the sr literature show this fading sr effect for periodic forcing signals when we plot either the signal to noise ratio or a signal correlation measure against the dispersion of the ensuremath alpha stable noise linear regression shows that an exponential law ensuremath gamma mathrm opt ensuremath alpha ca ensuremath alpha describes this relation between the impulsive index ensuremath alpha and the sr optimal noise dispersion ensuremath gamma mathrm opt the results show that sr is robust against noise outliers so sr may be more widespread in nature than previously believed such robustness also favors the use of sr in engineering systems we further show that an adaptive system can learn the optimal noise dispersion for two standard sr models the quartic bistable model and the fitzhugh nagumo neuron model for the signal to noise ratio performance measure this also favors practical applications of sr and suggests that evolution may have tuned the noise sensitive parameters of biological systems

computation of the real controllability radius and minimum norm perturbations of higher order descriptor and time delay lti systems a linear time invariant lti system s controllability radius measures the norm of the smallest parametric perturbation such that the perturbed system is uncontrollable and is of practical importance in this note we study the real controllability radii of i higher order systems ii descriptor systems and iii time delay systems where the perturbations are restricted to the set of real values and the spectral norm is considered formulas for these radii are presented using a framework involving generalized real perturbation values which has certain computational advantages over other formulations found in the literature in particular the formulas are readily more computable especially for higher dimensional systems and a minimum norm perturbation can also easily be obtained numerical examples are presented

between controllable and uncontrollable the distance between a system a b and the set of uncontrollable systems is the minimum of the smallest singular value of i a b with respect to

electromagnetic scattering from 3 d targets in a random medium using finite difference frequency domain in this paper a three dimensional 3 d numerical technique based on the finite difference frequency domain fdfd is implemented to calculate the scattering from arbitrary shaped objects embedded in a continuous random medium the total field scattered field tf sf algorithm is integrated with the fdfd to minimize the memory consumption and speed up the calculations for validation purposes the radar cross section of a 2 d conducting cylinder in random medium is calculated using the fdfd technique and compared to previously published data based on the current generator method an upgrade to a 3 d solver was then inspired once the idea of using the multi grid technique was introduced to accelerate the convergence rate of the bicgstab iterative solver thus allowing the fdfd technique to become a robust method to solve the scattering problem from large targets embedded in random medium therefore using the introduced simulating scheme one can easily elucidate any scattering information out of real life targets surrounded by random environmental effects

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

modeling radio wave propagation in tunnels with a vectorial parabolic equation to study radio wave propagation in tunnels we present a vectorial parabolic equation pe taking into account the cross section shape wall impedances slowly varying curvature and torsion of the tunnel axis for rectangular cross section two polarizations are decoupled and two families of adiabatic modes can be found explicitly giving a generalization of the known results for a uniform tunnel in the general case a boundary value problem arises to be solved by using finite difference finite element fd fe techniques numerical examples demonstrate the computational efficiency of the proposed method

the structured distance to uncontrollability under multi perturbations an approach using multi valued linear operators abstract in this paper we develop a unifying approach for computing the distance to uncontrollability of linear control systems by using multi valued linear operators in representing and estimating the system s equations and matrices we are able to derive computable formulas of the distance from a controllable linear system to the nearest uncontrollable system under the assumption that the system s matrices are subjected to structured multi perturbations and measured by arbitrary operator norms in the case of spectral norms the obtained results unify and extend some previous works as well as a recent interesting result in m karrow d kressner on the structured distance to uncontrollability systems control lett 58 2009 128 132 some illustrating examples are given

propagation modeling over irregular terrain by the improved two way parabolic equation method the two way split step parabolic equation 2w sspe method is improved to be able to model both forward and backward waves propagation over irregular perfectly electrical conducting pec surfaces and irregular lossy surfaces with impedance boundary conditions ibc the discrete mixed fourier transform dmft is applied in both forward and backward direction to deal with ibc and the boundary shift bs method is used to dispose of the irregular terrain effects numerical results obtained by the improved 2w sspe algorithm for several typical scenarios with different ground types are compared with those got by the previous 2w sspe method and the one way parabolic equation method

stochastic resonance in the mutual information between input and output spike trains of noisy central neurons abstract within the framework of information theory we investigate the phenomenon of stochastic resonance in a realistic stochastic model of central neurons the essential point is that for central neurons the input is not a continuous signal but a train of spikes which has in general a non periodic distribution the behavior of an individual neuron is modeled by a combination of an ornstein uhlenbeck diffusion process and a jump process corresponding to a discrete input spike train the input spike train is assumed to be given by a poisson distribution if the parameters are such that the input spike train provokes a membrane potential that hovers under the threshold the noise given by the diffusive part of the neuron model induces a maximum in the mutual information between the input and output spike trains thus the stochastic resonance effect results in a strategy for efficient information transmission of subthreshold signals

noise enhanced parameter estimation this paper investigates the phenomenon of noise enhanced systems for a general parameter estimation problem when the estimator is fixed and known the estimation performance before and after the addition of noise are evaluated performance comparisons are made between the original estimators and noise enhanced estimators based on different criteria the form of the optimal noise probability density function pdf is determined the results are further extended to the general case where the noise is introduced to the system via a transformation for the case where the estimator is fixed and unknown approaches are also proposed to find the optimum noise finally two illustrative examples are presented where the performance comparison is made between the optimal noise modified estimator and gaussian noise modified estimator

an adi pe approach for modeling radio transmission loss in tunnels alternate direction implicit adi method is used to study radio wave propagation in tunnels using the parabolic equation pe we formulate the adi technique for use in tunnels with rectangular circular and arched cross sections and with lossy walls the electrical parameters of the lossy walls are characterized by an equivalent surface impedance a vector pe is also formulated for use in tunnels with lossy walls it is shown that the adi is more computationally efficient than the crank nicolson method however boundary conditions become more difficult to model the boundary conditions of the adi intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced also when implementing the adi in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important to validate the adi pe we show simulation results for tunnel test cases with known analytical solutions furthermore the adi pe is used to simulate real tunnels in order to compare with experimental data it is shown that the pe models the electric fields most accurately in real tunnels at large distances where the lower order modes dominate

propagation modeling over terrain using the parabolic wave equation we address the numerical solution of the parabolic wave equation over terrain using the fourier split step approach the method referred to as a shift map generalizes that of beilis and tappert 1979 who introduced a coordinate transformation technique to flatten the boundary this technique is extended to a wide angle form allowing larger propagation angles with respect to the horizon a new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave pwe using the previously developed mixed fourier transform it is also shown by example that in many cases of interest the boundary may be approximated by discrete piecewise linear segments without affecting the field solution a more accurate shift map solution of the pwe for a piecewise linear boundary is therefore developed for modeling propagation over digitally sampled terrain data the shift map solution is applied to various surface types including ramps wedges curved obstacles and actual terrain where possible comparisons are made between the numerical solution and an exact analytical form the examples demonstrate that the shift map performs well for surface slopes as large as 10 15 spl deg and discontinuous slope changes on the order of 15 20 spl deg to accommodate a larger range of slopes it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well known terrain masking knife edge diffraction approximation

adaptive stochastic resonance in noisy neurons based on mutual information noise can improve how memoryless neurons process signals and maximize their throughput information such favorable use of noise is the so called stochastic resonance or sr effect at the level of threshold neurons and continuous neurons this work presents theoretical and simulation evidence that 1 lone noisy threshold and continuous neurons exhibit the sr effect in terms of the mutual information between random input and output sequences 2 a new statistically robust learning law can find this entropy optimal noise level and 3 the adaptive sr effect is robust against highly impulsive noise with infinite variance histograms estimate the relevant probability density functions at each learning iteration a theorem shows that almost all noise probability density functions produce some sr effect in threshold neurons even if the noise is impulsive and has infinite variance the optimal noise level in threshold neurons also behaves nonlinearly as the input signal amplitude increases simulations further show that the sr effect persists for several sigmoidal neurons and for gaussian radial basis function neurons

algebraic riccati equations and the distance to the nearest uncontrollable pair a connection is established between nearness to unstabilizability of a stabilizable pair a b of matrices and nearness to singularity of the symmetric positive definite solution to an associated algebraic riccati equation from this result computable upper and lower bounds are derived for the distance of a b to the nearest uncontrollable pair numerical tests confirm the validity of the method and potential applications are discussed

on the function space controllability of linear neutral systems a criterion is derived for approximate controllability of linear autonomous neutral functional differential equations in the sobolev space w 2 1 h 0 mathbb e n controllability conditions are based on an abstract evolution equation representation of the system an abstract criterion applicable to a general class of neutral systems is obtained and for the special case of matrix systems of the form frac d dt left x t a 1 x t h right e 0 x t e 1 x t h bu t this abstract criterion leads to an algebraic criterion

on the structured distance to uncontrollability this article is concerned with the structured distance to uncontrollability of a linear time invariant system and relates this concept to a variation of the value the developed framework is applied to derive computational expressions for the class of real perturbations as well as for hermitian symmetric and skew symmetric perturbations in a relatively simple manner examples demonstrate that the structured distance can differ from the standard unstructured distance to uncontrollability by an arbitrary amount it is also shown how systems of higher order can be addressed

the structured distance to non surjectivity and its application to calculating the controllability radius of descriptor systems abstract the classical eckart young formula for square matrices identifies the distance to singularity of a matrix the main purpose of this paper is to get generalizations of this formula we characterize the distance to non surjectivity of a linear operator w l x y in finite dimensional normed spaces x y under the assumption that the operator w is surjective i e w x y and subjected to structured perturbations of the form w m n as an application of these results we shall derive formulas of the controllability radius for a descriptor controllable system e a b e x a x b u t 0 under the assumption that systems matrices e a b are subjected to structured perturbations and to multi perturbations

a vector meshless parabolic equation method for three dimensional electromagnetic scatterings the three dimensional 3 d vector parabolic equation pe method was used for electromagnetic scattering with reasonable accuracy the algorithm was marched from plane to plane along some preferred directions so that the computational load could be reduced significantly however the traditional 3 d pe method was discretized with the finite difference fd scheme and was not flexible for complex geometries in this paper the radial point interpolation meshless rpim method is used for its flexibility to model complex geometries in each transverse plane when the pe method is applied to analyze the electromagnetic scattering from electrically large objects numerical results demonstrate that the proposed technique can efficiently give reasonably accurate results for the complex objects when compared with both the multilevel fast multipole method mlfmm and the traditional pe method

a curvilinear coordinate based split step parabolic equation method for propagation predictions over terrain propagation of radiowaves over irregular terrain and in an inhomogeneous atmosphere is solved by the parabolic equation method using the split step fourier algorithm on a terrain conformal mesh a piecewise continuous coordinate system is generated by the specification of 1 the terrain profile shape at discrete points and 2 an upper height the resulting mesh is conformal to the terrain at the lower boundary and gradually flattens off at the maximum height in addition to preserving the number of points on any vertical line between the terrain and the maximum height from one range step to another the coordinate transformation used in the paper produces a correction term in the refractive index whose gradient diminishes with height as a result the sampling requirements over steep terrain are relaxed when compared to the beilis tappert transformation formulation and results are given both for the horizontal and vertical polarizations

nonlinear dynamic elements with noisy sinusoidal forcing enhancing response via nonlinear coupling we consider a network of nonlinear dynamic elements with nonlinear global coupling subject to noise and a time periodic signal the system response characterized by its signal to noise ratio is computed via approximate analytic techniques and precise numerical simulations we find that cooperative effects arising from the noise and coupling lead to an enhancement of the response of the network over that of a single element

path loss predictions in the presence of buildings on flat terrain a 3 d vector parabolic equation approach starting from a parabolic approximation to the helmholtz equation a three dimensional 3 d vector parabolic equation technique for calculating path loss in an urban environment is presented the buildings are assumed to be polygonal in cross section with vertical sides and flat rooftops and the terrain is assumed to be flat both buildings and ground are allowed to be lossy and present impedance type boundary condition to the electromagnetic field vector fields are represented in terms of the two components of hertzian potentials and depolarization of the fields is automatically included in the formulation a split step algorithm is presented for marching the aperture fields along the range boundary conditions on the building surfaces are treated by using a local fourier representation of the aperture fields several test cases are considered to check the boundary treatment used in the technique as well as to validate the overall approach comparison is shown with uniform theory of diffraction utd exact solutions as well as with measurements

the analysis of exact controllability of neutral type systems by the moment problem approach the problem of exact null controllability is considered for a wide class of linear neutral type systems with distributed delay the main tool of the analysis is the application of the moment problem approach and the theory of the basis property of exponential families a complete characterization of this problem is given the minimal time of controllability is specified the results are based on the analysis of the riesz basis property of eigenspaces of the neutral type systems in hilbert space

controllability radii of linear neutral systems under structured perturbations abstractin this paper we shall deal with the problem of calculation of the controllability radii of linear neutral systems of the form x t a0x t a1x t h a 1x t h bu t we will derive the definition of exact controllability radius approximate controllability radius and euclidean controllability radius for this system by using multi valued linear operators the computable formulas for these controllability radii are established in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

information transmission in parallel threshold arrays suprathreshold stochastic resonance the information transmitted through a parallel summing array of noisy threshold elements with a common threshold is considered in particular using theoretical and numerical analysis a recently reported n g stocks phys rev lett 84 2310 2000 form of stochastic resonance termed suprathreshold stochastic resonance ssr is studied in detail ssr is observed to occur in arrays with two or more elements and unlike stochastic resonance sr in a single element gives rise to noise induced information gains that occur independent of the setting of the threshold or the size of the signal the transmitted information is maximized when all thresholds are set to coincide with the signal mean in this situation and for large arrays the noise can enhance performance up to approximately half the theoretical noiseless channel capacity the theory is tested against digital simulation

the robust covariation based music roc music algorithm for bearing estimation in impulsive noise environments this paper presents a new subspace based method for bearing estimation in the presence of impulsive noise which can be modeled as a complex symmetric alpha stable s spl alpha s process we define the covariation matrix of the array sensor outputs and show that eigendecomposition based methods such as the music algorithm can be applied to the sample covariation matrix to extract the bearing information from the measurements a consistent estimator for the marginals of the covariation matrix is presented and its asymptotic performance is studied the improved performance of the proposed source localization method in the presence of a wide range of impulsive noise environments is demonstrated via monte carlo experiments

an angle dependent impedance boundary condition for the split step parabolic equation method a formulation of the impedance boundary condition ibc is derived which can be used in split step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties the approach relies on ensuring that the plane wave decomposition of the field satisfies the appropriate ibc for each spectral component numerical experiments illustrating the robustness of the approach for low contrast interfaces and angles of incidence near the brewster angle are presented

an improved impedance boundary algorithm for fourier split step solutions of the parabolic wave equation a new implementation of the previously published mixed fourier transform mft method for including impedance boundaries in split step parabolic equation solutions is described and demonstrated the new algorithm is formulated entirely in the discrete domain which results in extended applicability and increased computation speed a brief review of the original mft solution is followed by a detailed description of the discrete formulation the performance of the new algorithm is then demonstrated with a few examples which rely heavily on the accuracy of the impedance boundary these examples include 10 mhz surface wave propagation over smooth and rough sea surfaces and 10 ghz calculations utilizing an effective rough surface impedance

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

a local learning rule that enables information maximization for arbitrary input distributions this note presents a local learning rule that enables a network to maximize the mutual information between input and output vectors the network s output units may be nonlinear and the distribution of input vectors is arbitrary the local algorithm also serves to compute the inverse c 1 of an arbitrary square connection weight matrix

necessary and sufficient conditions of approximate controllability for general linear retarded systems necessary and sufficient conditions of approximate controllability in the space r n times l 2 h 0 r n of general linear retarded systems are obtained it is shown that approximate controllability is equivalent to two conditions a spectral controllability and b the existence of linear feedback which transforms the original system into a system with a complete set of generalized eigenfunctions both conditions are expressed in algebraic form the proof of this result is based on recently obtained criteria of completeness of generalized eigenfunctions associated with retarded systems and on an algebraic approach to functional differential equations practical verifiability of the new conditions is demonstrated on several examples

measuring how far a controllable system is from an uncontrollable one the main goal of this note is to explore some of the properties of a controllable system which in a precise sense to be defined is near an uncontrollable pair to do this we first discuss the notion of the distance between a controllable system and the nearest uncontrollable one in both real and complex cases 2 5 when this distance is relatively small we show to what extent this distance corresponds to certain properties of the uncontrollability matrix and the reachability grammian

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

theory of the stochastic resonance effect in signal detection part ii variable detectors in part i of this paper ldquotheory of the stochastic resonance effect in signal detection part i fixed detectors rdquo ieee transactions on signal processing vol 55 no 7 pt 1 pp 3172 3184 the mechanism of the stochastic resonance sr effect for a fixed detector has been examined this paper analyzes the stochastic resonance sr effect under the condition that the detector structure or its parameters can also be changed the detector optimization problem with sr noise under both neyman pearson and bayesian criteria is examined in the bayesian approach when the prior probabilities are unknown the minimax approach is adopted the form of the optimal noise pdf along with the corresponding detector as well as the maximum achievable performance are determined the developed theory is then applied to a general class of weak signal detection problems under the assumptions that the sample size n is large enough and the test statistics satisfies the conditions of central limit theorem the optimal sr noise is shown to be a constant vector and independent of the signal strength for both neyman pearson and bayesian criteria illustrative examples are presented where performance comparisons are made between the original detector and the optimal sr noise modified detector for different types of sr noise

an alternative direction decomposition scheme and error analysis for parabolic equation model alternate direction implicit adi method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method in this paper a novel alternate direction decomposition add scheme is proposed to simplify the split step fourier transform ssft algorithm in parabolic equation pe model the spatial matrix of electric field is decoupled into two subarrays by add ssft algorithm which is used to decrease the computational burden for 3 d pe problems in a large scale environment after the integration order exchanged also the phase error of this approximation is analyzed to show the algorithm performance along with several numerical results including the tunnel cases with traditional modal theory and they are provided to validate and calibrate the pe model and demonstrate the efficiency and accuracy of the proposed algorithm moreover the scattering computation is investigated and compared with multilevel fast multipole method to show that add ssft is appropriate for solving the scattering problems for the electrical large target

stochastic resonance as dithering a direct correspondence is demonstrated between the phenomenon of stochastic resonance in static nonlinear systems and the dithering effect well known in the theory of digital waveform coding it is argued that many static systems displaying stochastic resonance are forms of dithered quantizers and that the existence or absence of stochastic resonance in such systems can be predicted from the effects of dither averaging upon their transfer characteristics also results are introduced regarding stochastic resonance in certain nonlinear systems with memory e g hysteretic systems