convergence of dynamical zeta functions i study poles and zeros of zeta functions in one dimensional maps numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function this describes convergence of the calculations of the first zero which is generally the physically interesting quantity some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow

chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

function approximation with spiked random networks examines the function approximation properties of the random neural network model or gnn the output of the gnn can be computed from the firing probabilities of selected neurons we consider a feedforward bipolar gnn bgnn model which has both positive and negative neurons in the output layer and prove that the bgnn is a universal function approximator specifically for any f spl isin c 0 1 sup s and any spl epsiv 0 we show that there exists a feedforward bgnn which approximates f uniformly with error less than spl epsiv we also show that after some appropriate clamping operation on its output the feedforward gnn is also a universal function approximator

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

the generalized milnor thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters this paper presents an answer to an open problem in the dynamical systems of three letters the generalized milnor thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two dimensional parameter plane the concept of equal topological entropy class is introduced by the dual star product which is a generalization of the derrida gervois pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval the algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the milnor thurston characteristic polynomials moreover the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space 3 of three letters

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

turning back time in markovian process algebra product form solutions in markovian process algebra mpa are constructed using properties of reversed processes the compositionality of mpas is directly exploited allowing a large class of hierarchically constructed systems to be solved for their state probabilities at equilibrium the paper contains new results on both reversed stationary markov processes as well as mpa itself and includes a mechanisable proof in mpa notation of jackson s theorem for product form queueing networks several examples are used to illustrate the approach

a devil s staircase into chaotic scattering abstract we discuss analytically the transition from a regular to a chaotic classical dynamics of a sample scattering model introduced earlier the transition arises as a homoclinic bifurcation of an atypical type in phase space as a discontinuity for the grammatical complexity and as an onset of a devil s staircase for the topological entropy

random neural networks with negative and positive signals and product form solution we introduce a new class of random neural networks in which signals are either negative or positive a positive signal arriving at a neuron increases its total signal count or potential by one a negative signal reduces it by one if the potential is positive and has no effect if it is zero when its potential is positive a neuron fires sending positive or negative signals at random intervals to neurons or to the outside positive signals represent excitatory signals and negative signals represent inhibition we show that this model with exponential signal emission intervals poisson external signal arrivals and markovian signal movements between neurons has a product form leading to simple analytical expressions for the system state

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

stability of the random neural network model in a recent paper 1 we have introduced a new neural network model called the random network in which negative or positive signals circulate modelling inhibitory and excitatory signals they are summed at the input of each neuron and constitute its signal potential the state of each neuron in this model is its signal potential while the network state is the vector of signal potentials at each neuron if its potential is positive a neuron fires and sends out signals to the other neurons of the network or to the outside world as it does so its signal potential is depleted we have shown that in the markovian case this model has product form i e the steady state probability distribution of its potential vector is the product of the marginal probabilities of the potential at each neuron the signal flow equations of the network which describe the rate at which positive or negative signals arrive to each neuron are non linear so that their existence and uniqueness is not easily established except for the case of feedforward or backpropagation networks 1 we examine two sub classes of networks balanced and damped networks and obtain stability conditions in each case a hardware implementation of these networks is also suggested

indices in classical mechanics the authors discuss the linearization of the equations of motion around a periodic orbit in a conservative two degree of freedom hamiltonian system with an emphasis on computationally useful formulae indices are defined using singularities in projections conjugate points self conjugate points and the possibility of assigning a symbolic code are discussed

robust decoupling of discrete systems using spl lscr sup 1 optimization we address a control design problem related to decoupling namely robust decoupling the idea of robust decoupling is defined and shown to be equivalent to robust performance of a hadamard weighted spl lscr sup 1 performance measure a necessary and sufficient condition to determine if a compensator satisfies robust performance of a hadamard weighted spl lscr sup 1 performance measure in the presence of structured uncertainty is given this condition can be used to determine if a compensator robustly decouples a plant finally an iterative scheme for synthesizing a robust decoupling controller is discussed

approximation of stochastic processes by t s fuzzy systems fuzzy systems can provide us with universal approximation models of deterministic input output relationships but in the stochastic environment few achievements related to the subject have so far achieved in the paper a novel stochastic takagi sugeno t s fuzzy system is introduced to represent approximately existing randomness in many real world systems by recapitulating the general architecture of the stochastic t s fuzzy rule based system we analyze systematically approximating capability of the stochastic system to a class of stochastic processes by the canonical representation of the stochastic processes the stochastic fuzzy system is capable of with arbitrary accuracy providing the approximation to the stochastic processes in mean square sense finally an efficient algorithm for the stochastic t s fuzzy system is developed a simulation example demonstrates how a stochastic t s fuzzy system can be constructed to realize the given stochastic process approximately

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

classical chaotic scattering periodic orbits symmetries multifractal invariant sets the infinite set of periodic orbits of a chaotic system is investigated their globally exact symbolic dynamics is related to birkhoff s method of iterated symmetry lines which produces symmetric periodic orbits of various classes a complete picture emerges as to what part of the periodic orbits can be obtained by means of the birkhoff lines in addition the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

controlling transient chaos the method of ott grebogi and yorke 1990 is extended to control transient chaos the controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor the time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime the number of controlled trajectories however decreases the maximum perturbation according to a power law applicability to experimental situations and comparison with permanent chaotic cases are discussed

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

propagation of periodic and chaotic action potential trains along nerve fibers abstract in this paper we report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically first spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded then the data were divided into two groups and analyzed with different methods phase space representation spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency due to the serious influence of the measurement noise no reliable conclusion can be drawn from them for the second group of data of the interspike intervals isi which seem to convey more rich and important information nonlinear forecasting method the surrogate data and the plot of isi n 1 vs isi n were used in the analysis good results have been obtained which confirm with those from the cell model the largest lyapunov exponent lle were calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos

a simple solution to the l 1 optimization problem abstract it is pointed out in this brief paper that the l 1 optimization problem min q l qp 1 h u q v 1 h l mn 1 u l mq 1 v l pn 1 can be solved in one step rather than two the solution of the dual problem is obviated by the direct solution of the primal problem via linear programming the method here is applicable to finite dimensional problems or approximating finite dimensional problems in the general case

a numerical method for locating stable periodic orbits in chaotic systems abstract a numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented the method is based on the selection of trajectory segments which are characterized by a relatively low positive local lyapunov exponent once such a selection is made convergence to stable or weakly unstable periodic orbits is obtained by a newton method the algorithm is rather general and can be used for systems with more than two degrees of freedom the proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field in the latter case new stable periodic orbits are found in the region of strong chaotic motion the possible quantum localization on these orbits is discussed briefly

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

training the random neural network using quasi newton methods abstract training in the random neural network rnn is generally specified as the minimization of an appropriate error function with respect to the parameters of the network weights corresponding to positive and negative connections we propose here a technique for error minimization that is based on the use of quasi newton optimization techniques such techniques offer more sophisticated exploitation of the gradient information compared to simple gradient descent methods but are computationally more expensive and difficult to implement in this work we specify the necessary details for the application of quasi newton methods to the training of the rnn and provide comparative experimental results from the use of these methods to some well known test problems which confirm the superiority of the approach

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

entropy of hidden markov processes via cycle expansion hidden markov processes hmp is one of the basic tools of the modern probabilistic modeling the characterization of their entropy remains however an open problem here the entropy of hmp is calculated via the cycle expansion of the zeta function a method adopted from the theory of dynamical systems for a class of hmp this method produces exact results both for the entropy and the moment generating function the latter allows to estimate via the chernoff bound the probabilities of large deviations for the hmp more generally the method offers a representation of the moment generating function and of the entropy via convergent series

an m g 1 retrial g queue with preemptive resume and feedback under n policy subject to the server breakdowns and repairs an m g 1 retrial g queue with preemptive resume and feedback under n policy vacation subject to the server breakdowns and repairs is investigated in this paper arrivals of both positive customers and negative customers are two independent poisson processes and positive customers receive service immediately if the server is free upon their arrivals otherwise they may enter a retrial orbit and try their luck after a random time interval all positive customers must receive preliminary first phase of service and primary second phase of service while at the preliminary service the server may push out the customer undergoing such service to the orbit to commence preliminary service of an arriving positive customer negative customers not only remove the customer being in service but also make the server under repair the server leaves for an n policy vacation as soon as the system empties by applying the supplementary variables method we obtain the steady state solutions for both queueing measures and reliability quantities the effects of various parameters on the system performance are analyzed numerically

bridging genetic networks and queueing theory one of the main challenges facing biology today is the understanding of the joint action of genes proteins and rna molecules interwoven in intricate interdependencies commonly known as genetic networks to this end several mathematical approaches have been introduced to date in addition to developing the analytical tools required for this task anew one can utilize knowledge found in existing disciplines specializing in the representation and analysis of systems featuring similar aspects we suggest queueing theory as a possible source of such knowledge this discipline which focuses on the study of workloads forming in a variety of scenarios offers an assortment of tools allowing for the derivation of the statistical properties of the inspected systems we argue that a proper adaptation of modeling techniques and analytical methods used in queueing theory can contribute to the study of genetic regulatory networks this is demonstrated by presenting a queueing inspired model of a genetic network of arbitrary size and structure for which the probability distribution function is derived this model is further applied to the description of the lac operon regulation mechanism in addition we discuss the possible benefits stemming for queueing theory from the interdisciplinary dialogue with molecular biology in particular the incorporation of various dynamical behaviours into queueing networks

neural and delay based heuristics for the steiner problem in networks high speed networks such as the b isdn must be adequately equipped to handle multipoint communication in a fast and economical manner multicast applications include desktop video conferencing distance learning distributed database applications etc in networks employing the asynchronous transfer mode atm technology routing a multicast is achieved by constructing a tree that spans the source and all the destinations for the purpose of routing the network is modeled as a weighted undirected graph the graph theoretic solution is to find a minimum steiner tree for the graph given a set of destinations this formulation suffices for building multicast trees with a single optimization constraint as would be the xcase for best effort transport for real time traffic however it is necessary to ensure that the delay between the sender and each of the receivers is bounded in this case the network is modeled as an undirected graph where the edges have both a cost and a delay associated with them the graph theoretic solution is then to find a constrained minimum steiner tree such that the delay between the source and each of the destinations does not violate the specified bound both of these problems are np complete in this paper we review prior work on the multipoint routing problem and discuss the formulation of the unconstrained and constrained steiner problems we use the random neural network rnn to significantly improve the quality of trees found by the two existing best heuristics for finding steiner trees the minimum spanning tree heuristic and the average distance heuristic we also develop a new heuristic for finding delay constrained steiner trees experimental results are presented which show that the new heuristics improve significantly over existing ones

random neural networks with multiple classes of signals by extending the pulsed recurrent random neural network rnn discussed in gelenbe 1989 1990 1991 we propose a recurrent random neural network model in which each neuron processes several distinctly characterized streams of signals or data the idea that neurons may be able to distinguish between the pulses they receive and use them in a distinct manner is biologically plausible in engineering applications the need to process different streams of information simultaneously is commonplace e g in image processing sensor fusion or parallel processing systems in the model we propose each distinct stream is a class of signals in the form of spikes signals may arrive to a neuron from either the outside world exogenous signals or other neurons endogenous signals as a function of the signals it has received a neuron can fire and then send signals of some class to another neuron or to the outside world we show that the multiple signal class random model with exponential interfiring times