an investigation into the characteristics of non linear frequency response functions part 1 understanding the higher dimensional frequency spaces the characteristics of generalized frequency response functions gfrfs of non linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques it is shown how a systematic analysis can be achieved for a wide class of non linear systems in the frequency domain using the proposed methods the paper is divided into two parts in part 1 the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi dimensional frequency spaces the visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time in part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non linear systems described by the narmax class of models case studies are included in part 2 to illustrate the application of the new methods

can one make any crash prediction in finance using the local hurst exponent idea abstract we apply the hurst exponent idea for investigation of djia index time series data the behavior of the local hurst exponent prior to drastic changes in financial series signal is analyzed the optimal length of the time window over which this exponent can be calculated in order to make some meaningful predictions is discussed our prediction hypothesis is verified with examples of 1929 and 1987 crashes as well as with more recent phenomena in stock market from the period 1995 to 2003 some interesting agreements are found

characterization of large price variations in financial markets statistics of drawdowns loss from the last local maximum to the next local minimum plays an important role in risk assessment of investment strategies as they incorporate higher two order correlations they offer a better measure of real market risks than the variance or other cumulants of daily or some other fixed time scale of returns previous results have shown that the vast majority of drawdowns occurring on the major financial markets have a distribution which is well represented by a stretched exponential while the largest drawdowns are occurring with a significantly larger rate than predicted by the bulk of the distribution and should thus be characterized as outliers eur phys j b 1 1998 141 j risk 2001 in the present analysis the definition of drawdowns is generalized to coarse grained drawdowns or so called e drawdowns and a link between such e outliers and preceding log periodic power law bubbles previously identified quantitative finance 1 2001 452 is established

log periodic crashes revisited we revisit the finding that crashes can be deterministic and governed by log periodic formulas d sornette a johansen significance of log periodic precursors to financial crashes quant finance 1 2001 452 471 d sornette w x zhou the us 2000 2002 market descent how much longer and deeper quant finance 2 2002 468 481 one and two harmonic equations are usually employed to fit daily data during bubble episodes but a three harmonics has been shown to fit anti bubbles a johansen d sornette financial anti bubbles log periodicity in gold and nikkei collapses int j mod phys c 10 1999 563 575 here we show that the three harmonic formula can work for bubble episodes as well as anti bubbles this is illustrated with daily data from the brazilian real us dollar exchange rate and we also show that the three harmonics can fit an intraday data set from that foreign exchange rate

gain bounds of higher order nonlinear transfer functions bounds on the gain for high order nonlinear transfer functions of a wide class of nonlinear systems are derived it is shown that the gain bounds can be expressed explicitly in terms of the coefficients of discrete time nonlinear models the bounds can be used to check the convergence of the transfer function series and hence to determine the truncation order in practical applications

output frequencies of nonlinear systems an explicit expression for the output frequency range of nonlinear systems for general inputs is derived this reveals the relationship between the input and output frequency ranges of a nonlinear system and extends the well known concept in linear systems where the output frequency range is exactly the same as that of the corresponding input the new results provide an important and basic relationship that can be applied in nonlinear system frequency domain analysis and design

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

a digital method of modeling quadratically nonlinear systems with a general random input without assuming particular statistics of the input a practical digital method of estimating linear and quadratic transfer functions of a nonlinear time invariant system that can be described by volterra series of up to second order is presented the method is tested and validated by analyzing input output data of a known quadratically nonlinear system it is used when there is little knowledge about the input statistics or the input is non gaussian it is also noted that the ordinary coherence functions cannot be used in explaining the input output power transfer relationship of a quadratic system excited by a non gaussian input signal with respect to the practical application of the method the relationship between the mean square errors involved in the transfer function estimates and the number of averages taken from the spectral estimation is qualitatively discussed

frequency response functions for nonlinear rational models a recursive algorithm which maps a general class of nonlinear rational model defined as the ratio of two polynomial functions into the frequency domain is derived using the harmonic expansion method the new algorithm provides for the first time a direct analytic map from the time domain rational model parameters to the higher order frequency response functions complex nonlinear time domain behaviours can be analysed and interpreted in the frequency domain and simulated examples are included to illustrate the concepts involved

the reaction of stock markets to crashes and events a comparison study between emerging and mature markets using wavelet transforms abstract we study here the behaviour of the first three eigenvalues 1 2 3 and their ratios 1 2 1 3 2 3 of the covariance matrices of the original return series and of those rebuilt from wavelet components for emerging and mature markets it has been known for some time that the largest eigenvalue 1 contains information on the risk associated with the particular assets of which the covariance matrix is comprised here we wish to ascertain whether the subdominant eigenvalues 2 3 hold information on the risk of the stock market and also to measure the recovery time for emerging and mature markets to do this we use the discrete wavelet transform which gives a clear picture of the movements in the return series by reconstructing them using each wavelet component our results appear to indicate that mature markets respond to crashes differently to emerging ones in that emerging markets may take up to two months to recover while major markets take less than a month to do so in addition the results appears to show that the subdominant eigenvalues 2 3 give additional information on market movement especially for emerging markets and that a study of the behaviour of the other eigenvalues may provide insight on crash dynamics

viscoelastic behavior of stock indices abstract the scattering diagram of a stock index results in a complex network structure which can be used to analyze the viscoelastic properties of the index the change along x or y direction of the diagram corresponds to purely elastic or spring like movement whereas the diagonal change at an angle of 45 corresponds to purely viscous or dashpot like movement the viscous component pushes the price from its current value to any other value while the elastic component acts like a restoring force four indices namely dji s p 500 nasdaq 100 and nasdaq composite were studied for the period of 2001 2009 nasdaq composite displayed very high elasticity while nasdaq 100 displayed the highest fluidity in the time period considered the fluidity of dji and s p 500 came out to be close to each other and they are almost the same in the second half of the period

a hierarchical model of financial crashes abstract we follow up our previous conjecture that large stock market crashes are analogous to critical points in statistical physics the term critical refers to regimes of cooperative behavior such as magnetism at the curie temperature and liquid gas transitions and is characterized by the singular mathematical behavior of relevant observables to illustrate the concept of criticality we present a simple hierarchical model of traders exhibiting crowd behavior and show that it has a well defined critical point whose mathematical signature is a power law dependence of the price modulated by log periodic structures as recently found in market data by several independent groups

application of higher order spectral analysis to cubically nonlinear system identification in this study digital higher order spectral analysis and frequency domain volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time invariant systems up to third order the primary focus is on consideration of random excitation of nonlinear systems and thus the approach makes extensive use of higher order spectral analysis to determine the frequency domain volterra kernels which correspond to linear quadratic and cubic transfer functions although the volterra model is nonlinear in terms of its input it is linear in terms of its unknown transfer functions thus a least squares approach is used to determine the optimal in a least squares sense set of linear quadratic and cubic transfer functions of particular practical note is the fact that the approach of this paper is valid for non gaussian as well as gaussian random excitation it may also be utilized for multitone inputs the complexity of the problem addressed in this paper arises from two principal causes 1 the necessity to work in a 3d frequency space to describe cubically nonlinear systems and 2 the necessity to characterize the non gaussian random excitation by computing higher order spectral moments up to sixth order a detailed description of the approach used to determine the nonlinear transfer functions including considerations necessary for digital implementation is presented

filtrations on the knot contact homology of transverse knots we construct a new invariant of transverse links in the standard contact structure on mathbb r 3 this invariant is a doubly filtered version of the knot contact homology differential graded algebra dga of the link see ekholm et al knot contact homology arxiv 1109 1542 2011 ng duke math j 141 2 365 406 2008 here the knot contact homology of a link in mathbb r 3 is the legendrian contact homology dga of its conormal lift into the unit cotangent bundle s mathbb r 3 of mathbb r 3 and the filtrations are constructed by counting intersections of the holomorphic disks of the dga differential with two conormal lifts of the contact structure we also present a combinatorial formula for the filtered dga in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links

inhomogeneous scaling behaviors in malaysian foreign currency exchange rates in this paper we investigate the fractal scaling behaviors of foreign currency exchange rates with respect to malaysian currency ringgit malaysia these time series are examined piecewise before and after the currency control imposed in 1st september 1998 using the monofractal model based on fractional brownian motion the global hurst exponents are determined using the r s analysis the detrended fluctuation analysis and the method of second moment using the correlation coefficients the limitation of these monofractal analyses is discussed the usual multifractal analysis reveals that there exists a wide range of hurst exponents in each of the time series a new method of modelling the multifractal time series based on multifractional brownian motion with time varying hurst exponents is studied

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

frequency domain analysis for non linear volterra systems with a general non linear output function for non linear volterra systems which include a non linear state equation and a general non linear output function the system frequency response functions and some related frequency response characteristics are developed and discussed in this study these new results establish the frequency response functions for this general form of non linear systems by extending some existing theory and provide an analytical insight into the relationship between model parameters and frequency response functions and the relationship between model parameters and the magnitude bound of frequency response functions several examples are given to illustrate the new results

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

a complete knot invariant from contact homology we construct an enhanced version of knot contact homology and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup in particular it completely determines a knot up to smooth isotopy the enhancement consists of the fully noncommutative legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere along with a product on a filtered part of this homology as a corollary we obtain a new holomorphic curve proof of a result of the third author that the legendrian isotopy class of the conormal torus is a complete knot invariant

a novel characteristic parameter approach for analysis and design of linear components in nonlinear systems nonlinearity always brings difficulties to the analysis and design of practical systems this study shows that for a wide class of nonlinear systems described by nonlinear differential equations the nonlinear output spectrum can be expressed as a polynomial function with respect to linear model parameters based on the recently developed concepts of parametric characteristic analysis and nonlinear characteristic output spectrum ncos function the coefficients of the new ncos function are independent of the linear parameters of interest and can be directly determined with proposed procedures this reveals an explicit and analytical relationship between the output spectrum of a nonlinear system and its linear components and therefore provides a convenient tool for the analysis and design of nonlinear systems using only linear components or linear feedback control while simultaneously fully considering inherent nonlinear influence this result is a useful complement to the ncos function based analysis and design recently developed in the literature the effectiveness and usefulness of the results are demonstrated through an application to the analysis of an amplifier with a multi tone input for suppressing intermodulation distortion it is shown that the new results reveal a straightforward insight into the parametric optimization of the amplifier for suppression of intermodulation distortion

a duality exact sequence for legendrian contact homology we establish a long exact sequence for legendrian submanifolds l p r where p is an exact symplectic manifold which admit a hamiltonian isotopy that displaces the projection of l to p off of itself in this sequence the singular homology h maps to linearized contact cohomology ch which maps to linearized contact homology ch which maps to singular homology in particular the sequence implies a duality between ker ch h and ch im h furthermore this duality is compatible with poincare duality in l in the following sense the poincare dual of a singular class which is the image of a ch maps to a class ch such that a 1 the exact sequence generalizes the duality for legendrian knots in r3 see 26 and leads to a refinement of the arnold conjecture for double points of an exact lagrangian admitting a legendrian lift with linearizable contact homology first proved in 7

combinatorial knot contact homology and transverse knots abstract we give a combinatorial treatment of transverse homology a new invariant of transverse knots that is an extension of knot contact homology the theory comes in several flavors including one that is an invariant of topological knots and produces a three variable knot polynomial related to the a polynomial we provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots including knots that cannot be distinguished by the heegaard floer transverse invariants or other previous invariants

nonlinear influence in the frequency domain alternating series the nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have volterra series expansion it is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities the magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series sufficient and necessary conditions in which the output spectrum can be cast into an alternating series are studied these results reveal a novel frequency domain insight into the nonlinear influence on a system and provide a new method for the analysis and design of nonlinear systems in the frequency domain examples are given to illustrate the results

legendrian contact homology in p times mathbb r a rigorous foundation for the contact homology of legendrian submanifolds in a contact manifold of the form p x r where p is an exact symplectic manifold is established the class of such contact manifolds includes 1 jet spaces of smooth manifolds as an application contact homology is used to provide smooth isotopy invariants of submanifolds ofr and more generally invariants of self transverse immersions into r n up to restricted regular homotopies when n 3 this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of ng

finite time singularity in the dynamics of the world population economic and financial indices contrary to common belief both the earth s human population and its economic output have grown faster than exponential i e in a super malthusian mode for most of the known history these growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052 10 signaling an abrupt transition to a new regime the degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in 1970 i e about 80 years before the predicted singular time corresponding to approximately 4 of the studied time interval over which the acceleration is documented this rounding off of the finite time singularity is probably due to a combination of well known finite size effects and friction and suggests that we have already entered the transition region to a new regime as theoretical support a multivariate analysis coupling population capital r d and technology shows that a dramatic acceleration in the population growth during most of the timespan can occur even though the isolated dynamics do not exhibit it possible scenarios for the cross over and the new regime are discussed

a bound for the magnitude characteristics of nonlinear output frequency response functions part 1 analysis and computation a bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs it is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included the paper is divided into two parts in part 1 an expression for the output magnitude bound is derived properties of the result are discussed and general procedures for the practical computation of the bound are developed in part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed

nonlinear system analysis and identification from random data 1991 nonlinear system analysis and identification from random data technometrics vol 33 no 4 pp 482 483

characterizing abrupt changes in the stock prices using a wavelet decomposition method abstract abrupt changes in the stock prices either upwards or downwards are usually preceded by an oscillatory behavior with frequencies that tend to increase as the moment of transition becomes closer the wavelet decomposition methods may be useful for analysis of this oscillations with varying frequencies because they provide simultaneous information on the frequency scale and localization in time translation however in order to use the wavelet decomposition certain requirements have to be satisfied so that the linear and cyclic trends are eliminated by standard least squares techniques the coefficients obtained by the wavelet decomposition can be represented in a graphical form a threshold can then be established to characterize the likelihood of a short time abrupt change in the stock prices actual data from the sao paulo stock exchange bolsa de valores de sao paulo were used in this work to illustrate the proposed method

generalized frequency response function matrix for mimo non linear systems recursive algorithms are derived to compute the generalized frequency response function matrix of multi input multioutput mimo non linear systems as an analytical map from both non linear differential equation models and narx non linear auto regressive models with exogenous inputs models of the system the algorithm is computationally compact and exposes the explicit relationship between the model parameters and the elements of the generalized frequency response function matrix and can thus provide important insights into the behaviour of non linear systems

a method for detection of abrupt changes in the financial market combining wavelet decomposition and correlation graphs abstract the objective of this work is to propose a new methodology to detect the imminence of abrupt changes in the stock market by combining a numerical indicator based on the wavelet decomposition technique with a measure of the interdependency of the markets using graph theory while the indicator based on wavelet decomposition is based on a single time series an approach based on network representation can provide information on the interdependency of the various markets more specifically the stock market indices are associated with nodes of a network and the correlation between pairs of nodes with links results from the theory of graphs can then be used to indicate numerically the connectivity of this network experimentations with a variety of financial time series shows that the connectivity varies as trends of the financial time series varies combining the indicator based on the wavelet decomposition with the proposed measure of the connectivity of the network it was possible to refine the authors previous results in terms of detecting abrupt changes in the stock market in order to illustrate the methodology a case study involving twelve stock market indices was presented

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

an investigation into the characteristics of non linear frequency response functions part 2 new analysis methods based on symbolic expansions and graphical techniques in part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi dimensional frequency spaces the visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented in this the second part symbolic expansion techniques are introduced and together with the results achieved in part 1 yield new methods for analysing the properties of generalized frequency response functions case studies are included to illustrate the application of the new methods

optimal investment horizons for stocks and markets abstract the inverse statistics is the distribution of waiting times needed to achieve a predefined level of return obtained from detrended historic asset prices i simonsen m h jensen a johansen eur phys j 27 2002 583 m h jensen a johansen i simonsen physica a 234 2003 338 such a distribution typically goes through a maximum at a time coined the optimal investment horizon which defines the most likely waiting time for obtaining a given return by considering equal positive and negative levels of return we reported in m h jensen a johansen i simonsen physica a 234 2003 338 on a quantitative gain loss asymmetry most pronounced for short horizons in the present paper the inverse statistics for 2 3 of the individual stocks presently in the djia is investigated we show that this gain loss asymmetry established for the djia surprisingly is not present in the time series of the individual stocks nor their average this observation points towards some kind of collective movement of the stocks of the index synchronization

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples