the relationship between the minimum variance and minimax disparity rim quantifier problems recently liu and lou on the equivalence of some approaches to the owa operator and rim quantifier determination fuzzy sets and systems 159 2007 1673 1688 investigated the equivalence of solutions to the minimum variance and minimax disparity rim quantifier problems however their proofs are very sensitive to the assumption and some are mathematically incomplete in this regard this paper provides a counterexample of the minimax disparity rim quantifier problem for the case in which generating functions are continuous the paper also provides a correct proof of the minimax disparity rim quantifier problem for the case in which generating functions are absolutely continuous and a generalized result for the minimum variance rim quantifier problem for the case in which generating functions are lebesgue integrable based on the results the paper provides a correct relationship between the minimum variance and minimax disparity rim quantifier problems

on the maximum entropy parameterized interval approximation of fuzzy numbers this paper extends the interval valued weighted possibilistic mean of a fuzzy number of fuller and majlender to a general weighting function without the monotonic increasing assumption this weighting function determines a weighted average aggregation of the cuts of the fuzzy number according to the preference of a decision maker some properties of the weighting function are provided and a preference index that qualifies this aggregation and can serve as a parameter for the definition of interval approximation of a fuzzy number is proposed a special class of parameterized weighting functions satisfying the maximal entropy principle is proposed

controlling chaos experimentally in systems exhibiting large effective lyapunov exponents we investigate experimentally the performance of the ott grebogi and yorke phys rev lett 64 1196 1989 feedback concept to control chaotic motion the experimental systems are a driven pendulum and a driven bronze ribbon both setups have unstable periodic orbits characterized by large effective lyapunov exponents all control vectors for the feedback control are extracted from the experimental data to do this for the pendulum a global model obtained by the flow field analysis of cremers and h ubler z naturforsch 42a 797 1987 is used and for the bronze ribbon linear approximations in embedding space are exploited we analyze the problems that arise due to the amplification of noise by large effective lyapunov exponents in the determination of the control values as well as in the performance of the experimental control successful control can be achieved in our experiments by applying the local control method which allows a quasicontinuous adjustment of the control parameter in contrast to adjusting the control parameter only once per return time of the poincar e map

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

an analytic approach for obtaining maximal entropy owa operator weights one important issue in the theory of ordered weighted averaging owa operators is the determination of the associated weights one of the first approaches suggested by o hagan determines a special class of owa operators having maximal entropy of the owa weights for a given level of orness algorithmically it is based on the solution of a constrained optimization problem in this paper using the method of lagrange multipliers we shall solve this constrained optimization problem analytically and derive a polynomial equation which is then solved to determine the optimal weighting vector

on proving the extended minimax disparity owa problem determining associated weights in the theory of ordered weighted averaging owa operators is an important issue wang and parkan information sciences 175 2005 20 29 proposed a minimax disparity approach for obtaining owa operator weights which is based on the solution of a linear program lp model for a given degree of orness recently amin and emrouznejad computers industrial engineering 50 2006 proposed an extended minimax disparity model and improved their proposed lp to obtain weights that would be as close as possible they also left an open problem related to the extended model for an even number of arguments there would be an interval for the degree of orness such that the optimal solution of the extended model could be obtained by a compact mathematical form in this paper we completely prove this open problem mathematically for all orness levels whatever the number of arguments

on the issue of obtaining owa operator weights we first investigate the issue of obtaining the weights associated with the owa aggregation in the situation when we have observed data on the arguments and the aggregated value we next introduce a family of owa operators called exponential owa operators finally we look at a simple procedure for generating the weights given a required degree of orness

on the optimality of the ott grebogi yorke control scheme abstract some of the characteristics of the ott grebogi yorke ogy control technique are presented as applied to nonlinear flows as distinct from nonlinear maps specifically we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function the choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the ogy controller an algorithm for designing the optimal basis function is presented the optimal basis function is shown to be defined by a step function with potentially several jumps thus revealing the intrinsic power of a standard ogy technique that uses a single step function as a basis function two numerical applications of the optimal design technique to a duffing oscillator are also presented to show that the standard ogy technique may be significantly improved by making an optimal choice of a basis function copyright

compact representations of kohn sham invariant subspaces we present a method to compute a hierarchical approximate representation of the solutions of the kohn sham equations the method is based on a recursive bisection algorithm and yields one particle wave functions localized on subdomains of varying sizes the accuracy of the representation is set a priori by specifying the highest acceptable error in 2 norm for any solution applications to large systems are used to illustrate the achievable reduction in data size implications for the development of linear scaling methods and for the acceleration of conventional iterative methods are discussed

on obtaining minimal variability owa operator weights one important issue in the theory of ordered weighted averaging owa operators is the determination of the associated weights one of the first approaches suggested by o hagan determines a special class of owa operators having maximal entropy of the owa weights for a given level of orness algorithmically it is based on the solution of a constrained optimization problem another consideration that may be of interest to a decision maker involves the variability associated with a weighting vector in particular a decision maker may desire low variability associated with a chosen weighting vector in this paper using the kuhn tucker second order sufficiency conditions for optimality we shall analytically derive the minimal variability weighting vector for any level of orness

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

optimal periodic orbits of chaotic systems occur at low period 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 that indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor in particular our results indicate that if we consider that the function to be optimized depends on a parameter gamma then the lebesgue measure in gamma corresponding to optimal periodic orbits of period ital p or greater decreases exponentially with increasing ital p furthermore the set of parameter values for which optimal orbits are nonperiodic typically has zero lebesgue measure copyright ital 1996 the american physical society

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

reconstruction of dynamical and geometrical properties of chaotic attractors from threshold crossing interspike intervals we reconstruct the largest lyapunov exponent and fractal dimension of a chaotic attractor using thresholdcrossing interspike intervals alone we show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor we also give an explanation of this possibility based on the concept of instantaneous frequency s1063 651x 98 51207 x pacs number s 05 45 1b the problem of extracting information about a dynamical system from experimental data is of great importance for all those who study the systems of natural origin that may be thought of as black boxes the traditional way of obtaining experimental data is to fix the sampling stepdt and store the values of some state variable s t corresponding to the time moments idt the fundamental works 1 state that the delay method applied to such time series often allows one to reconstruct a set being topologically equivalent to the initial system attractor on the base of these works a series of methods have been developed allowing one to evaluate attractor dimension 2 and the largest lyapunov characteristic exponent lce 3 to detect unstable equilibrium points 4 and unstable periodic orbits 5 and finally to create in some cases global models 6 or different predictors 7 another way of getting experimental data which is often preferred by biologists is to generate spike train data 8 one imposes a certain condition on the value of state variable s t and records the interspike intervals isi ti between the time moments for which the condition is fulfilled there are two general methods for recording spike train data the first one is to generate a spike when s t crosses some threshold level u in one direction and measure isi as an alternative one may measure the time intervals between the local maxima of s t fig 1 a the values of ti can be treated as the times of trajectory return to the surface defined by the condition imposed to the initial s t the condition s t 5u means the definition of a plane in the phase space while the condition s t 50 s t being a realization x j t o f a dynamical system dx w dt5f

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

control of nmr laser chaos in high dimensional embedding space the unstable periodic orbits of a chaotic nmr laser have been identified and classified according to their stability properties the dynamical control method by ott grebogi and yorke ogy phys rev lett 64 1196 1990 has been successfully applied in a six dimensional embedding space as required for a proper reconstruction of the attractors an improved method has then been implemented in order to circumvent the occurrence of several possibly complex unstable eigenvalues in the standard ogy procedure and has shown superior performance finally an extension to several control stations per poincar e time is discussed

a minimax disparity approach for obtaining owa operator weights one of the key issues in the theory of ordered weighted averaging owa operators is the determination of owa operator weights in this paper a minimax disparity approach for obtaining owa operator weights is proposed this involves the formulation and solution of a linear programming lp model for a given degree of orness the proposed approach generates the owa operator weights by minimizing the maximum difference between any two adjacent weights a numerical example is provided to illustrate the application of the proposed approach in generating viable owa operator weights in a simple and effective way

a note on the solution equivalence to general models for rim quantifier problems abstract this note presents a counterexample that illustrates why the lack of absolute continuity destroys the correctness of a main result of liu 2008 and provides a modified model for the minimax rim quantifier problem and the correct formulation of liu s result

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

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

improving minimax disparity model to determine the owa operator weights determining the ordered weighted averaging owa operator weights is important in decision making applications several approaches have been proposed in the literature to obtain the associated weights this paper provides an alternative disparity model to identify the owa operator weights the proposed mathematical model extends the existing disparity approaches by minimizing the sum of the deviation between two distinct owa weights the proposed disparity model can be used for a preference ranking aggregation a numerical example in preference ranking and an application in search engines prove the usefulness of the generated owa weights

compressed modes for variational problems in mathematics and physics this article describes a general formalism for obtaining spatially localized sparse solutions to a class of problems in mathematical physics which can be recast as variational optimization problems such as the important case of schrodinger s equation in quantum mechanics sparsity is achieved by adding an regularization term to the variational principle which is shown to yield solutions with compact support compressed modes linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

differentiability implies continuity in neuronal dynamics recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory troublesome in this work were the significant periods of time where no periodic orbits were extracted dynamically dark regions tests for periodic orbit structure typically require that the underlying dynamics are differentiable since continuity of a mathematical function is a necessary but insufficient condition for differentiability regions of observed differentiability should be fully contained within regions of continuity we here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity first we introduce a null jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted second we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches 2001 published by elsevier science b v

on the equivalence of some approaches to the owa operator and rim quantifier determination the ordered weighted averaging owa operator is a widely used aggregation method and its determination is usually a prerequisite step in many related applications the regular increasing monotone rim quantifier can be seen as the continuous case of the owa operator with the quantifier aggregation method some approaches with optimization criteria for the determination of owa operator and rim quantifier were proposed although these problems look different at the first sight a deeper investigation can reveal the equivalence of solutions between them inspired by the solution equivalence of minimum variance problems and minimax disparity problem for owa operator we propose the minimax disparity rim quantifier problem and two minimax ratio problems for owa operator and rim quantifier respectively we investigate the equivalence of solutions for the maximum entropy and minimax ratio problems and solutions for the minimum variance and minimax disparity problems of owa operator and rim quantifier respectively by a theoretical point of view

the relationship between the maximum entropy and minimax ratio rim quantifier problems liu and lou fuzzy sets and systems 159 2007 1673 1688 investigated the equivalence of solutions to the maximum entropy and minimax ratio rim quantifier problems the paper improves liu and lou s theorems from a mathematical perspective because their theorems are not suitable for absolutely continuous generating functions the paper provides a correct proof of the minimax ratio rim quantifier problem for the case in which generating functions are absolutely continuous and a generalized solution to the maximum entropy rim quantifier problem for the case in which generating functions are lebesgue integrable based on the results the paper provides a correct relationship between the maximum entropy and minimax ratio rim quantifier problems

continuous extension of the geometric control method in this paper we give a continuous version valid for flows of the geometric control method introduced by toroczkai et al 1994 and constructed for mappings on 2d poincare sections this method does not require explicit knowledge of the dynamics just a rough location of the periodic orbit and a single parameter easily computed from four data points applicability in numerical and real experiments is discussed