research on dynamics in a resource extraction game with bounded rationality abstract based on bounded rationality a linear dynamic system is proposed in this work for the duopoly game of renewable resource extraction this dynamics reflects the decision makers dynamical output strategy consisting of all the nash equilibriums in every two adjacent periods the resource growth rate the discount rate and the expected extraction rate are the major parameters in this system the system stability and the equilibrium characteristics of the stable state are analyzed numerical simulations are made to show the influence of all the parameters on the speed for the system to approaches its stable equilibrium by the mathematical analysis we find that if the discount is smaller or the expected extraction rate is larger the system will converge in shorter time and approach a stable state with lager resource reserve and with larger resource output so we draw the conclusion that decision makers focusing on the long term profits is helpful to sustainable development on natural resource market in the long run

complex dynamics of duopoly game with heterogeneous players a further analysis of the output model abstract in this paper an output duopoly game with heterogeneous players is analyzed in order to study the influence of players different behavior on the dynamics of game two types of players are considered which are bounded rationality expectation and naive expectation player with naive expectation chooses an output level based on the market price of previous period while player with bounded rationality adjusts his output adaptively following a bounded rationality adjustment process based on a local estimate of the marginal profit of previous period the game model is also based on the assumption that demand and cost function are nonlinear the existence of equilibrium points and its local stability of the output game are investigated the complex dynamics bifurcations and chaos are displayed by numerical experiment numerical methods also show that the long run average profit achieved by player adopting naive expectation is higher than that achieved by player using self adaptive adjustment measure although players use similar production methods

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

relative perturbation theory for hyperbolic eigenvalue problem we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem hx jx where h is a positive definite matrix and j is a diagonal matrix of signs we consider two types of perturbations when a graded matrix h d ad is perturbed in a graded sense to h h d a a d and the multiplicative perturbations of the form h h i e h i e our bounds are simple to compute compare well to the classical results and can be used when analyzing numerical algorithms

the dynamics of a business game a 2d piecewise smooth nonlinear map abstract in this paper we give a proper investigation of the problem of remanufacturing duopoly game 1 a smooth piecewise map is introduced to describe such game a rich analysis either analytical or numerical is presented in this paper in order to investigate the stability of the fixed points of the game s model our obtained results show that those fixed points lose its stability due to flip and neimark sacker border bifurcations furthermore we study the stability of those points when there are periodic cycles with points jump from region to other region

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

complex dynamics analysis for a duopoly stackelberg game model with bounded rationality in view of the effect of differences between plan products and actual products a duopoly stackelberg model of competition on output is formulated the firms announce plan products sequentially in planning phase and act simultaneously in production phase backward induction is used to solve subgame nash equilibrium the equilibrium outputs and equilibrium profits are affected by cost coefficients for the duopoly stackelberg model a nonlinear dynamical system which describes the time evolution with bounded rationality is analyzed the equilibria of the corresponding discrete dynamical systems are investigated the local stability analysis has been carried out the stability of nash equilibrium gives rise to complex dynamics as some parameters of the model are varied numerical simulations were used to show bifurcation diagram stability region and chaos it is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos

a new relative perturbation theorem for singular subspaces abstract this note addresses the sensitivity of singular subspaces of a matrix under relative perturbations it employs a new technique of separating a multiplicative perturbation d into two components one is the distance of a scalar multiple of d to the nearest unitary matrix q and the other is the distance of q to the identity consequently the new bounds reflect the intrinsic differences in how left and right multiplicative perturbations affect left and right singular subspaces

collective motions in globally coupled tent maps with stochastic updating we study a generalization of globally coupled maps where the elements are updated with probability p when p is below a threshold p c the collective motion vanishes and the system is the stationary state in the large size limit we present the linear stability analysis

relative residual bounds for indefinite hermitian matrices abstract we prove several residual bounds for relative perturbations of the eigenvalues of indefinite hermitian matrix the bounds fall into two categories the weyl type bounds and the hofmann wielandt type bounds the bounds are expressed in terms of sines of acute principal angles between certain subspaces associated with the indefinite decomposition of the given matrix the bounds are never worse than the classical residual bounds and can be much sharper in some cases the bounds generalize the existing relative residual bounds for positive definite matrices to indefinite case

study on a 3 dimensional game model with delayed bounded rationality abstract a nonlinear dynamic triopoly game model is studied based on the theory of nonlinear dynamics and previous researches in this paper a lagged structure is introduced to the model to study stability conditions of the nash equilibrium under a local adjustment process when players price their products with delayed bounded rationality numerical simulations are provided to demonstrate the complexity of system evolvement and influence of the strategy of delayed bounded rationality on system stability we find that besides the lagged structure suitable delayed parameters are also important factors to eliminate chaos or expand the stable region of the system and various players adjustment parameters have different effect on stability of the system

tripoly stackelberg game model this paper is devoted to introduce and study a stackelberg game consisting of three competed firms the three firms are classified as a leader which is the first firm and the other two firms are called the followers a linear inverse demand function is used in addition a quadratic cost based on an actual and announced quantities is adopted based on bounded rationality a three dimensional discrete dynamical system is constructed for the system the backward induction is used to solve the system and to get nash equilibrium the obtained results are shown that nash equilibrium is unique and its stability is affected by the systems parameters by which the system behaves chaotically due to bifurcation and chaos appeared some numerical experiments are performed to portrays such chaotic behavior a control scheme is used to return the system back to its stability state and is supported by some simulations

effects of microscopic disorder on the collective dynamics of globally coupled maps this paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps our analysis is complementary to the approach taken by teramae and kuramoto j teramae y kuramoto strong desynchronizing effects of weak noise in globally coupled systems phys rev e 63 2001 036210 who pointed out the anomalous scaling properties preceding the loss of coherence we focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one using properly defined order parameters an analytical approach is proposed for describing the collective dynamics in terms of an approximate low dimensional system the systematic derivation of the macroscopic equations provides a link between the microscopic features of the population single element dynamics and noise distribution and the properties of the emergent behaviour the macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population

a note on the existence of the hyperbolic singular value decomposition abstract in this note we present a new matrix decomposition for a matrix pair a b with a hermitian the hyperbolic svd comes as a special case of the decomposition with a set to be the signature matrix

relative perturbation theory for hyperbolic singular value problem we give relative perturbation bounds for singular values and perturbation bounds for singular subspaces of a hyperbolic singular value problem for the pair g j where g is a full rank matrix and j is a diagonal matrix of signs we consider two types of relative perturbations g g b b d and g g d b b depending whether g has full column or full row rank respectively in both cases we also consider relative element wise perturbations of g which typically occur in numerical computations

a new method to control chaos in an economic system in this paper the method to control chaos by using phase space compression is applied to economic systems because of economic significance of state variable in economic dynamical systems the values of state variables are positive due to capacity constraints and financial constraints we can control chaos by adding upper bound or lower bound to state variables in economic dynamical systems which is different from the chaos stabilization in engineering or physics systems the knowledge about system dynamics and the exact variety of parameters are not needed in the application of this control method so it is very convenient to apply this method two kinds of chaos in the dynamic duopoly output systems are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method the results show that performance of the system is improved by controlling chaos in practice owing to capacity constraints financial constraints and cautious responses to uncertainty in the world the firm often restrains the output advertisement expenses research cost etc to confine the range of these variables fluctuation this shows that the decision maker uses this method unconsciously in practice

on bertrand duopoly game with differentiated goods utility function of the form u q 1 q 2 q 1 q 2 is introduced new dynamical bertrand games are derived stability conditions of equilibrium points are discussed chaos and bifurcation behavior are studied the paper investigates a dynamic bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period the demand functions are derived from an underlying ces utility function we investigate numerically the dynamical properties of the model we consider two specific parameterizations for the ces function and study the nash equilibrium and its local stability in the models the general finding is that the nash equilibrium becomes unstable as the speed of adjustment increases the nash equilibrium loses stability through a period doubling bifurcation and the system eventually becomes chaotic either through a series of period doubling bifurcations or after a neimark sacker bifurcation

a sin2θ theorem for graded indefinite hermitian matrices this paper gives double angle theorems that bound the change in an invariant subspace of an indefinite hermitian matrix in the graded form h d ad subject to a perturbation h h d a a d these theorems extend recent results on a definite hermitian

an overview of relative sin t theorems for invariant subspaces of complex matrices abstract relative perturbation bounds for invariant subspaces of complex matrices are reviewed with emphasis on bounding the sines of the largest principal angle between two subspaces i e sin theorems the goal is to provide intuition as well as an idea for why the bounds hold and why they look the way they do relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations in particular there are certain classes of relative perturbations to which subspaces of indefinite hermitian matrices can be more sensitive than subspaces of definite matrices

analysis of the dynamics of cournot team game with heterogeneous players in this paper we study a dynamical system of a two team cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation the equilibrium solutions and the conditions of their locally asymptotic stability are studied it is demonstrated that as some parameters in the model are varied the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation chaotic phenomenon periodic windows strange attractor and unpredictable trajectories will occur the great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis

chaotic dynamics in nonlinear duopoly game with heterogeneous players in this study we investigate the dynamics of a nonlinear discrete time duopoly game where the players have heterogeneous expectations two players with different expectations are considered one is boundedly rational and the other thinks with adaptive expectations the stability conditions of the equilibria are discussed we show how the dynamics of the game depend on the model parameters we demonstrate that as some parameters of the game are varied the stability of nash equilibrium is lost through period doubling bifurcation the chaotic features are justified numerically via computing lyapunov exponents sensitive dependence on initial conditions and the fractal dimension

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

new results on stable region of nash equilibrium of output game model we obtains that the real stable region of nash equilibrium of output game model is smaller than that in general because of the prisoners dilemma players would not select some of the values of parameters though those points of parameters orbit will tend to the nash equilibrium point that is players strategies are unstable in those values in order to discuss this problem we introduce a duopoly output dynamical model we use numerical simulation to analyze the initial output and the decisive parameters influences on the speed of the system s evolution convergence to nash equilibrium and we measure performances of the model in various period states and chaotic state by using the index of average profits it is found that the optimal strategy locates in the stable region of nash equilibrium and aggregate profits are not sensitive to initial output in certain neighborhood of nash equilibrium and the decisive parameters in their majority of stable region of nash equilibrium neither impact of the decisive parameters on the speed of the initial output convergent to nash equilibrium and the prisoners dilemma in game theory are the main reason yielding uneven parts in both sides of 3 dimensions simulation figure the projection of the even parts of the 3 dimensions simulation on the plane of parameters is the real stable region of the nash equilibrium point it is necessary to ascertain the stable region of nash equilibrium point by combining the jury condition with some performance indices in economic systems such as average profit aggregate profits aggregate sales revenues and so on we may determine the factual stable region by using the indifference character of utilities of players in the traditional stable region given by jury s condition we also obtain that players should pay attention to the performance of economic entity in various period states

stochastic perturbations for a duopoly stackelberg model abstract in this paper a duopoly stackelberg model of competition on output with stochastic perturbations is proposed sufficient conditions for the existence and the uniqueness of global positive solution of the stochastic system are derived for the stochastic stackelberg model ssm the authors obtained some sufficient conditions for the globally asymptotical stability of the nash equilibrium when ignoring the stochastic perturbations sufficient criteria for the corresponding deterministic stackelberg model dsm is also obtained in addition by applying milstein s numerical algorithm we solve the stochastic differential model then plot the time series diagram which can help us deduce the change regulation of the output of two firms in the competition game finally some discussions are given for the economic significance for the model

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

relative perturbation theory for a class of diagonalizable hermitian matrix pairs abstract veselic and slapnicar gave a general perturbation result for the eigenvalues of the hermitian matrix pair h k where k is positive definite in this paper their result is generalized to a wider class of hermitian matrix pairs especially estimates for the relative perturbation of eigenvalues of definite pairs are also obtained

analysis of a duopoly game with delayed bounded rationality a dynamic of cournot duopoly game is analyzed where players use different production methods and choose their quantities with bounded rationality a dynamic of nonlinear cournot duopoly game is analyzed where players choose quantities with delayed bounded rationality and similar methods of production the equilibria of the corresponding discrete dynamical systems are investigated the stability conditions of nash equilibrium under a local adjustment process are studied the stability of nash equilibrium as some parameters of the model are varied gives rise to complex dynamics such as cycles of higher order and chaos we show that firms using delayed bounded rationality have a higher chance of reaching nash equilibrium numerical simulations are used to show bifurcations diagrams stability regions and chaos

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

the hyberbolic singular value decomposition and applications a new generalization of the singular value decomposition svd the hyperbolic svd is advanced and its existence is established under mild restrictions the hyperbolic svd accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays sliding rectangular windowing and array calibration two algorithms for effecting this decomposition are detailed one is sequential and follows a similar pattern to the sequential bidiagonal based svd algorithm the other is for parallel implementation and mimics hestenes svd algorithm 1958 numerical examples demonstrate that like its conventional counterpart the hyperbolic svd exhibits superior numerical behavior relative to explicit formation and solution of the normal equations furthermore the hyperbolic svd applies in problems where the conventional svd cannot be employed

an implementation of the dqds algorithm positive case abstract the dqds algorithm was introduced in 1994 to compute singular values of bidiagonal matrices to high relative accuracy but it may also be used to compute eigenvalues of tridiagonal matrices this paper discusses in detail the issues that have to be faced when the algorithm is to be realized on a computer criteria for accepting a value for splitting the matrix and for choosing a shift to reduce the number of iterations as well as the relative advantages of using ieee arithmetic when available ways to avoid unnecessary over underflows are described in addition some new formulae are developed to approximate the smallest eigenvalue from a twisted factorization of a matrix the results of extensive testing are presented at the end the list of contents is a valuable guide to the reader interested in specific features of the algorithm

analysis of nonlinear triopoly game with heterogeneous players a nonlinear triopoly game with heterogeneous players is presented we consider three types of players boundedly rational adaptive and naive a triopoly game is modelled by a three dimensional discrete dynamical system the stability conditions of the equilibrium points are analyzed numerical simulations are used to show bifurcation diagrams phase portraits sensitive dependence on initial conditions and fractal dimension the chaotic behavior of the model has been stabilized on the nash equilibrium point by the use of the pyragas delay feedback control method

absolute and relative perturbation bounds for invariant subspaces of matrices absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices in the two norm and in the frobenius norm the absolute bounds can be considered extensions of davis and kahan s sin theorem to general matrices and invariant subspaces of any dimension the relative bounds are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation when the perturbed subspace has dimension one the relative bound is implied by the absolute bound

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

on delayed dynamical duopoly delayed duopoly model with gradient dynamics is considered linear stability constraints are derived it is shown that delay increases stability hence firms using duopoly model with gradient method in optimized theory with delay has a higher chance of reaching nash equilibrium cycles and chaos regions are determined

an oligopoly model with best response and imitation rules abstract imitation based behaviors are considered in economics with significant contributions in reference to homogeneous populations where players are characterized by the same decisional processes see for example 42 48 however the presence of imitation behaviors is detected in experimental oligopolies coexisting with rational like rules this motivates us to consider an heterogeneous population where best responders and imitators coexist and compete in a deterministic oligopoly framework the model we formulate is characterized by two stationary states specifically the cournot nash equilibrium and a further production level at which best responders are inactive and imitators produce at the marked clearing price the heterogeneities among players give to the model a nonlinear structure influence the stability properties of the cournot nash equilibrium and give rise to complex dynamic scenarios we found that the imitators relative fraction have an ambiguous role in determining the stability properties of the cournot nash equilibrium and provided intermediate values of the population size its variations may cause the occurrence of both flip and neimark sacker bifurcations chaotic dynamics and coexistent attractors characterized by not connected basins may also be observed we finally note that certain dynamic regimes described by the model are provided by analogous features as those characterizing experimental outcomes and several experiments can be reproduced with different parameters sets

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

perturbations of the eigenprojections of a factorized hermitian matrix abstract we give the perturbation bounds for the eigenprojections of a hermitian matrix h gjg where g has full column rank and j is nonsingular under the perturbations of the factor g our bounds hold for example when g is given with elementwise relative error our bounds contain relative gaps between the eigenvalues and may thus be much less pessimistic than the standard norm estimates

the dynamics of a quantum bertrand duopoly with differentiated products and heterogeneous expectations abstract based on the bounded rationality and naive expectation the paper sets up a dynamic model of quantum bertrand duopoly game with differentiated products by applying quantum game theory we analyze the effect of quantum entanglement on stability of the quantum equilibrium points and complex dynamic behaviors the result shows that the quantum entanglement can enhance the stability of the system the adjustment speeds of bounded rational player can lead to chaotic characteristics of the system the quantum entanglement can control the system s chaos numerical simulations demonstrate the chaotic features via bifurcation largest lyapunov exponent strange attractors fractal dimensions and sensitivity to initial conditions

analytical solution of metapopulation dynamics in a stochastic environment we study a discrete stochastic linear metapopulation model to understand the effect of risk spreading by dispersion we calculate analytically the stable distribution of populations in different habitats the simultaneous distribution of populations in habitats has a complicated self similar structure but the population in each habitat follows a log normal distribution a class of discrete stochastic matrix models was mostly dealt with numerically our analytical predictions are robust in the wide range of parameters qualitative predictions of the current results should hold in the case of multiple habitats we thus conclude that environmental stochasticity always promotes dispersal

chaotic dynamics in cournot duopoly model with bounded rationality based on relative profit delegation maximization abstract in this paper we maximize the utility function under budget constraint in an oligopoly market we consider a cournot duopoly model with boundedly rational players adopting the gradient adjustment mechanism to update their productions the dynamics of cournot duopoly model with relative profit delegation maximization is analyzed we explore the local stability of nash equilibrium point and the occurrence of complexity behaviors additionally the chaotic dynamics of this game via flip bifurcations and neimark sacker bifurcations are researched numerical simulations are presented to show bifurcation diagrams sensitive dependence on initial conditions strange attractors and chaos as to the fractal structure emerged in evolution process the fractal dimension is calculated further analysis of the simulation results indicates that the chaotic behavior of the system can be stabilized on the nash equilibrium point by using state variables feedback and parameter variation method