infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

scale free animal movement patterns levy walks outperform fractional brownian motions and fractional levy motions in random search scenarios the movement patterns of a diverse range of animals have scale free characteristics these characteristics provide necessary but not sufficient conditions for the presence of movement patterns that can be approximated by levy walks nevertheless it has been widely assumed that the occurrence of scale free animal movements can indeed be attributed to the presence of levy walks this is in part because it is known that the super diffusive properties of levy walks can be advantageous in random search scenarios when searchers have little or no prior knowledge of target locations however fractional brownian motions fbms and fractional levy motions flms are both scale free and super diffusive and so it is possible that these motions rather than levy walks underlie some or all occurrences of scale free animal movement patterns here this possibility is examined in numerical simulations through a determination of the searching efficiencies of fbm and flm searches it is shown that these searches are less efficient than levy walk searches this finding does not rule out the possibility that some animals with scale free movement patterns are executing fbm and flm searches but it does make levy walk searches the more likely possibility

sil nikov saddle node interaction near a codimension 2 bifurcation laser with injected signal abstract we describe a sil nikov saddle node interaction in the proximity of a hopf saddle node bifurcation both from numerical and theoretical viewpoints inspired by numerical observations on a three dimensional model of a laser with injected signal we describe the interaction near the codimension 2 point using a return map controlled by the parameters of the bifurcation the theoretical predictions compare well with numerical experiments finally we discuss the reasons for the existence of the sil nikov saddle node bifurcation in the laser with injected signal and correspondingly the characteristics of other systems that might display this bifurcation

fractal analysis of the short time series in a visibility graph method abstract the aim of this study is to evaluate the performance of the visibility graph vg method on short fractal time series in this paper the time series of fractional brownian motions fbm characterized by different hurst exponent h are simulated and then mapped into a scale free visibility graph of which the degree distributions show the power law form the maximum likelihood estimation mle is applied to estimate power law indexes of degree distribution and in this progress the kolmogorov smirnov ks statistic is used to test the performance of estimation of power law index aiming to avoid the influence of droop head and heavy tail in degree distribution as a result we find that the mle gives an optimal estimation of power law index when ks statistic reaches its first local minimum based on the results from ks statistic the relationship between the power law index and the hurst exponent is reexamined and then amended to meet short time series thus a method combining vg mle and ks statistics is proposed to estimate hurst exponents from short time series lastly this paper also offers an exemplification to verify the effectiveness of the combined method in addition the corresponding results show that the vg can provide a reliable estimation of hurst exponents

successive homoclinic tangencies to a limit cycle the dynamics near a perturbed degenerate homocfinic connection to a periodic orbit in three dimensions is modeled by a two parameter map one parameter controls the passage of the manifolds of the orbit through one another and the other breaks the degeneracy and causes the manifolds to intersect transversely an analysis of the map recovers the results of gaspard and wang 1987 relating to the accumulation of saddle node bifurcations of periodic orbits on a single homoclinic tangency and in addition shows that the local behavior of these orbits at the two tangencies can be linked together giving closed loops in period versus parameter plots these analytic results are then compared with numerical results from a three dimensional of ordinary differential equations

shape of growth rate distribution determines the type of non gibrat s property abstract in this study the authors examine exhaustive business data on japanese firms which cover nearly all companies in the mid and large scale ranges in terms of firm size to reach several key findings on profits sales distribution and business growth trends here profits denote net profits first detailed balance is observed not only in profits data but also in sales data furthermore the growth rate distribution of sales has wider tails than the linear growth rate distribution of profits in log log scale on the one hand in the mid scale range of profits the probability of positive growth decreases and the probability of negative growth increases symmetrically as the initial value increases this is called non gibrat s first property on the other hand in the mid scale range of sales the probability of positive growth decreases as the initial value increases while the probability of negative growth hardly changes this is called non gibrat s second property under detailed balance non gibrat s first and second properties are analytically derived from the linear and quadratic growth rate distributions in log log scale respectively in both cases the log normal distribution is inferred from non gibrat s properties and detailed balance these analytic results are verified by empirical data consequently this clarifies the notion that the difference in shapes between growth rate distributions of sales and profits is closely related to the difference between the two non gibrat s properties in the mid scale range

characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments we study correlation properties of the magnitude and the sign of the increments in the time intervals between successive heartbeats during light sleep deep sleep and rapid eye movement rem sleep using the detrended fluctuation analysis method we find short range anticorrelations in the sign time series which are strong during deep sleep weaker during light sleep and even weaker during rem sleep in contrast we find long range positive correlations in the magnitude time series which are strong during rem sleep and weaker during light sleep we observe uncorrelated behavior for the magnitude during deep sleep since the magnitude series relates to the nonlinear properties of the original time series while the sign series relates to the linear properties our findings suggest that the nonlinear properties of the heartbeat dynamics are more pronounced during rem sleep thus the sign and the magnitude series provide information which is useful in distinguishing between the sleep stages

on the first passage time and leapover properties of levy motions we investigate two coupled properties of levy stable random motions the first passage times fpts and the first passage leapovers fpls while in general the fpt problem has been studied quite extensively the fpl problem has hardly attracted any attention considering a particle that starts at the origin and performs random jumps with independent increments chosen from a levy stable probability law x the fpt measures how long it takes the particle to arrive at or cross a target the fpl addresses a different question given that the first passage jump crosses the target then how far does it get beyond the target these two properties are investigated for three subclasses of levy stable motions i symmetric levy motions characterized by levy index 0 2 and skewness parameter 0 ii one sided levy motions with 0 1 1 and iii two sided skewed levy motions the extreme case 1 2 1

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

extreme value statistics in records with long term persistence many natural records exhibit long term correlations characterized by a power law decay of the autocorrelation function c s ensuremath sim s ensuremath ensuremath gamma with time lag s and correlation exponent 0l ensuremath gamma l1 we study how the presence of such correlations affects the statistics of the extreme events i e the maximum values of the signal within time segments of the fixed duration r we find numerically that i the integrated distribution function of the maxima converges to a gumbel distribution for large r similar to uncorrelated signals ii the deviations for finite r depend on the initial distribution of the records and on their correlation properties iii the maxima series exhibit long term correlations similar to those of the original data and most notably iv the maxima distribution as well as the mean maxima significantly depend on the history in particular on the previous maximum the last item implies that conditional mean maxima and conditional maxima distributions with the value of the previous maximum as condition should be considered for an improved extreme event prediction we provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long term correlated records

spectral measure of the harmonic oscillator with α stable noise we study the coordinate velocity couple of the harmonic oscillator with stable noise as previously shown by sokolov et al the distribution of this couple is bivariate stable in this work we determine explicitly its associated spectral measure exhibiting directly both the non independence and non ellipticity of the coordinate velocity couple knowledge of the spectral measure allows to analyze and quantify the deviation from ellipticity

permutation entropy and detrend fluctuation analysis for the natural complexity of cardiac heart interbeat signals abstract we compute fractal dimension and permutation entropy for healthy and people who have experienced heart failure our result shows that permutation entropy is a suitable approach as well as detrend fluctuation analysis dfa the result of dfa shows that the fractal dimensions for healthy and heart failure are different as well as the permutation entropy result the fluctuation value for permutation entropy for an individual who has experienced heart failure is bigger than for a healthy person there is some specific change in the interbeat signal of a person who has experienced heart failure but there is not previous trend for a healthy person

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

scale invariance and universality organizing principles in complex systems this paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the eld of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on a broad class of complex phenomena we shall see that while scale invariance has been tested for many years universality is relatively more rarely discussed in particular we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems c 2000 elsevier science b v all rights reserved

equivalent low order model for a nonlinear diffusion equation abstract we present an equivalent low order model for a simple pde system that exhibits interesting low dimensional dynamics with a rich variety of homoclinic phenomena and whose effective dimension may be gradually increased by means of system parameters the system is a linear heat equation subject to a nonlinear and nonlocal boundary condition and the reduction procedure is based on a finite element method we will show that both the pde and ode systems have indentical stationary solution with a very similar linear stability behaviour and exhibit also very similar dynamics at least within parameter ranges corresponding to physical devices

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

long range dependencies in heart rate signals revisited abstract the arguments are given that local exponents obtained in multifractal analysis by two methods wavelet transform modulus maxima wtmm and multifractal detrended fluctuation analysis mdfa allow to separate statistically hearts of healthy people and subjects suffering from reduced left ventricle systolic function nyha i iii class proposed indices of fractality suggest that a signal of human heart rate is a mixture of two processes monofractal and multifractal ones

the gradually truncated levy flight for systems with power law distributions abstract power law distributions have been observed in various economical and physical systems levy flights have infinite variance which discourage a physical approach we introduce a class of stochastic processes the gradually truncated levy flight in which large steps of a levy flight are gradually eliminated it has finite variance and the system can be analyzed in a closed form we applied the present method to explain the distribution of a particular economical index the present method can be applied to describe time series in a variety of fields i e turbulent flow anomalous diffusion polymers etc

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

center manifolds normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions abstract we study the instabilities known to aeronautical engineers as flutter and divergence mathematically these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation making use of the center manifold and normal form theorems we concentrate on the situation in which flutter and divergence become coupled and show that there are essentially two ways in which this is likely to occur in the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator this system which may be analyzed by conventional phase plane techniques captures all the qualitative features of the full system we discuss the reduction and show how the nonlinear terms may be simplified and put into normal form invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics repeating the approach in the second case we show that the essential model is now three dimensional and that far more complex behavior is possible including nonperiodic and chaotic motions throughout we take a two degree of freedom system as an example but the general methods are applicable to multi and even infinite degree of freedom problems

assessment of the relative ratio of correlated and uncorrelated waiting times in the southern california earthquakes catalogue abstract in the present study we investigated the interevent time interval distribution of earthquakes in southern california we analyzed and compared datasets of waiting times between consecutive earthquakes and time structure distorted datasets the aim of this study was to determine the proportion of waiting time values in the original catalogue that can be regarded as statistically distinguishable or indistinguishable from the baseline time intervals datasets where the original structure of the temporal distribution of earthquakes was disrupted we compiled two types of time structure distorted baseline sequences which comprised mean values of a shuffled original interevent data and b interevent times data from time randomized catalogues to compare the dynamical characteristics of the original and time structure distorted baseline sequences we used several data analysis methods such as power spectrum regression detrended fluctuation and multifractal detrended fluctuation analysis we also tested the nonlinear structure of the original and baseline sequences using the magnitude and sign scaling analysis method we calculated the z s c o r e in order to assess whether the interevent time values in the original dataset shared statistical similarity or dissimilarity with the time structure distorted baseline interevent data sequence we compared the interevent time values in the original dataset with the mean baseline interevent times computed for two types of time structure distorted sequences the results showed that about 30 of the original interevent times were statistically indistinguishable from the mean of the shuffled dataset and 10 from the mean of the time structure distorted baseline interevent dataset we performed similar analyses for other catalogues obtained from different parts of the world according to the results of this analysis the proportion of events in the original catalogues that were indistinguishable from sequences with disturbed time structure did not contradict the results obtained for the southern california catalogue

directed transport induced by spatially modulated levy flights in this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative stable levy noise in the limit of a noise amplitude we compare the dynamics for ito and marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells a detailed analysis of the noise induced current is performed for the seebeck ratchet with a weak multiplicative noise for

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

the level crossing and inverse statistic analysis of german stock market index dax and daily oil price time series a b s t r a c t the level crossing and inverse statistics analysis of dax and oil price time series are given we determine the average frequency of positive slope crossings where t 1 is the average waiting time for observing the level again we estimate the probability p k which provides us the probability of observing k times of the level with positive slope in time scale t for analyzed time series we found that maximum k is about 6 we show that by using the level crossing analysis one can estimate how the dax and oil time series will develop we carry out the same analysis for the increments of dax and oil price log returns which is known as inverse statistics and provide the distribution of waiting times to observe some level for the increments 2011 elsevier b v all rights reserved stochastic processes occur in many natural and man made phenomena ranging from various indicators of economic activities in the stock market velocity fluctuations in turbulent flows and heartbeat dynamics etc 1 the level crossing analysis of stochastic processes has been introduced by rice 1944 1945 2 26 and used to describe the turbulence 16 rough surfaces 27 stock markets 28 burgers turbulence and kardar parisi zhang equation 29 30 the level crossing analysis of the data set has the advantage that it gives important global properties of the time series and do not need the scaling feature the almost of the methods in time series analysis are using the scaling features of time series and their applications are restricted to the time series with scaling properties our goal with the level crossing analysis is to characterize the statistical properties of the data set with the hope to better understand the underlying stochastic dynamics and provide a possible tool to estimate its dynamics the level crossing and inverse statistics analysis can be viewed as the complementary method to the other well known methods such as detrended fluctuation analysis dfa 31 detrended moving average dma 32 wavelet transform modulus maxima wtmm 33 rescaled range analysis r s 34 scaled windowed variance swv 35 langevin dynamics 36 detrended cross correlation analysis 37 multifactor analysis of

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

possible sets of autocorrelations and the simplex algorithm the problem of imposing a set of correlations of any order on binary sequences is addressed the entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle point method and it was observed that a finite fraction of sets are characterized by a non extensive entropy in this paper the region of finite entropy the allowed region of sets of correlations is found to be a convex hyper polygon in the space of correlation sets using the simplex algorithm outside of this region the simplex solution indicates that sequences obeying the correlations cannot be found therefore the entropy is in particular the boundaries of the allowed region for c1 cm are presented at the boundaries the entropy drops in a first order phase transition fashion and this drop can be explained from a combinatorial point of view finally we observe that the fraction of the volume occupied by allowed correlation sets drops exponentially with the number of correlations imposed and a qualitative explanation of this scaling phenomenon is provided

scale invariant fluctuations at different levels of organization in developing heart cell networks in the last few years it has been realized that the intervals of spontaneous spiking events in the intact heart exhibit coexisting scale invariant count fluctuations and anticorrelated interspike intervals fluctuations here we show experimentally that this feature is an intrinsic property of single isolated heart cells which is preserved when the cells couple into networks we present a model explaining this behavior at both the single cell and network levels

time evolution of stochastic processes with correlations in the variance stability in power law tails of distributions abstract we model the time series of the s p500 index by a combined process the ar garch process where ar denotes the autoregressive process which we use to account for the short range correlations in the index changes and garch denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long range correlations in the variance we study the ar garch process with an initial distribution of truncated levy form we find that this process generates a new probability distribution with a crossover from a levy stable power law to a power law with an exponent outside the levy range beyond the truncation cutoff we analyze the sum of n variables of the ar garch process and find that due to the correlations the ar garch process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n a feature which is observed in the probability distribution of the s p500 index we find that this power law stability depends on the characteristic scale in the correlations we also find that inclusion of short range correlations through the ar process is needed to obtain convergence to a limiting gaussian distribution for large n as observed in the data

a special adi model for the laplace tidal equations abstract an efficient method for numerical integration of the time dependent laplace tidal equations for a flat or curved earth is described the method is implicit uses alternating directions in two space dimensions with time staggering and a split sea height variable some special auxiliary features of the method are described and stability is discussed

correlation cascades of levy driven random processes abstract we explore the correlation structure of a large class of random processes driven by non gaussian levy noise sources with possibly infinite variances examples of such processes include levy motions levy driven ornstein uhlenbeck motions levy driven moving average processes fractional levy motions and fractional levy noises based on the fact that non gaussian levy noises are continuum superpositions of poisson noises we unveil an underlying cascade of levy correlation functions which characterize the process distribution and the correlation structure of the processes under consideration in the case where the driving levy noise sources are fractal the resulting cascade admits a unique scale free form

singularities on the boundary of the stability domain near 1 1 resonance abstract we study the linear differential equation x l x in 1 1 resonance that is x r 4 and l is 4 4 matrix with a semi simple double pair of imaginary eigenvalues i i i i we wish to find all perturbations of this linear system such that the perturbed system is stable since linear differential equations are in one to one correspondence with linear maps we translate this problem to gl 4 r in this setting our aim is to determine the stability domain and the singularities of its boundary the dimension of gl 4 r is 16 therefore we first reduce the dimension as far as possible here we use a versal unfolding of l i e a transverse section of the orbit of l under the adjoint action of gl 4 r repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4 a 3 sphere in this 4 dimensional space contains all information about the neighborhood of l in gl 4 r considering the 3 sphere as two 3 discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids one in each 3 disc the singularities of this surface are transverse self intersections whitney umbrellas and an intersection of self intersections where the surface has a self tangency a whitney stratification of the 3 sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of l and in particular identify the stability domain

the entwined wiggling of homoclinic curves emerging from saddle node hopf instabilities abstract an analysis is presented of the homoclinic bifurcations occurring in a generic unfolding of a saddle node hopf singularity also known as a gavrilov guckenheimer point specifically an explanation is given of previously numerically observed oscillations of loci of homoclinic orbits to two different saddle focus equilibria these oscillations occur within an exponentially thin wedge of parameter space that emerges from the codimension two point the frequency of oscillation tends to zero as the codimension two point is approached earlier theory by gaspard showed that homoclinic orbits must exist inside the parameter wedge this result is here extended to give the frequency and amplitude of the oscillations of the homoclinic loci within the wedge it is also shown how the two loci are related to each other and in the case of only cubic perturbations of the normal form that they are precisely out of phase the analysis is shown to agree with numerical results on perturbed normal forms and in two model systems arising in applications to atmospheric dynamics and to calcium wave propagation

distribution of natural trends in long term correlated records a scaling approach we estimate the exceedance probability w x ensuremath alpha l that in a long term correlated gaussian distributed sub record of length l characterized by a fluctuation exponent ensuremath alpha between 0 5 and 1 5 a relative increase ensuremath delta ensuremath sigma t of size larger than x occurs where ensuremath delta is the total observed increase measured by linear regression and ensuremath sigma t is the standard deviation around the regression line we consider l between 500 and 2000 which is the typical length scale of monthly local and reconstructed annual global temperature records we use scaling theory to obtain an analytical expression for w x ensuremath alpha l from this expression we can determine analytically for a given confidence probability q the boundaries ifmmode pm else textpm fi x q ensuremath alpha l of the confidence interval in the presence of an external linear trend the total observed increase is the sum of the natural and the external increase an observed relative increase ensuremath delta ensuremath sigma t is considered unnatural when it is above x q ensuremath alpha l in this case the size of the external relative increase is bounded by ensuremath delta ensuremath sigma t ifmmode pm else textpm fi x q ensuremath alpha l we apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends

statistical analysis of fluctuations in the ecg morphology abstract we investigate ecg recordings of healthy subjects h subjects and those with heart failure hf subjects based on a new measure derived from the shape of the ecg signal during each heart beat from this measure we construct sequences called morphograms which are complementary to sequences of successive interbeat time intervals commonly called tachograms the morphograms show long range anticorrelations between the increments of their successive elements analogous to the behavior found previously in the tachograms by studying the average strength of the fluctuations of the increments on certain scales we find that it is possible to completely separate the two groups of h and hf subjects the separation is comparable and in certain cases even more pronounced than that found earlier based on the tachograms

evolution towards ergodic behavior of stationary fractal random processes with memory application to the study of long range correlations of nucleotide sequences in dna abstract the possible occurrence of ergodic behavior for large times is investigated in the case of stationary random processes with memory it is shown that for finite times the time average of a state function is generally a random variable and thus two types of cumulants can be introduced for the time average and for the statistical ensemble respectively in the limit of infinite time a transition from the random to the deterministic behavior of the time average may occur resulting in an ergodic behavior the conditions of occurrence of this transition are investigated by analyzing the scaling behavior of the cumulants of the time average a general approach for the computation of these cumulants is developed explicit computations are presented both for short and long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized into products of functions depending on binary time differences in both cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time differences increase to infinity the analysis leads to the surprising conclusion that the scaling behavior of the cumulants of the time average is relatively insensitive to the type of memory considered both for short and long memory the cumulants of the time average obey inverse different from zero for large time differences then the time averaage is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds the theory is applied to the study of long range correlations of nucleotide sequences in dna in this case the length t of a sequence of nucleotides plays the role of the time variable a proportionality relationship is established between the cumulants of the pyrimidine excess in a sequence of length t and the cumulants of the time length average of the probability of occurrence of a pyrimidine it is shown that the statistical analysis of the dna data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengths the implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional brownian motion are also investigated

long correlations and truncated levy walks applied to the study latin american market indices abstract this work is devoted to the study of long correlations and other statistical properties of latin american market indices we concluded that the behavior of the return is compatible with a slow convergence to a gaussian distribution we also detected long range correlations in the absolute value of the return analyzing the effects of working with short data series this fact has relevant consequences in the volatility dynamics

path integral solutions of the governing equation of sdes excited by levy white noise abstract in this paper the probability density functions pdfs of scalar stochastic differential equations sdes subject to stable levy white noise are investigated the path integral pi method is extended to solve one dimensional space fractional fokker planck kolmogorov fpk equations which are the governing equations corresponded to scalar sdes excited by stable levy white noise first we derive a short time solution of the one dimensional space fractional fpk equation which is used in the chapman kolmogorov smoluchowski cks equation to obtain the pi solution then the accuracy of the pi solution is analyzed theoretically in terms of its characteristic function our results demonstrate that the pi method has a higher accuracy than the first order finite difference method for one step iteration in time finally several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the pi method for solving one dimensional space fractional fpk equations we find that the pi solution agrees well with the exact solution or the monte carlo one

irreversible processes the generalized affinities within tsallis statistics abstract within the generalized statistical mechanics introduced recently by tsallis the generalized form of the affinities the quantities which drive a process in the theory of irreversible thermodynamics was derived the affinities were obtained by considering changes in the generalized entropy s q aub of a system composed by two subsystems a and b the non extensive character of the tsallis entropy is taken into account at the end the equilibrium condition is discussed and the zeroth law in the framework of the generalized thermodynamics is consistently recovered as in the usual case two systems which are in equilibrium with a third one are necessarily in equilibrium among them and share the same temperature

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

signal modeling with self similar spl alpha stable processes the fractional levy stable motion model the purpose of the correspondence is the introduction of fractional levy stable motion flsm as a model for signals with long memory and high variability commonly encountered in natural processes we present a concise description of this model from a signal processing viewpoint and its successful application to real world infrared signals for the purpose of resolution enhancement via stochastic interpolation

fractal fluctuations in cardiac time series human heart rate controlled by complex feedback mechanisms is a vital index of systematic circulation however it has been shown that beat to beat values of heart rate fluctuate continually over a wide range of time scales herein we use the relative dispersion the ratio of the standard deviation to the mean to show by systematically aggregating the data that the correlation in the beat to beat cardiac time series is a modulated inverse power law this scaling property indicates the existence of long time memory in the underlying cardiac control process and supports the conclusion that heart rate variability is a temporal fractal we argue that the cardiac control system has allometric properties that enable it to respond to a dynamical environment through scaling

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

a trade investment model for distribution of wealth econophysics provides a strategy for understanding the potential mechanisms underlying the anomalous distribution of wealth found in real societies we present a computational nonlinear stochastic model for the distribution of wealth that depends upon three parameters and two mechanisms trade and investment to avoid economic paradoxes the trade mechanism is assumed to be related to the poorer trader s wealth and to statistically advantage the poorer of the two traders the two mechanisms together are shown to generate a distribution that reproduces the full range of the empirical wealth distribution and not only the inverse power law tail that pareto found in western societies at the end of the 19th century

interaction of steady and periodic bifurcating modes with imperfection effects in reaction diffusion systems asymptotic solutions for two dimensional reaction diffusion systems are obtained under suitable conditions a trivial solution independent of both time and space is assumed to exist and the linearized system about this solution is considered the analysis is restricted to the case where this linearized system possesses one real eigenvalue and a complex pair of eigenvalues all with real part approximately equal to zero steady periodic and doubly periodic bifurcating solutions are found with special attention given to the effects of imperfection in one of the parameters

a complexity measure for heart rate signals abstract in this paper we present an adaptive measurement to quantify the complexity of time series called q sample entropy difference mean q sediffave and apply this method to heart rate time series it quantifies the structural information of the signals and helps identify different conditions of the heart systems experiments show that compared with the traditional sample entropy sampen q sediffave has significant advantages in reflecting the inherent information of sequences it also reveals that the q sediffave curves of healthy subjects are significantly different from the atrial fibrillation af and congestive heart failure chf groups because the chf signals tend to fluctuate between narrow ranges while the atrial fibrillation af signals are random and the signals of healthy subjects are usually mixed by signals with different dynamic characteristics showing higher complexity in addition the most important parameter in the model q can enlarge the differences between q sediffave curves of healthy and pathological states

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

complex fractal dimension of the bronchial tree the architecture of the mammalian lung has been shown to be correctly described using a fractal model with a complex dimension related to a cantor set with random errors for four different species here we provide an interpretation of that model that has implications for biological evolution we argue that fractals are more error tolerant than other structures and therefore have an evolutionary advantage

multifractal estimates of monofractality in rr heart series in power spectrum ranges abstract two popular estimators of multifractal properties the wavelet transform modulus maxima method and multifractal detrended fluctuation analysis are applied to investigate signals consisting of normal rr series in 39 healthy subjects and 90 patients suffering from systolic dysfunction of the left ventricle however differently from standards for multifractal analysis the scaling is performed separately in intervals corresponding to standard power spectral bands low lf very low vlf and ultra low frequencies ulf tests on fractional brownian motions fbm are done to quantify properties of the estimators as detectors of monofractality in lf vlf and ulf bands arguments are given that multifractal analysis of rr series performed in these bands has a physiological meaning the increased activation of the sympathetic nervous system caused by heart disease is detected evidently only by analysis in lf the transition in multifractal characteristics between diurnal and nocturnal activity takes place when the analysis moves from lf and vlf to ulf only in ulf the diurnal heart rate variability can be approximated by fbm with a self similarity index of h 0 20

stochastic model for heart rate fluctuations a normal human heart rate shows complex fluctuations in time which is natural because the heart rate is controlled by a large number of different feedback control loops these unpredictable fluctuations have been shown to display fractal dynamics long term correlations and 1 f noise these characterizations are statistical and they have been widely studied and used but much less is known about the detailed time evolution dynamics of the heart rate control mechanism here we show that a simple one dimensional langevin type stochastic difference equation can accurately model the heart rate fluctuations in a time scale from minutes to hours the model consists of a deterministic nonlinear part and a stochastic part typical to gaussian noise and both parts can be directly determined from the measured heart rate data studies of 27 healthy subjects reveal that in most cases the deterministic part has a form typically seen in bistable systems there are two stable fixed points and one unstable one

deviations from uniform power law scaling in nonstationary time series a classic problem in physics is the analysis of highly nonstationary time series that typically exhibit long range correlations here we test the hypothesis that the scaling properties of the dynamics of healthy physiological systems are more stable than those of pathological systems by studying beat to beat fluctuations in the human heart rate we develop techniques based on the fano factor and allan factor functions as well as on detrended fluctuation analysis for quantifying deviations from uniform power law scaling in nonstationary time series by analyzing extremely long data sets of up to n 10 5 beats for 11 healthy subjects we find that the fluctuations in the heart rate scale approximately uniformly over several temporal orders of magnitude by contrast we find that in data sets of comparable length for 14 subjects with heart disease the fluctuations grow erratically indicating a loss of scaling stability