quasiperiodic dynamics in hamiltonian 1 1 2 degree of freedom systems far from integrability abstract the subject of this paper is two quasiperiodicity in a large class of one and a half degree of freedom hamiltonian systems the main result is that such systems have invariant tori for any internal frequency that is of constant type and sufficiently large relative to the forcing frequency an explicit bound on the minimum value of the internal frequency is presented the systems under consideration are not required to be small perturbations of integrable ones

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

design of interpretable fuzzy rule based classifiers using spectral analysis with structure and parameters optimization this paper presents a design method for fuzzy rule based systems that performs data modeling consistently according to the symbolic relations expressed by the rules the focus of the model is the interpretability of the rules and the model s accuracy such that it can be used as tool for data understanding the number of rules is defined by the eigenstructure analysis of the similarity matrix which is computed from data the rule induction algorithm runs a clustering algorithm on the dataset and associates one rule to each cluster each rule is selected among all possible combinations of one dimensional fuzzy sets as the one nearest to a cluster s center the rules are weighted in order to improve the classifier performance and the weights are computed by a bounded quadratic optimization problem the model complexity is minimized in a structure selection search performed by a genetic algorithm that selects simultaneously the most representative subset of variables and also the number of fuzzy sets in the fuzzy partition of the selected variables the resulting model is evaluated on a set of benchmark datasets for classification problems the results show that the proposed approach produces accurate and yet compact fuzzy classifiers the resulting model is also evaluated from an interpretability point of view showing how the rule weights provide additional information to help data understanding and model exploitation

integration of an index to preserve the semantic interpretability in the multiobjective evolutionary rule selection and tuning of linguistic fuzzy systems in this paper we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions mfs is performed the proposed index is the aggregation of three metrics that preserve the original meanings of the mfs as much as possible while a tuning of their definition parameters is performed additionally rule selection mechanisms can be used to reduce the model complexity which involves another important interpretability aspect to this end we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy rule based systems with three objectives accuracy semantic interpretability maximization and complexity minimization we tested our approach on nine real world regression datasets in order to analyze the interaction between the fuzzy rule selection approach and the tuning approach these are also individually proved in a multiobjective framework and compared with their respective single objective counterparts we compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective based approaches results confirm the effectiveness of our approach and a wide range of solutions is obtained which are not only more interpretable but are also more accurate

effective discommensurations in the incommensurate ground states of the extended frenkel kontorowa models abstract for the frenkel kontorowa model and its extensions in several dimensions with several neighbor interactions etc it is proven that the hull function of an incommensurate ground state is purely discrete when the phonon spectrum exhibits a non zero gap the same result also holds when the lyapunov coefficient of the corresponding set of trajectories in the associated twist map when it is definable is strictly positive when this theorem applies the fourier coefficients of the incommensurate modulation can be described by an analytic hull function this is in some sense a dual result to the kolmogorov arnol d moser theorem which proves under different hypotheses that incommensurate modulation in real space instead of reciprocal space can be described by an analytic hull function the physical implication of this theorem is that the incommensurate ground states can be decomposed into a linear superposition of localized effective discommensurations the shape of these discommensurations depends on the model parameters and on their density approaching the transition by breaking of analyticity tba from above the size of these discommensurations diverges below the tba these discommensurations cannot be described anymore as localized objects in the analytic phase these results confirm that tba in non linear models extends the concept of localization of eigenstates which up to now was only meaningful for linear operators

internal modes of discrete solitons near the anti continuum limit of the dnls equation discrete solitons of the discrete nonlinear schrodinger dnls equation are compactly supported in the anti continuum limit of the zero coupling between lattice sites eigenvalues of the linearization of the dnls equation at the discrete soliton determine its spectral stability small eigenvalues bifurcating from the zero eigenvalue near the anti continuum limit were characterized earlier for this model here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti continuum limit this result rules out the existence of internal modes neutrally stable eigenvalues of the discrete spectrum near the anti continuum limit

a proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings however since the accuracy and the interpretability of the obtained model are contradictory properties the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled to solve this problem one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level in this paper a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability to do so we consider a new rule representation scheme base on the linguistic 2 tuples representation model which allows the lateral variation of the involved labels furthermore the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper presenting results on different real applications the obtained results show the good performance of the proposed approach in high dimensional problems and its ability to cooperate with methods to remove unnecessary rules

nonlinear dynamics from the wilson lagrangian a nonlinear hamiltonian dynamics is derived from the wilson action in lattice gauge theory let d be a linear space of lattice dirac operators d a defined by some lattice gauge field a we consider the lagrangian d tr d a im 4 on d where m c is a mass parameter critical points of this functional are given by solutions of a nonlinear discrete wave equation which describe the time evolution of the gauge fields a in the simplest case the dynamical system is a cubic henon map in general it is a symplectic coupled map lattice we prove the existence of non trivial critical points in two examples

analysis of the separatrix map in hamiltonian systems abstract a one degree of freedom hamiltonian system with time periodic perturbation generally display rich dynamics near the separatrix the understanding of which is important in studying transport and diffusion we introduce the exact separatrix map which provides an effective way to describe dynamics near the separatrix we show that the separatrix map and its inverse satisfy twist conditions with logarithmic singularities and therefore they have an infinite twist near the transversal intersections of the stable and unstable manifolds of the hyperbolic equilibrium using the variational formalism with the generating functions we show the dynamics near the singularities correspond to the anti integrable limit as a consequence we show the nonexistence of rotational invariant circles with sufficiently large rotation number first obtained by lazutkin in standard maps moreover the existence of a uniformly hyperbolic chaotic invariant set is shown in connection with the theory of the anti integrable limit the simplest form of the separatrix map is studied in detail to illustrate some of the above results explicitly

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

topological entropy of standard type monotone twist maps we study invariant measures of families of monotone twist maps s q p 2q p v q q with periodic morse potential v we prove that there exist a constant c c v such that the topological entropy satisfies htop s log c 3 in particular htop s for we show also that there exist arbitrary large such that s has nonuniformly hyperbolic invariant measures with positive metric entropy for large the measures are hyperbolic and for a class of potentials which includes v q sin q the lyapunov exponent of the map s with invariant measure grows monotonically with

large memory capacity in chaotic artificial neural networks a view of the anti integrable limit in the literature it was reported that the chaotic artificial neural network model with sinusoidal activation functions possesses a large memory capacity as well as a remarkable ability of retrieving the stored patterns better than the conventional chaotic model with only monotonic activation functions such as sigmoidal functions this paper from the viewpoint of the anti integrable limit elucidates the mechanism inducing the superiority of the model with periodic activation functions that includes sinusoidal functions particularly by virtue of the anti integrable limit technique this paper shows that any finite dimensional neural network model with periodic activation functions and properly selected parameters has much more abundant chaotic dynamics that truly determine the model s memory capacity and pattern retrieval ability to some extent this paper mathematically and numerically demonstrates that an appropriate choice of the activation functions and control scheme can lead to a large memory capacity and better pattern retrieval ability of the artificial neural network models

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions the globally positive diffeomorphisms of the 2n dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite for these globally positive diffeomorphisms an aubry mather theory was developed by garibaldi and thieullen that provides the existence of some minimizing measures using the two green bundles g and g that can be defined along the support of these minimizing measures we will prove that there is a deep link between the angle between g and g along the support of the considered measure mu the size of the smallest positive lyapunov exponent of mu the tangent cone to the support of mu

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map

a tabu harmony search based approach to fuzzy linear regression we propose an unconstrained global continuous optimization method based on tabu search and harmony search to support the design of fuzzy linear regression flr models tabu and harmony search strategies are used for diversification and intensification of flr respectively the proposed approach offers the flexibility to use any kind of an objective function based on client s requirements or requests and the nature of the dataset and then attains its minimum error moreover we elaborate on the error produced by this method and compare it with the errors resulting from the other known estimation methods to study the performance of the method three categories of datasets are considered numeric inputs symmetric fuzzy outputs symmetric fuzzy inputs symmetric fuzzy outputs and numeric inputs asymmetric fuzzy outputs through a series of experiments we demonstrate that in terms of the produced error with different model fitting measurements the proposed method outperforms or is pareto equivalent to the existing methods reported in the literature

on the stability of interpolatory variable stepsize adams methods in nordsieck form this paper considers the stability of adams methods that vary the stepsize using the interpolation technique of nordsieck sufficient conditions sometimes necessary on the stepsize ratios h n 1 h n that imply the stability of adams methods with different strategies of stepsize variation are obtained this study extends previous results on this subject by gear and tu siamj numer anal 11 1974 pp 1025 1043 and skeel and jackson siam j numer anal 20 1983 pp 840 853

hyperbolic cantori have dimension zero it is proved that cantori for area preserving twist maps have dimension zero only finitely many orbits of gaps and gap widths go to zero exponentially in each orbit if they are hyperbolic

homogenization of fully overdamped frenkel kontorova models in this paper we consider the fully overdamped frenkel kontorova model this is an infinite system of coupled first order odes each ode represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential after a proper rescaling a macroscopic model describing the evolution of densities of particles is obtained we get this homogenization result for a general class of frenkel kontorova models the proof is based on the construction of suitable hull functions in the framework of viscosity solutions

rigorous diffusion properties for the sawtooth map we get a rigorous bound for the diffusion constant of the hamiltonian dynamical system generated by a sawtooth map on a cylinder the momentum variable properly renormalized then behaves almost like a brownian motion in the limit of infinite coupling constants the strategy of the proof is a rigorous reformulation of the random phase approximation

rotationally ordered periodic orbits for multiharmonic area preserving twist maps abstract the poincare birkhoff theorem guarantees existence of at least two rotationally ordered periodic orbits of each rational rotation number for each area preserving twist map for many maps however there are more than two we prove this for maps near a non degenerate multi well anti integrable limit and deduce an intricate bifurcation diagram for rotationally ordered periodic orbits in so called multiharmonic families our results are motivated and supported by numerical investigations of the reversible 2 harmonic family we believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families which numerically exhibits a cantor set of cusps

mixed dynamics in a parabolic standard map abstract we use numerical and analytical tools to provide arguments in favor of the existence of a family of smooth symplectic diffeomorphisms of the two dimensional torus that have both a positive measure set with positive lyapunov exponent and a positive measure set with zero lyapunov exponent the family we study is the unfolding of an almost hyperbolic diffeomorphism on the boundary of the set of anosov diffeomorphisms proposed by lewowicz

stability of symbolic embeddings for difference equations and their multidimensional perturbations abstract in this paper we study complexity of solutions of a high dimensional difference equation of the form x i m x i 1 x i x i 1 x i n 0 i z where is a c 1 function from r l m n 1 to r l our main result provides a sufficient condition for any sufficiently small c 1 perturbation of to have symbolic embedding that is to possess a closed set of solutions that is invariant under the shift map such that the restriction of the shift map to is topologically conjugate to a subshift of finite type the sufficient condition can be easily verified when depends on few variables including the logistic and henon families to prove the result we establish a global version of the implicit function theorem for perturbed equations the proof of the main result is based on the brouwer fixed point theorem and the proof of the global implicit function theorem is based on the contraction mapping principle and other ingredients our novel approach extends results in 2 3 8 15 21

comparison and design of interpretable linguistic vs scatter frbss gm3m generalization and new rule meaning index for global assessment and local pseudo linguistic representation this work is devoted to defining more general interpretability indexes to be applied to any scatter or linguistic model implemented by any type of membership functions they are based on metrics that should take into account the semantic and inference issues the semantic issue in order to preserve the meaning of the linguistic labels and the inference issue since this can influence the behavior of the rules on the other hand these metrics have been designed to be intuitive in order to support the analysis or selection of a final model and to favor a low computational cost within an optimization process in order to check their usefulness a multi objective evolutionary algorithm simultaneously performing a rule selection and an adjustment of the fuzzy partitions is guided by the proposed indexes on several benchmark data sets to obtain models with different degrees of accuracy and interpretability in addition using these metrics a local analysis can be carried out between models of a different nature this local analysis through the model components gives support to the user to make the best choice from among the models

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

destruction of invariant tori in pendulum type equations abstract in this paper we prove that if there exists an invariant torus with the rotation number 1 in the pendulum type equation x q 0 x t x for a given potential q 0 q 0 t x c t 2 and is a liouville number then for any neighborhood n q 0 of q 0 in the c topology there exists a potential q q t x n q 0 such that the system x q x t x does not admit any invariant torus with the rotation number 1 this confirms j moser s suggestion in bol soc brasil mat 20 1981 29 45

the stability of variable stepsize nordsieck methods conditions are given for the stability of multistep methods that vary the stepsize using the interpolation technique of nordsieck if the stepsize selection function is variation bounded then these methods are stable alternatively there exist constants a b c depending on the method and satisfying a leqq b 1 c such that if the stepsize ratio h n 1 h n satisfies either 0 h n 1 h n leqq a or b h n 1 h n leqq c then the method is stable as the frequency of stepsize changing decreases the methods become more stable in the sense that the intervals defined by a b and c are larger tables and graphs are given for the adams and backward differentiation methods indicating how stability changes with frequency of changing stepsize in particular it is shown that k 1 value adams formulas k leqq 6 are stable for 0 h n 1 h n leqq c that is it is shown that stability is maintained whenever the stepsize is decreased by an arb

equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for frenkel kontorova models there is a well known correspondence between the dynamics of symplectic twist maps which represent an important class of hamiltonian systems and the equilibrium states of a class of variational problems in solid state physics known as frenkel kontorova models in this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent this allows one to transfer many ideas between the two and hence to deduce new results for example we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so called non degenerate anti integrable limits by aubry and abramovici using the concept of phonon gap and deduce that they have measure zero we prove existence of many multi defect equilibrium states for a frenkel kontorova system if it has a single non degenerate defect equilibrium state via standard results in the theory of uniform hyperbolicity

angular dynamical spectra a new method for determining frequencies weak chaos and cantori we define the invariant spectra of rotation angles and twist angles angular dynamical spectra and study the properties of their moments angular moments in a model two dimensional map the angular moments give the main frequencies of the orbits a main frequency is defined both for a regular and chaotic orbit for kam curves around a centre this frequency corresponds to the rotation number inside islands of stability we obtain both the main frequency rotation number and the epicyclic frequency a fast detection of thin chaotic layers is obtained on the basis of the behaviour of the frequency curve we explore the resonant structure near a last kam boundary the secondary islands of various multiplicities forming farey sequences and the noble tori between them are located we find a criterion to determine whether various resonant chaotic zones communicate or are separated by invariant tori

modulational and numerical solutions for the steady discrete sine gordon equation in two space dimensions abstract we develop a modulation theory based on a suitably averaged lagrangian for steady solutions of the sine gordon equation in a two dimensional lattice these lump solutions are nonzero in circular or polygonal regions and zero elsewhere the modulation theory gives approximate solutions away from small perturbations of the exact anti continuum solutions for both radial and polygonal solutions we show how the peierls nabarro potential determines the shape of the boundary between excited sites and the zero solution these solutions are compared with the corresponding numerical solutions and significant agreement is found moreover we show that solutions with a large radius more than sixteen lattice sites can be explained using a continuous trial function for the averaged lagrangian while smaller polygonal solutions can be constructed using a trial function which takes into account the angular variation of the boundary imposed by the lattice finally the ideas of equivariant bifurcation theory are used to obtain a full numerical description of the solution branches as functions of the coupling parameter between neighboring sites the results of this work can be used to study steady solutions for other types of lattice equations

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

approximate reasoning on a basis of z number valued if then rules approximate reasoning is about reasoning with imperfect information nowadays a large diversity of approaches to approximate reasoning with fuzzy information and fuzzy type 2 information exists it should be stressed however that real world imperfect information is characterized by combination of fuzzy and probabilistic uncertainties which is referred to as bimodal information in view of this zadeh introduced the concept of a z number regarded as an ordered pair z a b of fuzzy numbers a and b where a is a linguistic value of a variable of interest and b is a linguistic value of probability measure of a playing a role of its reliability unfortunately up to day there is no research on approximate reasoning realized on the basis of if then rules with z number valued antecedents and consequents briefly z rules zadeh addressed this problem as related to an uncharted territory in this paper a new approach is developed to study approximate reasoning with z rules on a basis of linear interpolation we provide an application of the approach to job satisfaction evaluation and to students educational achievement evaluation problems related to psychological and perceptual issues naturally characterized by imperfect information the obtained results show applicability and validity of the proposed approach

discrete breathers in aperiodic diatomic fpu lattices with long range order this study presents various aspects of discrete breathers in diatomic fpu lattices with masses varying according to the aperiodic fibonacci and thue morse sequences we investigate the existence of time periodic breathers starting from the anti continuous limit and consider excitations of isolated light atoms hence obtaining the domain of unique existence for these breathers we also perform a linear stability analysis by studying the floquet operator the found exact solutions are used slightly perturbed as initial conditions for long time simulations of the breathers these breathers turn out to be robust finally we consider how initial excitations of two consecutive light atoms evolve depending on the properties of the phase space for the two atom system at the anti continuous limit we obtain different behaviour of these excitations especially we find that the aperiodic lattices can support localized excitations with a continuous frequency distribution for the timescales we consider while a periodic lattice is unable to these excitations are referred to as chaotic breathers c 2000 elsevier science b v all rights reserved

multivariable gaussian evolving fuzzy modeling system this paper introduces a class of evolving fuzzy rule based system as an approach for multivariable gaussian adaptive fuzzy modeling the system is an evolving takagi sugeno ets functional fuzzy model whose rule base can be continuously updated using a new recursive clustering algorithm based on participatory learning the fuzzy sets of the rule antecedents are multivariable gaussian membership functions which have been adopted to preserve information between input variable interactions the parameters of the membership functions are estimated by the clustering algorithm a weighted recursive least squares algorithm updates the parameters of the rule consequents experiments considering time series forecasting and nonlinear system identification are performed to evaluate the performance of the approach proposed the multivariable gaussian evolving fuzzy models are compared with alternative evolving fuzzy models and classic models with fixed structures the results suggest that multivariable gaussian evolving fuzzy modeling is a promising approach for adaptive system modeling

interface interactions in modulated phases and upsilon points certain features in frenkel kontorova and other models of phases with a one dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases and treating the free energy to create interfaces as well as the interaction of two three or more interfaces as phenomenological parameters a strategy employed by fisher and szpilka for interacting defects can be extended to the case of interfaces allowing a systematic study of the phase diagram by ignoring all interface interactions and then successively taking into account pair triple and higher order terms the possible phase diagrams which can occur near the point where 0 include various sorts of endpoints analogous to critical endpoints an accumulation point of first order transitions and triple points and a self similar structure which we call an upsilon point which turns out to be an accumulation point of an infinite number of segments of first order transition lines each of which terminates in two upsilon points

discrete breathers in a two dimensional fermi pasta ulam lattice using asymptotic methods we investigate whether discrete breathers are supported by a two dimensional fermi pasta ulam lattice a scalar one component two dimensional fermi pasta ulam lattice is shown to model the charge stored within an electrical transmission lattice a third order multiple scale analysis in the semi discrete limit fails since at this order the lattice equations reduce to the 2 1 dimensional cubic nonlinear schrodinger nls equation which does not support stable soliton solutions for the breather envelope we therefore extend the analysis to higher order and find a generalised 2 1 dimensional nls equation which incorporates higher order dispersive and nonlinear terms as perturbations we find an ellipticity criterion for the wave numbers of the carrier wave numerical simulations suggest that both stationary and moving breathers are supported by the system calculations of the energy show the expected threshold behaviour whereby the energy of breathers does em not go to zero with the amplitude we find that the energy threshold is maximised by stationary breathers and becomes arbitrarily small as the boundary of the domain of ellipticity is approached

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

homoclinic bifurcations for the henon map abstract chaotic dynamics can be effectively studied by continuation from an anti integrable limit we use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations we find a bound on the parameter range for which the henon map exhibits a complete binary horseshoe as well as a subshift of finite type we classify homoclinic bifurcations and study those for the area preserving case in detail simple forcing relations between homoclinic orbits are established we show that a symmetry of the map gives rise to constraints on certain sequences of homoclinic bifurcations our numerical studies also identify the bifurcations that bound intervals on which the topological entropy is apparently constant

on topological entropy of billiard tables with small inner scatterers abstract we present in this paper an approach to studying the topological entropy of a class of billiard systems in this class any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers combining the concept of anti integrable limit with the theory of lyusternik shnirel man we show that a billiard system in this class generically admits a set of non degenerate anti integrable orbits which corresponds bijectively to a topological markov chain of arbitrarily large topological entropy the anti integrable limit is the singular limit when scatterers shrink to points in order to get around the singular limit and so as to apply the implicit function theorem on auxiliary circles encircling these scatterers we define a length functional whose critical points are well defined at the anti integrable limit and give rise to billiard orbits when the scatterers are not points consequently we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

stability of multistep methods on variable grids this paper is concerned with the stability of multistep methods for ordinary initial value problems on grids with variable mesh sizes a necessary and sufficient condition for stability is given from which generalizations of recent results by gear et al and by zlatev can be obtained as special cases as an application the stability of the variable bdf formulas is treated

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

on the influence of the kinetic energy on the stability of equilibria of natural lagrangian systems natural lagrangian systems t on r2 described by the equation frac d dt frac partial t partial dot q frac partial t partial q frac partial pi partial q are considered where t q q is a positive definite quadratic form in q and q has a critical point at 0 it is constructively proved that there exist a c potential energy and two c kinetic energies t and t such that the equilibrium q t 0 is stable for the system t and unstable for the system t equivalently it is established that for c natural systems the kinetic energy can influence the stability in the analytic category this is not true

highly continuous interpolants for one step ode solvers and their application to runge kutta methods we suggest a general method for the construction of highly continuous interpolants for one step methods applied to the numerical solution of initial value problems of odes of arbitrary order for the construction of these interpolants one uses along with the numerical data of the discrete solution of a problem provided by a typical one step method at endstep points high order derivative approximations of this solution this approach has two main advantages it allows an easy way of construction of high order runge kutta and nystrom interpolants with reduced cost in additional function evaluations that also preserve the one step nature of the underlying discrete ode solver moreover for problems which are known to possess a solution of high smoothness the approximating interpolant resembles this characteristic a property that on occasion might be desirable an analysis of the stability behavior of such interpolatory processes is carried out in the general case a new numerical technique concerning the accurate determination of the stability behavior of numerical schemes involving higher order derivatives and or approximations of the solution from previous grid points over nonequidistant meshes is presented this technique actually turns out to be of a wider interest as it allows us to infer in certain cases more accurate results concerning the stability of for example the bdf formulas over variable stepsize grids moreover it may be used as a framework for analyzing more complex and supposedly more promising types of methods as they are the general linear methods for first and second order differential equations many particular variants of the new method for first order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics a detailed application concerning the construction of c 2 and c 3 continuous extensions for some fifth and sixth order runge kutta pairs supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type is also provided various numerical examples show in these cases several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior in comparison with others currently in use

fingrams visual representations of fuzzy rule based inference for expert analysis of comprehensibility since zadeh s proposal and mamdani s seminal ideas interpretability is acknowledged as one of the most appreciated and valuable characteristics of fuzzy system identification methodologies it represents the ability of fuzzy systems to formalize the behavior of a real system in a human understandable way by means of a set of linguistic variables and rules with a high semantic expressivity close to natural language interpretability analysis involves two main points of view readability of the knowledge base description regarding complexity of fuzzy partitions and rules and comprehensibility of the fuzzy system regarding implicit and explicit semantics embedded in fuzzy partitions and rules as well as the fuzzy reasoning method readability has been thoroughly treated by many authors who have proposed several criteria and metrics unfortunately comprehensibility has usually been neglected because it involves some cognitive aspects related to human reasoning which are very hard to formalize and to deal with this paper proposes the creation of a new paradigm for fuzzy system comprehensibility analysis based on fuzzy systems inference maps so called fuzzy inference grams fingrams by analogy with scientograms used for visualizing the structure of science fingrams show graphically the interaction between rules at the inference level in terms of co fired rules i e rules fired at the same time by a given input the analysis of fingrams offers many possibilities measuring the comprehensibility of fuzzy systems detecting redundancies and or inconsistencies among fuzzy rules identifying the most significant rules etc some of these capabilities are explored in this study for the case of fuzzy models and classifiers

a genetic learning process for the scaling factors granularity and contexts of the fuzzy rule based system data base abstract in this contribution we propose a new method to automatically learn the knowledge base of a fuzzy rule based system by finding an appropriate data base using a genetic algorithm and considering a simple generation method to derive the rule base our genetic process learns all the components of the data base number of labels working ranges and membership function shapes for each linguistic variable using a non linear scaling function to adapt the fuzzy partition contexts

recent developments in the theory of hamiltonian systems area preserving mappings of an annulus occur as poincare mappings of hamiltonian systems they were studied extensively by g d birkhoff recently aubry and mather investigated the subclass of so called monotone twist mappings for which they constructed independently closed invariant cantor sets their work led to important new results their theory is discussed and related to hamiltonian systems satisfying a legendre condition the connection of this theory with the stability problem kam theory and in particular the disintegration of invariant tori is discussed various constructions of mather sets are explained in which minimal solutions of variational problems play a central role this theory has a close relation to the differential geometric investigations by morse and hedlund on geodesics on two dimensional surfaces

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis

model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

topological horseshoes in travelling waves of discretized nonlinear wave equations applying the concept of anti integrable limit to coupled map lattices originated from space time discretized nonlinear wave equations we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions in particular the coupled map lattices display spatio temporal chaos on the horseshoes

existence of dynamical hull functions with two variables for the ac driven frenkel kontorova model abstract in this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the frenkel kontorova model with periodic driving force we show in the infinite dimensional phase space there exists an ordered circle on which both the poincare map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers and respectively the latter characterizing the mean spacing of particles furthermore we prove that the average velocity for the particle chain with irrational exists and t where t denotes the period of the driving force and it is continuous and nondecreasing with respect to f the average of the external driving force if the shift map on the invariant ordered circle is ergodic we demonstrate the existence of two variable dynamical hull functions and that there are no mode locking behaviors

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

chaotic orbits for differentiable maps near anti integrable limits abstract in this work we study the dynamical system which can be transformed into a difference equation of the form g x t n 2 x t x t n 1 g 0 x t 0 where r is a parameter g r n 1 n 2 1 r and g 0 r r are both c 1 functions and n 1 n 2 are both nonnegative integers suppose that the function g 0 has k simple zeros where k 2 we give criteria of existence of some kinds of chaotic orbits and construct infinite ones explicitly as applications of these results we establish snap back repellers and heteroclinical repellers for the modified mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the henon maps the arneodo coullet tresser maps and the quadratic volume preserving maps

cantori for multiharmonic maps abstract we compute all the cantori and their gap and turnstile structures for area preserving twist maps near non degenerate anti integrable limits with arbitrarily many wells per period the results imply a rich bifurcation diagram for cantori of families of maps containing degenerate anti integrable limits we conjecture the structure of this bifurcation diagram in the case of the two harmonic family

one dimensional dynamics for traveling fronts in coupled map lattices multistable coupled map lattices typically support traveling fronts separating two adjacent stable phases we show how the existence of an invariant function describing the front profile allows a reduction of the infinitely dimensional dynamics to a one dimensional circle homeomorphism whose rotation number gives the propagation velocity the mode locking of the velocity with respect to the system parameters then typically follows we study the behavior of fronts near the boundary of parametric stability and we explain how the mode locking tends to disappear as we approach the continuum limit of an infinite density of sites

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

on the stability of variable stepsize nordsieck bdf methods in this paper the stability of backward differentiation methods that vary the stepsize using the interpolation technique of nordsieck is considered conditions for the stability of the methods are obtained the study is an extension of previous results given by gear and tu 3 and skeet and jackson 14

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

energy landscape properties studied using symbolic sequences abstract we investigate a classical lattice system with n particles the potential energy v of the scalar displacements is chosen as a 4 on site potential plus interactions its stationary points are solutions of a coupled set of nonlinear equations starting with aubry s anti continuum limit it is easy to establish a one to one correspondence between the stationary points of v and symbolic sequences 1 n with n 0 we prove that this correspondence remains valid for interactions with a coupling constant below a critical value c and that it allows the use of a thermodynamic formalism to calculate statistical properties of the so called energy landscape of v this offers an explanation for why topological quantities of v may become singular like in phase transitions in particular we find that the saddle index distribution is maximum at a saddle index n s max 1 3 for all c furthermore there exists an interval v v max in which the saddle index n s as a function of the average energy v is analytical in v and it vanishes at v above the ground state energy v gs whereas the average saddle index n s as a function of the energy v is highly nontrivial it can exhibit a singularity at a critical energy v c and it vanishes at v gs only close to v gs n s v exhibits power law behavior which even holds for noninteracting particles

aubry mather sets and birkhoff s theorem for geodesic flows on the two dimensional torus in this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of riemannian metrics on the two dimensional torus also our method gives a new proof of birkhoff s theorem for twist maps of the cylinder we prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the aubry mather set with this rotation number

a self organizing network in the weak coupling limit abstract we prove the existence of spatially localized ground states of the diffusive haken model this model describes a self organizing network whose elements are arranged on a d dimensional lattice with short range diffusive coupling the network evolves according to a competitive gradient dynamics in which the effects of diffusion are counteracted by a localizing potential that incorporates an additional global coupling term in the absence of diffusive coupling the ground states of the system are strictly localized i e only one lattice site is excited for sufficiently small non zero diffusive coupling it is shown analytically that localized ground states persist in the network with the excitations exponentially decaying in space numerical results establish that localization occurs for arbitrary values of in one dimension but vanishes beyond a critical coupling c d when d 1 the one dimensional localized states are interpreted in terms of instanton solutions of a continuum version of the model

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

the convergence of variable stepsize variable formula multistep methods in this paper variable stepsize variable formula multistep methods for the numerical solution of ordinary differential equations are considered some more general results on stability than are given in the papers of gear tu siam j numer anal 11 1974 pp 1025 1043 and gear watanabe siam j numer anal 11 1974 pp 1044 1058 are obtained further the asymptotic behavior of the error in the variable stepsize fixed formula case is studied finally the above study is extended to predictor corrector methods

some new qualitative insights into quality of fuzzy rule based models rules in fuzzy rule based models convey essential knowledge about the system under discussion as such they capture the essence of relationships occurring among input and output variables while the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules although to a limited extent the quality of rules being regarded as generic pieces of knowledge has not been studied so far in this study we formulate and investigate this problem by looking at the quality of rules including aspects of a stability b generalizability and c conflict we identify a concept of rule multiplicity conflict and study an emergence of rule generalization a number of pertinent performance indices are developed and their usage is presented through a series of experimental studies

soliton energies in the standard map beyond the chaotic threshold the birkhoff moser theorem guarantees the existence of an analytical reduction to its normal form of an area preserving map in the neighbourhood of an unstable fixed point the stable and unstable manifolds emanating from this point are then simply the images of the axes in the normal coordinates for sufficiently high parameter values of the standard map the series for the manifolds can be resummed into a closed form providing good approximations to the first loops of the manifolds the authors thus obtain simple approximations for a pair of homoclinic orbits for parameters where most orbits are chaotic the frenkel kontorova model consists of an infinite sequence of equal springs and masses under the action of a periodic potential the configuration where all the masses lie at the minima of the potential corresponds to the unstable fixed point of the standard map a soliton or discommensuration where one of the minima is missed is represented by a homoclinic orbit using the normal form they calculate the soliton energy and its pinning energy good agreement is found with the pinning energy of pokrovsky 1981 obtained as a perturbation from the continuum approximation

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

critical phonons of the supercritical frenkel kontorova model renormalization bifurcation diagrams abstract the phonon modes of the frenkel kontorova model are studied both at the pinning transition as well as in the pinned cantorus phase we focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction using an exact decimation scheme the eigenfunctions are shown to have non trivial scaling properties not only at the pinning transition point but also in the cantorus regime therefore the phonons defy localization and remain critical even where the associated area preserving map has a positive lyapunov exponent in this region the critical scaling properties vary continuously and are described by a line of renormalization limit cycles interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti integrable limit both of these limits are described by a trivial decimation fixed point very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the trivial fixed point at these degeneracy points the phonon hull is represented by an infinite series of step functions this novel behavior persists in the extended version of the model containing two harmonics additional richnesses of this extended model are the one to two hole transition line characterized by a divergence in the renormalization cycles non exponentially localized phonons and the preservation of critical behavior all the way upto the anti integrable limit

amount of rotation about a point and the morse index for the case of exact area preserving monotone twist diffeomorphisms we give formulas relating the amount of rotation about an orbit and certain morse indices

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

a fuzzy system constructed by rule generation and iterative linear svr for antecedent and consequent parameter optimization this paper proposes a new fuzzy regression model i e the fuzzy system constructed by rule generation and iterative linear support vector regression fs rglsvr for structural risk minimization the fs rglsvr is composed of takagi sugeno ts type fuzzy if then rules these rules are automatically constructed by a self splitting rule generation algorithm that introduces the self splitting technique to the k means clustering algorithm this new algorithm regards a cluster as a fuzzy rule where no preassignment of the cluster rule number is necessary the cost function for parameter learning is defined based on structural risk instead of empirical risk minimization in order to achieve generalizability tuning all of the free parameters in the fs rglsvr using linear support vector regression svr is proposed to minimize the cost function each of the consequent and antecedent part parameters is expressed as a linear combination coefficient in a transformed input space so that the linear svr is applicable this paper introduces iterative linear svr to tune antecedent and consequent parameters this paper demonstrates the capabilities of fs rglsvr by two simulated and four practical regression examples comparisons with fuzzy systems with different types of learning algorithms verify the performance of the fs rglsvr

a 0 stability of variable stepsize bdf methods abstract in this paper the a0 stability of interpolatory variable coefficient and fixed leading coefficient bdf formulas is considered a comparison of the stability properties of the three methods based on their spectral radius is presented in addition we study the effect of some strategies of stepsize variation employed in usual codes on the stability of these methods

discrete breathers in disordered media abstract we describe some simple physical models where discrete breathers nonlinear localised modes in a lattice exist together with spatial disorder as the models are translation invariant both spatial localisation and spatial disorder are only due to the interplay of nonlinearity and discreteness this result is obtained as a straightforward application of a general result of existence of discrete breathers j a sepulchre r s mackay localised oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators nonlinearity 10 1997 1 35