poincare recurrences from the perspective of transient chaos we obtain a description of the poincare recurrences of chaotic systems in terms of the ergodic theory of transient chaos it is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble this ensemble is atypical in terms of the natural measure of the leaked system the conditionally invariant measure accordingly for general initial ensembles the average recurrence and escape times are different however we show that the decay rate of these distributions is always the same our results remain valid for hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and nonhyperbolic components

extremum statistics in scale free network models we investigate the statistics of the most connected node in scale free networks for a scale free network model with homogeneous nodes we show by means of extensive simulations that the exponential truncation due to the finite size of the network of the degree distribution governs the scaling of the extreme values and that the distribution of maxima follows the gumbel statistics for a scale free network model with heterogeneous nodes we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maxima distribution

extreme value laws for fractal intensity functions in dynamical systems minkowski analysis typically in the dynamical theory of extremal events the function that gauges the intensity of a phenomenon is assumed to be convex and maximal or singular at a single or at most a finite collection of points in phase space in this paper we generalize this situation to fractal landscapes i e intensity functions characterized by an uncountable set of singularities located on a cantor set this reveals the dynamical role of classical quantities like the minkowski dimension and content whose definition we extend to account for singular continuous invariant measures we also introduce the concept of extremely rare event quantified by non standard minkowski constants and we study its consequences to extreme value statistics limit laws are derived from formal calculations and are verified by numerical experiments

iterative learning control for discrete nonlinear systems with randomly iteration varying lengths abstract this note proposes ilc for discrete time affine nonlinear systems with randomly iteration varying lengths no prior information on the probability distribution of random iteration length is required prior for controller design the conventional p type update law is used with a modified tracking error because of randomly iteration varying lengths a novel technical lemma is proposed for the strict convergence analysis in pointwise sense an illustrative example verifies the theoretical results

the global statistics of return times return time dimensions versus generalized measure dimensions we investigate return times in dynamical systems i e the time required by a trajectory to complete a return journey to a neighborhood of the initial position in particular we study the relations holding between the scaling exponents of phase space moments of return times in balls of diminishing radius on the one side and the generalized dimensions of invariant measures on the other because of a heuristic use of kac theorem the former have been used in place of the latter in numerical and experimental investigations to mark the distinction we call them return time dimensions we derive a full set of inequalities linking generalized dimensions of invariant measures and return time dimensions we comment on their optimality with the aid of two maps due to von neumann kakutani and to gaspard wang we conjecture a formula for the return time dimensions in a typical system we only assume that the dynamical system under investigation is ergodic and that motion takes place in a compact finite dimensional space

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

connection between energy spectrum self similarity and specific heat log periodicity as a first step towards the understanding of the thermodynamical properties of quasiperiodic structures we have performed both analytical and numerical calculations of the specific heats associated with successive hierarchical approximations to multiscale fractal energy spectra we show that in a certain range of temperatures the specific heat displays log periodic oscillations as a function of the temperature we exhibit scaling arguments that allow for relating the mean value as well as the amplitude and the period of the oscillations to the characteristic scales of the spectrum

multivariable norm optimal iterative learning control with auxiliary optimisation the paper describes a substantial extension of norm optimal iterative learning control noilc that permits tracking of a class of finite dimensional reference signals whilst simultaneously converging to the solution of a constrained quadratic optimisation problem the theory is presented in a general functional analytical framework using operators between chosen real hilbert spaces this is applied to solve problems in continuous time where tracking is only required at selected intermediate points of the time interval but simultaneously the solution is required to minimise a specified quadratic objective function of the input signals and chosen auxiliary state variables applications to the discrete time case including the case of multi rate sampling are also summarised the algorithms are motivated by practical need and provide a methodology for reducing undesirable effects such as payload spillage vibration tendencies and actuator wear whilst maintaining the desired tracking accuracy necessary f

electronic specific heat properties in one dimensional quasicrystals abstract in this paper we investigate the thermodynamical properties of electrons in one dimensional quasiperiodic potentials arranged in accordance with the pure and generalized fibonacci sequences the sequences are described in terms of a series of generations that obey peculiar recursion relations and have controlled disorder i e they are neither random nor periodic their electronic energy spectra are calculated using the one dimensional schrodinger equation in a tight binding approximation we present both analytical and numerical studies on the temperature dependence of the electron s specific heat associated with their multiscale fractal energy spectra we show that when t 0 the specific heat displays oscillations and when t the specific heat goes to zero with t 2

discrete time quantum walk on the apollonian network this work discusses the discrete time quantum walk dtqw on the first five generations of the apollonian network this structure is constructed in a geometric recurrent way and is characterized by among other features a scale free distribution of node degrees the dtqw formalism requires a node dependent coin operator that for each node has as many different output states as that node degree so that each local coin operator is expressed by the fourier operator the dtqw time evolution matrix has a larger rank than that of the network adjacency matrix or the matrix representation of the continuous time quantum walk ctqw results for the time evolution return time to specific nodes and asymptotic probability of site occupancy are obtained such dtqw specific features are discussed and compared to those obtained for the classical random walk and ctqw on the same lattice

the singularity of the canonical model of compact kahler manifolds soit x une variete de kahler compacte dont l anneau canonique 7b r m 0 h 0 o x mk x est finement genere alors il existe un q diviseur effectif sur s proj 7b r tel que s est log terminal

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

exponential distribution for the occurrence of rare patterns in gibbsian random fields we study the distribution of the occurrence of rare patterns in sufficiently mixing gibbs random fields on the lattice d d 2 a typical example is the high temperature ising model this distribution is shown to converge to an exponential law as the size of the pattern diverges our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns a similar result holds for the repetition of a rare pattern we apply these results to the fluctuation properties of occurrence and repetition of patterns we prove a central limit theorem and a large deviation principle

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

multifractal statistics of the local order parameter at random critical points application to wetting transitions with disorder disordered systems present multifractal properties at criticality in particular as discovered by ludwig a w w ludwig nucl phys b 330 639 1990 in the case of a diluted two dimensional potts model the moments overline ensuremath rho q r of the local order parameter ensuremath rho r scale with a set x q of nontrivial exponents x q ensuremath ne qx 1 we reexamine these ideas to incorporate more recent findings i whenever a multifractal measure w r normalized over space ensuremath sum r w r 1 occurs in a random system it is crucial to distinguish between the typical values and the disorder averaged values of the generalized moments y q ensuremath sum r w q r since they may scale with different generalized dimensions d q and stackrel ifmmode tilde else fi d q and ii as discovered by wiseman and domany s wiseman and e domany phys rev e 52 3469 1995 the presence of an infinite correlation length induces a lack of self averaging at critical points for thermodynamic observables in particular for the order parameter after this general discussion valid for any random critical point we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples we study the bidimensional wetting or the poland scheraga dna model with loop exponents c 1 5 marginal disorder and c 1 75 relevant disorder finally we argue that the presence of finite griffiths ordered clusters at criticality determines the asymptotic value x q ensuremath rightarrow ensuremath infty d and the minimal value ensuremath alpha min d q ensuremath rightarrow ensuremath infty d ensuremath x 1 of the typical multifractal spectrum f ensuremath alpha

fractal and multifractal properties of exit times and poincarerecurrences systems with chaotic dynamics possess anomalous statistical properties and their trajectories do not correspond to the gaussian process this property imposes description of such time characteristics as the distribution of exit times or poincar acute e recurrences by introducing a multi fractal time scale in order to satisfy the observed powerlike tails of the distributions we introduce a corresponding phase space time partitioning and spectral function for dimensions and make a connection between dimensions and transport exponent that defines the anomalous open quotes strange close quotes kinetics copyright ital 1997 ital the american physical society

poincare recurrences and transient chaos in systems with leaks in order to simulate observational and experimental situations we consider a leak in the phase space of a chaotic dynamical system we obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos this expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble with a subtle choice of the initial ensemble we obtain an equivalence to the classical problem of poincare recurrences in closed systems which is treated in the same framework finally we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components related respectively to the intermediate exponential and asymptotic power law decays of the survival probability

numerical estimates of local dimension by waiting time and quantitative recurrence abstract waiting time indicators are defined by measuring the time needed for a point x to approach a given point y our aim is to test the use of waiting time indicators and quantitative recurrence ones to numerically estimate the local dimension of attractors in dynamical systems as suggested by some rigorous results we test this approach on a collection of examples whose dimensions are already known rigorously or with a large amount of precision the main outcome is that the use of waiting time compared to recurrence increases largely the speed of convergence of the estimators

spectral properties of the apollonian network abstract a first characterization of the spectral properties of the apollonian network is presented results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network the main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues the increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation

multifractal properties of return time statistics the global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions comparison with the usual multifractal analysis of measures is presented and the difference between the two corresponding sets of dimensions is established theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented

return times for stochastic processes with power law scaling an analytical study of the return time distribution of extreme events for stochastic processes with power law correlation has been carried out the calculation is based on an expansion in the correlation exponent c t ensuremath mid t ensuremath mid ensuremath 1 the fixed point of the theory is associated with stretched exponential scaling of the distribution analytical expressions have been provided in the preasymptotic regime also the permanence time distribution appears to be characterized by stretched exponential scaling the conditions for application of the theory to non gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed

critical properties of contact process on the apollonian network we investigate an epidemic spreading process by means of a computational simulation on the apollonian network which is simultaneously small world scale free euclidean space filling and matching graphs an analysis of the critical behavior of the contact process cp is presented using a monte carlo method our model shows a competition between healthy and infected individuals in a given biological or technological system leading to a continuous phase transition between the active and inactive states whose critical exponents and 1 are calculated employing a finite size scaling analysis we show that the continuous phase transition belongs to the mean field directed percolation universality class in regular lattices

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

distribution of first return times in correlated stationary signals we present an analytical expression for the first return time frt probability density function of a stationary correlated signal precisely we start by considering a stationary discrete time ornstein uhlenbeck ou process with exponential decaying correlation function the first return time distribution for this process is derived by adopting a well known formalism typically used in the study of the frt statistics for nonstationary diffusive processes then by a subordination approach we treat the case of a stationary process with power law tail correlation function and diverging correlation time we numerically test our findings obtaining in both cases a good agreement with the analytical results we notice that neither in the standard ou nor in the subordinated case a simple form of waiting time statistics like stretched exponential or similar can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

brief paper iterative learning control for large scale nonlinear systems with observation noise the iterative learning control ilc is constructed for the discrete time large scale systems each subsystem is affine nonlinear and its observation equation is with noise subsystems are nonlinearly connected via the large state vector of the whole system the possibility of data missing and communication delay is taken into account it is proved that ilc given in the paper with probability one converges to the optimal one minimizing the tracking error the simulation results are consistent with theoretical analysis

generalized dimensions large deviations and the distribution of rare events abstract generalized dimensions of multifractal measures are usually seen as static objects related to the scaling properties of suitable partition functions or moments of measures of cells when these measures are invariant for the flow of a chaotic dynamical system generalized dimensions take on a dynamical meaning as they provide the rate function for the large deviations of the first hitting time which is the average time required to connect any two different regions in phase space we prove this result rigorously under a set of stringent assumptions as a consequence the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions numerical examples presented along with the theory suggest that the validity of this technique reaches far beyond the range covered by the theorem we state our result within the framework of extreme value theory this approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure this suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension we show an application of this technique to experimental data from climate dynamics

approximate adjoint based iterative learning control this paper characterises stochastic convergence properties of adjoint based gradient based iterative learning control ilc applied to systems with load disturbances when provided only with approximate gradient information and noisy measurements specifically conditions are discussed under which the approximations will result in a scheme which converges to an optimal control input both the cases of time invariant step sizes and cases of decreasing step sizes as in stochastic approximation are discussed these theoretical results are supplemented with an application on a sequencing batch reactor for wastewater treatment plants where approximate gradient information is available it is found that for such case adjoint based ilc outperforms inverse based ilc and model free p type ilc both in terms of convergence rate and measurement noise tolerance

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

competitive cluster growth in complex networks in this work we propose an idealized model for competitive cluster growth in complex networks each cluster can be thought of as a fraction of a community that shares some common opinion our results show that the cluster size distribution depends on the particular choice for the topology of the network of contacts among the agents as an application we show that the cluster size distributions obtained when the growth process is performed on hierarchical networks e g the apollonian network have a scaling form similar to what has been observed for the distribution of a number of votes in an electoral process we suggest that this similarity may be due to the fact that social networks involved in the electoral process may also possess an underlining hierarchical structure

a novel markov chain based ilc analysis for linear stochastic systems under general data dropouts environments this technical note contributes to the convergence analysis for iterative learning control ilc for linear stochastic systems under general data dropout environments i e data dropouts occur randomly at both the measurement and actuator sides data updating in the memory array is arranged in such a way that data at every time instance is updated independently which allows successive data dropouts both in time and iteration axes the update mechanisms for both the computed input and real input are proposed and then the update process of both inputs is shown to be a markov chain by virtue of markov modeling a new analysis method is developed to prove the convergence in both mean square and almost sure senses an illustrative example verifies the theoretical results

recurrence dimensions and lyapunov exponents we show that the poincare return time of a typical cylinder is at least its length for one dimensional maps we express the lyapunov exponent and dimension via return times

three dimensional wavelet based multifractal method the need for revisiting the multifractal description of turbulence dissipation data we generalize the wavelet transform modulus maxima wtmm method to multifractal analysis of 3d random fields this method is calibrated on synthetic 3d monofractal fractional brownian fields and on 3d multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration then we apply the 3d wtmm method to the dissipation field issued from 3d isotropic turbulence simulations we comment on the need to revisit previous box counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures

iterative learning control for networked stochastic systems with random packet losses the iterative learning control is constructed for discrete time stochastic systems with random measurement losses modelled by a stochastic sequence a simple p type update law is used and the almost sure convergence is strictly proved for both linear case and nonlinear case based on stochastic approximation illustrative examples show the effectiveness of the proposed approach

multifractal geometry in stock market time series it has been recently noticed that time series of returns in stock markets are of multifractal multiscaling character in that context multifractality has been always evidenced by its statistical signature i e the scaling exponents associated to a related variable however a direct geometrical framework much more revealing about the underlying dynamics is possible in this paper we present the techniques allowing the multifractal decomposition we will show that there exists a particular fractal component the most singular manifold msm which contains the relevant information about the dynamics of the series it is possible to reconstruct the series at a given precision from the msm we analyze the dynamics of the msm which shows revealing features about the evolution of this type of series