robust h_ infty based synchronization of the fractional order chaotic systems by using new self evolving nonsingleton type 2 fuzzy neural networks in this paper a novel h infty based adaptive fuzzy control is presented for the synchronization of fractional order chaotic systems a self evolving nonsingleton type 2 fuzzy neural network se nst2fnn is proposed for the estimation of the unknown functions in the dynamics of the system the effects of the approximation error and the external disturbances are eliminated by designing an adaptive compensator such that the h infty norm of the synchronization error is minimized and asymptotically stability is achieved the consequent parameters of se nst2fnn are tuned based on the adaptation laws that are derived from lyapunov stability analysis the antecedent part and the rule database of se nst2fnn are optimized based on a clustering method and the modified invasive weed optimization algorithm respectively the effectiveness of proposed control scheme is verified by simulation examples

sequential adaptive fuzzy inference system safis for nonlinear system identification and prediction in this paper a sequential adaptive fuzzy inference system called safis is developed based on the functional equivalence between a radial basis function network and a fuzzy inference system fis in safis the concept of influence of a fuzzy rule is introduced and using this the fuzzy rules are added or removed based on the input data received so far if the input data do not warrant adding of fuzzy rules then only the parameters of the closest in a euclidean sense rule are updated using an extended kalman filter ekf scheme the performance of safis is compared with several existing algorithms on two nonlinear system identification benchmark problems and a chaotic time series prediction problem results indicate that safis produces similar or better accuracies with less number of rules compared to other algorithms

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

sofmls online self organizing fuzzy modified least squares network in this paper an online self organizing fuzzy modified least square sofmls network is proposed the algorithm has the ability to reorganize the model and adapt itself to a changing environment where both the structure and learning parameters are performed simultaneously the network generates a new rule if the smallest distance between the new data and all the existing rules the winner rule is more than a prespecified radius the major contributions of this paper are as follows 1 a new network is proposed in which unidimensional membership functions are used and only two parameters for each rule are employed thus reducing the number of parameters the network avoids the singularity produced by the widths in the antecedent part for online learning 2 a new pruning algorithm based on the density is proposed where the density is the number of times each rule is used in the algorithm the rule that has the smallest density the looser rule in a selected number of iterations is pruned if the value of its density is smaller than a prespecified threshold and 3 the stability of the proposed algorithm is proven and the bound for the average of the identification error is found the condition that led the algorithm to avoid the local minimum is found and it is proven that the parameter error is bounded by the initial parameter error three simulations give the effectiveness of the suggested algorithm

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

an approach for on line extraction of fuzzy rules using a self organising fuzzy neural network this paper presents a hybrid neural network called the self organising fuzzy neural network sofnn to extract fuzzy rules from the training data the first hidden layer of this network consists of ellipsoidal basis function ebf neurons every ebf neuron in the sofnn has both a centre vector and a width vector neurons are organised by the network itself the methods of the structure and parameter learning based on new adding and pruning techniques and a recursive learning algorithm are simple and effective with a high accuracy and a compact structure simulations show that the sofnn has the capability to encode fuzzy rules in the resulting network

vector majorization via nonnegative definite doubly stochastic matrices of maximum rank abstract it is known that for real n vectors x and y y majorizes x if and only if ay x for some doubly stochastic matrix a of order n suppose that the coordinates of each of x and y are in nonincreasing order then the matrix a can be chosen to be nonnegative definite if there is no coincidence and if the coordinates of x are not all equal then a can be chosen to be positive definite in this paper we obtain a simple formula for calculating the maximum rank of all nonnegative definite doubly stochastic matrices a such that ay x

state space black box modelling via markov parameters based on evolving type 2 neural fuzzy inference system for nonlinear multivariable dynamic systems abstract in this paper an approach for state space interval type 2 neural fuzzy identification of multivariable dynamic systems in an evolving and incremental learning context is proposed in combination with type 2 fuzzy systems the adopted methodology regards the following aspects recursive learning of the footprint of uncertainty and recursive learning of the rule consequents part by a recursive linear state space approaches where markov parameters are incrementally and robustly learnt in sample wise manner the efficiency and applicability of the proposed methodology are demonstrated through experimental and computational results and compared with prominent learning algorithms of the literature

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

liouvillian dynamics of the hopf bifurcation two dimensional vector fields undergoing a hopf bifurcation are studied in a liouville equation approach the liouville equation rules the time evolution of statistical ensembles of trajectories issued from random initial conditions but evolving under the deterministic dynamics the time evolution of the probability densities of such statistical ensembles can be decomposed in terms of the spectrum of the resonances i e the relaxation rates of the liouvillian operator or the related frobenius perron operator the spectral decomposition of the liouvillian operator is explicitly constructed before at and after the hopf bifurcation because of the emergence of time oscillations near the hopf bifurcation the resonance spectrum turns out to be complex and defined by both relaxation rates and oscillation frequencies the resonance spectrum is discrete far from the bifurcation and becomes continuous at the bifurcation this continuous spectrum is caused by the critical slowing down of the oscillations occurring at the hopf bifurcation and it leads to power law relaxation as 1 square root of t of the probability densities and statistical averages at long times t infinity moreover degeneracy in the resonance spectrum is shown to yield a jordan block structure in the spectral decomposition

the doubly stochastic matrices of a vector majorization abstract let y be majorized by x we investigate the polytope of doubly stochastic matrices d for which y dx we determine when there is a positive d and when there is a fully indecomposable d we calculate the dimension of the polytope and as a result determine when d is uniquely determined

applications of m matrices to majorization abstract m matrices are used to give a characterization of majorization for any vectors x y in the n dimensional euclidean space such that x 1 x 2 x n and y 1 y 2 y n x is majorized by y if and only if x ay for some n n nonnegative definite doubly stochastic matrix a an explicit formula for finding such an a is given

an online self constructing neural fuzzy inference network and its applications a self constructing neural fuzzy inference network sonfin with online learning ability is proposed in this paper the sonfin is inherently a modified takagi sugeno kang tsk type fuzzy rule based model possessing neural network learning ability there are no rules initially in the sonfin they are created and adapted as online learning proceeds via simultaneous structure and parameter identification in the structure identification of the precondition part the input space is partitioned in a flexible way according to an aligned clustering based algorithm as to the structure identification of the consequent part only a singleton value selected by a clustering method is assigned to each rule initially afterwards some additional significant terms selected via a projection based correlation measure for each rule will be added to the consequent part incrementally as learning proceeds the combined precondition and consequent structure identification scheme can set up an economic and dynamically growing network a main feature of the sonfin in the parameter identification the consequent parameters are tuned optimally by either least mean squares or recursive least squares algorithms and the precondition parameters are tuned by a backpropagation algorithm to enhance the knowledge representation ability of the sonfin a linear transformation for each input variable can be incorporated into the network so that much fewer rules are needed or higher accuracy can be achieved

experimental study of intelligent controllers under uncertainty using type 1 and type 2 fuzzy logic uncertainty is an inherent part in control systems used in real world applications the use of new methods for handling incomplete information is of fundamental importance type 1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems type 2 fuzzy sets that are used in type 2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom this paper deals with the design of control systems using type 2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise etc the experimental results are divided in two classes in the first class simulations of a feedback control system for a non linear plant using type 1 and type 2 fuzzy logic controllers are presented a comparative analysis of the systems response in both cases was performed with and without the presence of uncertainty for the second class a non linear identification problem for time series prediction is presented based on the experimental results the conclusion is that the best results are obtained using type 2 fuzzy systems

design of interval type 2 fuzzy sliding mode controller in this paper an interval type 2 fuzzy sliding mode controller it2fsmc is proposed for linear and nonlinear systems the proposed it2fsmc is a combination of the interval type 2 fuzzy logic control it2flc and the sliding mode control smc which inherits the benefits of these two methods the objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed loop system the lyapunov stability method is adopted to verify the stability of the interval type 2 fuzzy sliding mode controller system the design procedure of the it2fsmc is explored in detail a typical second order linear interval system with 50 parameter variations an inverted pendulum with variation of pole characteristics and a duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method the simulation results show that the it2fsmc achieves the best tracking performance in comparison with the type 1 fuzzy logic controller t1flc the it2flc and the type 1 fuzzy sliding mode controller t1fsmc

applications of type 2 fuzzy logic systems to forecasting of time series abstract in this paper we begin with a type 1 fuzzy logic system fls trained with noisy data we then demonstrate how information about the noise in the training data can be incorporated into a type 2 fls which can be used to obtain bounds within which the true noisefree output is likely to lie we do this with the example of a one step predictor for the mackey glass chaotic time series m c mackey l glass oscillation and chaos in physiological control systems science 197 1977 287 280 we also demonstrate how a type 2 fls can be used to obtain better predictions than those obtained with a type 1 fls

simplified interval type 2 fuzzy neural networks this paper describes a self evolving interval type 2 fuzzy neural network fnn for various applications as type 1 fuzzy systems cannot effectively handle uncertainties in information within the knowledge base we propose a simple interval type 2 fnn which uses interval type 2 fuzzy sets in the premise and the takagi sugeno kang tsk type in the consequent of the fuzzy rule the tsk type consequent of fuzzy rule is a linear combination of exogenous input variables given an initially empty the rule base all rules are generated with on line type 2 fuzzy clustering instead of the time consuming k m iterative procedure the design factors ql and qr are learned to adaptively adjust the upper and lower positions on the left and right limit outputs using the parameter update rule based on a gradient descent algorithm simulation results demonstrate that our approach yields fewer test errors and less computational complexity than other type 2 fnns

vector majorization via positive definite matrices abstract it is well known that for real n vectors y and x y majorizes x if and only if a y x for some doubly stochastic matrix a of order n if the components of each of y and x are in nonincreasing order then it is known that the matrix a can be chosen to be positive semidefinite symmetric we characterize when there is a positive definite doubly stochastic matrix a such that a y x

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

a neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

neuro fuzzy systems for function approximation abstract we present a neuro fuzzy architecture for function approximation based on supervised learning the learning algorithm is able to determine the structure and the parameters of a fuzzy system the approach is an extension to our already published nefcon and nefclass models which are used for control or classification purposes the proposed extended model which we call nefprox is more general and can be used for any application based on function approximation

global stability analysis using the eigenfunctions of the koopman operator we propose a novel operator theoretic framework to study global stability of nonlinear systems based on the spectral properties of the so called koopman operator our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems the main results establish the necessary and sufficient relationship between the existence of specific eigenfunctions of the koopman operator and the global stability property of hyperbolic fixed points and limit cycles these results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space

a hybrid learning algorithm for a class of interval type 2 fuzzy neural networks in real life information about the world is uncertain and imprecise the cause of this uncertainty is due to deficiencies on given information the fuzzy nature of our perception of events and objects and on the limitations of the models we use to explain the world the development of new methods for dealing with information with uncertainty is crucial for solving real life problems in this paper three interval type 2 fuzzy neural network it2fnn architectures are proposed with hybrid learning algorithm techniques gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation at the antecedents layer a interval type 2 fuzzy neuron it2fn model is used and in case of the consequents layer an interval type 1 fuzzy neuron model it1fn in order to fuzzify the rule s antecedents and consequents of an interval type 2 takagi sugeno kang fuzzy inference system it2 tsk fis it2 tsk fis is integrated in an adaptive neural network in order to take advantage the best of both models this provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy if then rules in addition to handling uncertainty and imprecision on the other hand neural networks are highly adaptable with learning and generalization capabilities experimental results are divided in two kinds in the first one a non linear identification problem for control systems is simulated here a comparative analysis of learning architectures it2fnn and anfis is done for the second kind a non linear mackey glass chaotic time series prediction problem with uncertainty sources is studied finally it2fnn proved to be more efficient mechanism for modeling real world problems