measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

on the set of imputations induced by the k additive core an extension to the classical notion of core is the notion of k additive core that is the set of k additive games which dominate a given game where a k additive game has its mobius transform or harsanyi dividends vanishing for subsets of more than k elements therefore the 1 additive core coincides with the classical core the advantages of the k additive core is that it is never empty once k geq 2 and that it preserves the idea of coalitional rationality however it produces k imputations that is imputations on individuals and coalitions of at most k individuals instead of a classical imputation therefore one needs to derive a classical imputation from a k order imputation by a so called sharing rule the paper investigates what set of imputations the k additive core can produce from a given sharing rule

secondary resonances and approximate models of routes to chaotic motion in non linear oscillators in two types of non linear oscillators with single equilibrium positions the question of transition to chaos is investigated by making use of a computer simulation and approximate analytical methods to study periodic solutions and their stability the zone of chaotic motion is found as a transition zone between qt periodic sub or subultraharmonic solution and the t periodic solution thus replacing the classical jump phenomenon two routes to chaos are studied the gentle one via a cascade of period doubling bifurcations which appears to be characteristic in unsymmetrical oscillators and a sharp route which proves to accompany a stability limit of a symmetrical periodic solution an approximate mathematical model of the pre chaotic motion is proposed

integrity measures quantifying the erosion of smooth and fractal basins of attraction abstract the phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors which typically include point equilibria periodic oscillations harmonic and subharmonic quasi periodic solutions and chaotic attractors each attractor is embedded in its own domain or basin of attraction bounded by a separatrix associated with an unstable saddle solution under the variation of a control parameter as the attractors move and bifurcate the basins also undergo corresponding changes and metamorphoses these changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes along with its basin at a saddle node bifurcation associated with the homoclinic tangling of the invariant manifolds of the saddle solution basin boundaries can also change in nature from smooth to fractal in this paper the escape of a driven oscillator from a cubic potential well as an archetypal example is used to explore the engineering significance of the basin erosions that occur under increased forcing various measures of engineering integrity of the constrained attractor are introduced a global measure assesses the overall basin area a local measure assesses the distance from the attractor to the basin boundary and a velocity measure is related to the size of impulse that could be sustained without failure since engineering systems may be subjected to pulse loads of finite duration attention is given to both the absolute and transient basin boudaries the significant erosion of these at homoclinic tangencies is particularly highlighted in the present study the fractal basins having a severely reduced integrity under all three criteria

an examination of initial condition maps for the sinusoidally excited buckled beam modeled by the duffing s equation abstract the initial condition problem is studied for the dynamics of the buckled beam by using the forced holmes duffing s equation in an attempt to understand the route to chaos earlier results generated by moon 1 for different parameter values are reconsidered and new techniques are used in order to understand more of the global behavior of the system in particular fractal basin boundaries are determined for two types of basins of attraction these are correlated with previous results to help provide an explanation for observed chaotic system behavior

ordinal criteria in stochastic multicriteria acceptability analysis smaa abstract we suggest a method for providing descriptive information about the acceptability of decision alternatives in discrete co operative group decision making problems the new smaa o method is a variant of the stochastic multicriteria acceptability analysis smaa smaa o is designed for problems where criteria information for some or all criteria is ordinal that is experts or decision makers have ranked the alternatives according to each ordinal criterion considerable savings can be obtained if rank information for some or all the criteria is sufficient for making decisions without significant loss of quality the approach is particularly useful for group decision making when the group can agree on the use of an additive decision model but only partial preference information or none at all is available

classifying efficient alternatives in smaa using cross confidence factors abstract stochastic multicriteria acceptability analysis smaa is a family of methods for aiding multicriteria group decision making these methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one the main results of the analysis are rank acceptability indices central weight vectors and confidence factors for different alternatives the rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative the central weight vectors represent the typical preferences favouring each alternative and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision in some cases when the problem involves a large number of efficient alternatives the analysis may fail to discriminate between them this situation is revealed by low confidence factors in this paper we develop cross confidence factors which are based on computing confidence factors for alternatives using each other s central weight vectors the cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives these sets are related to the concept of reference sets in data envelopment analysis dea but generalized for stochastic models forming these sets is useful when trying to identify one or more most preferred alternatives or suitable compromise alternatives the reference sets can also be used for evaluating whether criteria need to be measured more accurately and at which alternatives the measurements should be focused this may cause considerable savings in measurement costs we demonstrate the use of the cross confidence factors and reference sets using a real life example

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

stochastic multicriteria acceptability analysis using the data envelopment model abstract data envelopment analysis dea and stochastic multicriteria acceptability analysis smaa 2 are methods for evaluating alternatives based on multiple criteria while dea is mainly an ex post tool used for classifying alternatives into efficient and inefficient ones smaa 2 is an ex ante tool for supporting multiple criteria decision making both methods use a kind of value function where the importance of criteria is modeled using weights unlike many other methods neither dea nor smaa 2 requires decision makers weights as input instead these so called non parametric methods explore the weight space in order to identify weights favorable for each alternative this paper introduces the smaa d method which is a combination of dea and smaa 2 smaa d can be characterized as an extension of dea to handle uncertain or imprecise data to provide stochastic efficiency measures alternatively the combined method can be seen as a variant of smaa 2 with a dea type value function

quasiperiodic resonance and bifurcations of tori in the weakly nonlinear duffing oscillator abstract we study analytically and numerically the weakly nonlinear quasiconservative duffing oscillator under 2 periodic forcing utilizing the secondary averaging approach we derive approximate analytical expressions for the quasiperiodic response of the oscillator and for conditions of torus bifurcations the theoretical results are in good agreement with the numerical ones the structure of the bifurcation sets in the oscillator parameter space is similar to that of a one dimensional map with two extrema

experimental mimicry of duffing s equation abstract extensive analytical and numerical investigations have focused on duffing s equation however experimental work in a mechanics context has been limited to studying systems the stiffness characteristics of which can be approximated by a non linear cubic restoring force e g a buckled beam excited transversely or a rigid pendulum undergoing moderately large amplitude motion this work describes a novel experimental approach whereby a particle rigid body is contrived to mimic the behavior of duffing s equation this is a direct extension of the concept of a ball rolling on a double well potential energy surface both free and forced oscillations are considered illustrating familiar non linear dynamics features including competing steady state attractors hysteresis sensitivity to initial conditions subharmonic oscillations and chaos

multivariate gaussian criteria in smaa abstract we consider stochastic multicriteria decision making problems with multiple decision makers in such problems the uncertainty or inaccuracy of the criteria measurements and the partial or missing preference information can be represented through probability distributions in many real life problems the uncertainties of criteria measurements may be dependent however it is often difficult to quantify these dependencies also most of the existing methods are unable to handle such dependency information in this paper we develop a method for handling dependent uncertainties in stochastic multicriteria group decision making problems we measure the criteria their uncertainties and dependencies using a stochastic simulation model the model is based on decision variables and stochastic parameters with given distributions based on the simulation results we determine for the criteria measurements a joint probability distribution that quantifies the uncertainties and their dependencies we then use the smaa 2 stochastic multicriteria acceptability analysis method for comparing the alternatives based on the criteria distributions we demonstrate the use of the method in the context of a strategic decision support model for a retailer operating in the liberated european electricity market

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

implementing stochastic multicriteria acceptability analysis abstract stochastic multicriteria acceptability analysis smaa is a family of methods for aiding multicriteria group decision making in problems with inaccurate uncertain or missing information these methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one or that would give a certain rank for a specific alternative the main results of the analysis are rank acceptability indices central weight vectors and confidence factors for different alternatives the rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative the central weight vectors represent the typical preferences favouring each alternative and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision the computations in smaa require the evaluation of multidimensional integrals that must in practice be computed numerically in this paper we present efficient methods for performing the computations through monte carlo simulation analyze the complexity and assess the accuracy of the presented algorithms we also test the efficiency of these methods empirically based on the tests the implementation is fast enough to analyze typical sized discrete problems interactively within seconds due to almost linear time complexity the method is also suitable for analysing very large decision problems for example discrete approximations of continuous decision problems

on the transition from regular to chaotic behaviour in the duffing oscillator a numerical procedure is introduced to study the transition from regular periodic motion to chaotic behaviour in the duffing oscillator having a soft characteristics this method is based on the computation of very accurate high order approximate periodic solutions of galerkin s type in conjunction with the corresponding stability analysis which is performed by computing the eigenvalues of a certain matrix related to the first variational equations the results obtained by this numerical method are in good agreement with those found previously by huberman and crutchfield who used an analog computer to obtain the phase portraits

a method for stochastic multiple criteria decision making based on dominance degrees this paper proposes a method for solving the stochastic multiple criteria decision making smcdm problem where consequences of alternatives with respect to criteria are represented by random variables with probability distributions firstly definitions and related analysis of dominance degree of one probability distribution over another are given then by calculating the dominance degrees the dominance degree matrix of alternative pairwise comparisons with respect to each criterion is built further using promethee ii method an overall dominance degree matrix of alternative pairwise comparisons is constructed and a net flow of each alternative is calculated based on the obtained net flows a ranking of alternatives is determined finally numerical examples for the three cases are given to illustrate the use of the proposed method

ordinal data in multicriteria decision making a stochastic dominance approach to siting nuclear power plants abstract this paper addresses decision problems with ordinal data weights criterion scores a random sampling approach is proposed to generate quantitative values which are consistent with the underlying ordinal information an attractive feature of the approach is that it is applicable with mixed quantitative ordinal data another feature is that the approach can be extended to rankings with degrees of difference the outcome of the approach is a distribution of performance scores stochastic dominance concepts are proposed to arrive at a final ranking of alternatives an application of these procedures is given for a location study of nuclear power plants

k balanced games and capacities in this paper we present a generalization of the concept of balanced game for finite games balanced games are those having a nonempty core and this core is usually considered as the solution of the game based on the concept of k additivity we define the so called k balanced games and the corresponding generalization of core the k additive core whose elements are not directly imputations but k additive games we show that any game is k balanced for a suitable choice of k so that the corresponding k additive core is not empty for the games in the k additive core we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion moreover we look for necessary and sufficient conditions for a game to be k balanced for the general case it is shown that any game is either balanced or 2 balanced finally we treat the special case of capacities

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

pseudo criteria versus linear utility function in stochastic multi criteria acceptability analysis abstract stochastic multi criteria acceptability analysis smaa is a multi criteria decision support method for multiple decision makers dms in discrete problems smaa does not require explicit or implicit preference information from the dms instead the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one partial preference information can be represented in the weight space analysis through weight distributions in this paper we compare two variants of the smaa method using randomly generated test problems with 2 12 criteria and 4 12 alternatives in the original smaa a utility or value function models the dms preference structure and the inaccuracy or uncertainty of the criteria is represented by probability distributions in smaa 3 electre iii type pseudo criteria are used instead both methods compute for each alternative an acceptability index measuring the variety of different valuations that supports this alternative and a central weight vector representing the typical valuations resulting in this decision we seek answers to three questions 1 how similar are the results provided by the decision models 2 what kind of systematic differences exists between the models and 3 how could one select indifference and preference thresholds of the pseudo criteria model to match a utility model with given probability distributions

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

smaa 2 stochastic multicriteria acceptability analysis for group decision making stochastic multicriteria acceptability analysis smaa is a multicriteria decision support method for multiple decision makers in discrete problems in smaa the decision makers need not express their preferences explicitly or implicitly instead the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis in this paper we introduce the smaa 2 method which extends the original smaa by considering all ranks in the analysis in situations where the elitistic smaa may assess large acceptability only for extreme alternatives without sufficient majority support the more holistic smaa 2 analysis can be used to identify good compromise candidates the results are presented graphically we consider also situations where partial preference information is available we demonstrate the new method using a real life decision problem

chaotic phenomena triggering the escape from a potential well this paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold the aim is to show how the well known atoms of dissipative dynamics saddle node folds period doubling flips cascades to chaos boundary crises etc assemble to form molecules of overall response hierarchies of cusps incomplete feigenbaum trees etc particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle the latter being accurately predicted by a melnikov analysis after escape chaotic transients are shown to conform to recent scaling laws analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

on necessary and sufficient conditions for chaos to occur in duffing s equation an heuristic approach many different explanations have been offered for the occurrence of chaos in duffing s equation holmes has derived a criterion for the onset of chaos using melnikov theory that is based upon the initial entanglement of the unstable manifold with the stable limit cycle moon offered a criterion based on physical principles by considering how much energy a trajectory had to possess to get out of the potential well of one of the stable limit cycles in this paper conclusive numerical evidence is offered that in order for steady state chaos to occur in this system there must be a near intersection of the stable and unstable steady state limit cycles in the phase space of duffing s equation

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

doubling bifurcations of destroyed t 2 tori abstract two duffing oscillators driven with incommensurate frequencies and coupled additively are investigated with this system we observe a host of complex behavior which compete with each other in parameter space in particular the period doubling of an attracting t 2 torus and the destruction of a t 2 torus as the parameters are varied along a curve in parameter space we observe period doubling bifurcations of a destroyed t 2 torus this phenomenon is studied in greater detail via a simple recursive map

second order averaging and chaos in quasiperiodically forced weakly nonlinear oscillators abstract using the averaging method and melnikov s technique we study the chaotic dynamics of a class of quasiperiodically forced weakly nonlinear oscillators averaging is carried out to second order and the existence of transverse homoclinic tori resulting in chaotic dynamics for quasiperiodically forced oscillators is detected we also give an example for a duffing type weakly nonlinear oscillator subjected to combined parametric and external forcing in the case of double resonance in which primary external resonance and subharmonic or superharmonic parametric resonance exist simultaneously

smaa stochastic multiobjective acceptability analysis stochastic multiobjective acceptability analysis smaa is a multicriteria decision support technique for multiple decision makers based on exploring the weight space inaccurate or uncertain input data can be represented as probability distributions in smaa the decision makers need not express their preferences explicitly or implicitly instead the technique analyses what kind of valuations would make each alternative the preferred one the method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative a central weight vector representing the typical valuations resulting in that decision and a confidence factor measuring whether the input data is accurate enough for making an informed decision

power spectra of chaotic vibrations of a buckled beam abstract many of the strange attractors associated with low dimensional differential equations are known to contain transverse homoclinic orbits in particular the duffing equation with negative linear stiffness which models the behavior of a buckled beam subject to transverse excitation possesses such orbits on the assumption that the chaotic time history of a single variable in such an equation can be represented by the random superposition of deterministic structures each being a single excursion around a homoclinic orbit the power spectrum of the variable in question is predicted good agreement is demonstrated among our predictions numerical simulations of duffing s equation and experimental spectra obtained from a cantilever beam buckled by magnetic forces

applying young diagrams to 2 symmetric fuzzy measures with an application to general fuzzy measures abstract in this paper we apply young diagrams a well known object appearing in combinatorics and group representation theory to study some properties of the polytope of 2 symmetric fuzzy measures with respect to a given partition a subpolytope of the set of fuzzy measures the main result in the paper allows to build a simple and fast algorithm for generating points on this polytope in a random fashion besides we also study some other properties of this polytope as for example its volume in the last section we give an application of this result to the problem of identification of general fuzzy measures

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

on chaos and fractal behavior in a generalized duffing s system abstract in this paper three duffing s equations their attractors and their initial condition maps are examined for elements of universality in the literature all three equations have been examined briefly before but their behavior has only been incompletely described this paper more completely examines these system and answers important questions concerning the reason for chaotic system behavior

bispectral analysis of possessing chaotic motion abstract bispectral analyses of the equations producing a chaotic rossler attractor and the chaotic trajectories of a one mode galerkin approximation of the magnetically buckled beam are presented yielding interesting consequences concerning chaotic system identification for driven oscillators and autonomous systems bicoherence spectra isolate the phase coupling between increasing numbers of triads of fourier modes for a pure period doubling sequence route to chaos for the rossler equations bicoherences from a period doubling intermittency catastrophe route to chaotic motion are also observed for single frequency excitation for the buckled beam although the bicoherence successfully characterizes important aspects of the non linearity of the rossler attractor in the nonchaotic and chaotic regimes where the primary non linearity is quadratic it only partially characterizes the magnetically buckled beam system because the dominant non linearity is cubic

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions