fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

entropy rate estimates from mutual information we show how to estimate the kolmogorov sinai entropy rate for chaotic systems using the mutual information function easily obtainable from experimental time series we state the conditions under which the relationship is exact and explore the usefulness of the approach for both maps and flows we also explore refinements of the method and study its convergence properties as a function of time series length

direction of arrival estimation using parametric signal models we consider the problem of estimating the directions of arrival doas of narrowband sources with known center frequency the paper evaluates the potential improvement in estimation accuracy by using spatial temporal processing for signals obeying a deterministic parametric model one would expect that prior information about the temporal structure of the signals will yield some gain in performance by deriving the cramer rao bound crb on the doa estimates we quantify this gain and identify the cases for which the gain is significant we show that for the single source case spatial temporal processing does not yield any gain in performance relative to conventional spatial processing for multiple noncoherent signals incorporating temporal processing can achieve the single source performance yielding a significant gain for the case of multiple sources with small spatial separation relative to the beamwidth of the array however spatial temporal processing cannot yield any gain in performance for multiple coherent signals

long range order in the scaling behaviour of hyperbolic dynamical systems abstract detailed statistical analyses of chaotic signals have been so far limited to 2 d maps with one expanding and one contracting direction here by studying filtered signals we provide evidence that the existence of a further stable direction can transform self similarity into self affinity and generate phase transitions according to the thermodynamic formalism of dynamical systems in a hyperbolic case the dimension spectrum is computed exactly as a function of the filter cut off frequency moreover the validity of the kaplan yorke formula and of its generalization are rigorously proven finally the scaling function is shown to satisfy a stochastic recursive relation whose lyapunov exponent allows to control the extension of the range of interactions in the associated hamiltonian

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

chaotic and non linear dynamic analysis of a two axis rate gyro with feedback control mounted on a space vehicle abstract an analysis is presented in this paper for a two axis rate gyro subjected to linear feedback control mounted on a space vehicle which is spinning with uncertain angular velocity z t about its spin of the gyro for the autonomous case in which z t is steady the stability analysis of the system is studied by routh hurwitz theory for the non autonomous case in which z t is sinusoidal function this system is a strongly non linear damped system subjected to parametric excitation by varying the amplitude of sinusoidal motion periodic and chaotic responses of this parametrically excited non linear system are investigated using the numerical simulation some observations on symmetry breaking bifurcations period doubling bifurcations and chaotic behavior of the system are investigated by various numerical techniques such as phase portraits poincare maps average power spectra and lyapunov exponents in addition some discussions about chaotic motions of this system can be suppressed and changed into regular motions by a suitable constant motor torque are included

calculation of lyapunov exponents avoiding spurious elements abstract a new approach to evaluate lyapunov exponents from time series is discussed with the help of this approach the number of calculated lyapunov exponents can be reduced to the physically relevant ones calculations in high dimensional embedding spaces are performed on smaller data bases with improved speed more direct information is obtained from the dynamics of the underlying dynamical system

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

a scaling law how an attractor s volume depends on noise level abstract we investigate the meaning of the dimension of a strange attractor for systems with noise more specifically we investigate the effect of adding noise of magnitude e to a deterministic system with d degrees of freedom if the attractor has dimension d and d d then its volume is zero the addition of noise may be an important physical probe for experimental situations useful for determining how much of the observed phenomena in a system is due to noise already present when the noise is added the attractor a e has positive volume we conjecture that the generalized volume of a e is proportional to e d d for e near 0 and show this relationship is valid in several cases for chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

subspace based estimation of time delays and doppler shifts this paper considers the problem of estimating the time delays and doppler shifts of a known waveform received via several distinct paths by an array of antennas the general maximum likelihood estimator is presented and is shown to require a 2d dimensional nonlinear minimization where d is the number of received signal reflections two alternative solutions based on signal and noise subspace fitting are proposed requiring only a d dimensional minimization in particular we show how to decouple the required search into a two step procedure where the delays are estimated and the dopplers solved for explicitly initial conditions for the time delay search can be obtained by applying generalizations of the music and esprit algorithms which are also outlined in the paper simulation examples are included to illustrate the algorithms performance relative to the cramer rao bound

liapunov exponents of simple smooth flows an attempt is made to introduce the notion of a simple dynamical system as one where liapunov exponents may be obtained by purely algebraic methods specifically by exploiting the lie algebra relations between vector fields thus in a simple system one need not solve the dynamical equations but is often faced with an equally difficult task of unravelling the algebraic structure the theory is presented within the framework of differential geometry several examples illustrate the usefulness of the proposed concept

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

computationally efficient angle estimation for signals with known waveforms this paper presents a large sample decoupled maximum likelihood deml angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise the deml estimator decouples the multidimensional problem of the exact ml estimator to a set of 1 d problems and hence is computationally efficient we shall derive the asymptotic statistical performance of the deml estimator and compare the performance with its cramer rao bound crb i e the best possible performance for the class of asymptotically unbiased estimators we will show that the deml estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms we will also show that for moderately correlated signals with known waveforms the deml estimator is no longer a large sample maximum likelihood ml estimator but the deml estimator may still be used for angle estimation and the performance degradation relative to the crb is small we shall show that the deml estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix finally several numerical examples showing the performance of the deml estimator are presented in this paper

ergodic properties of a kicked damped particle we investigate a class of nonlinear dynamical systems describing the movement of a particle in a viscous medium under the influence of a kick force these systems can be regarded as a generalization of the langevin approach to brownian motion in the sense that the fluctuating force on the particle is not gaussian white noise but an arbitrary non gaussian process generated by a nonlinear dynamical system we investigate how certain properties of the force periodicity ergodicity mixing property transfer to the velocity of the particle moreover the relaxation properties of the system are analysed

experiments and analysis on chaotic vibrations of a shallow cylindrical shell panel abstract detailed experimental results and analytical results are presented on chaotic vibrations of a shallow cylindrical shell panel subjected to gravity and periodic excitation the shallow shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained by in plain springs in the experiment the cylindrical shallow shell panel with thickness 0 24 mm square form of length 140 mm and mean radius 5150 mm is used for the test specimen all edges around the shell boundary are simply supported by adhesive flexible films first to find fundamental properties of the shell panel linear natural frequencies and characteristics of restoring force of the shell panel are measured these results are compared with the relevant analytical results then geometrical parameters of the shell panel are identified exciting the shell panel with lateral periodic acceleration nonlinear frequency responses of the shell panel are obtained by sweeping the frequency of periodic acceleration in typical ranges of the exciting frequency predominant chaotic responses are generated time histories of the responses are recorded for inspection of the chaos in the analysis the donnell equation with lateral inertia force is introduced assuming mode functions the governing equation is reduced to a set of nonlinear ordinary differential equations by the galerkin procedure periodic responses are calculated by the harmonic balance method chaotic responses are integrated numerically by the runge kutta gill method the chaotic responses which are obtained by the experiment and the analysis are inspected with the fourier spectra the poincare projections the maximum lyapunov exponents and the lyapunov dimension it is found that the dominant chaotic responses of the shell panel are generated from the responses of the sub harmonic resonance of 1 2 order and of the ultra sub harmonic resonance of 2 3 order by the convergence of the maximum lyapunov exponent to the embedding dimension the number of predominant vibration modes which contribute to the chaos is found to be three or four fairly good agreements are obtained between the experimental results and the analytical results

approach to gaussian stochastic behavior for systems driven by deterministic chaotic forces we consider skew product dynamical systems that describe the stroboscopic dynamics of a damped particle subjected to a chaotic kick force in a suitable scaling limit the dynamics converges to the ornstein uhlenbeck process we investigate the deterministic chaotic corrections in the vicinity of this gaussian limit case for various examples of chaotic forces we present numerical evidence that for certain classes of chaotic forces the deterministic chaotic corrections of the invariant density are universal we provide analytical results for forces generated by tchebyscheff maps and sketch a renormalization group theory in the space of probability densities

subspace based adaptive generalized likelihood ratio detection subspace based adaptive detection performance is examined for the generalized likelihood ratio detector based on wilks spl lambda statistic the problem considered here is detecting the presence of one or more signals of known shape embedded in gaussian distributed noise with unknown covariance structure the data is mapped into a subspace prior to detection the probability of false alarm is independent of the subspace transformation and depends only on subspace dimension the probability of detection depends on the subspace transformation through a nonadaptive signal to noise ratio snr parameter subspace processing results in an snr loss that tends to decrease performance and a gain in statistical stability that tends to increase performance it is shown that the statistical stability effect dominates the snr loss for short data records and subspace detectors can require substantially less snr than full space detectors for equivalent performance a method for designing the subspace transformation to minimize the snr loss is proposed and illustrated through simulations

efficient maximum likelihood doa estimation for signals with known waveforms in the presence of multipath we present a large sample maximum likelihood ml algorithm for estimating the directions of arrival doa s and signal amplitudes of known possibly coherent signals impinging on an array of sensors the algorithm is an extension of the deml method of li et al that handles coherent multipath that may be present in the signals the algorithm is computationally efficient because the nonlinear minimization step decouples into a set of minimizations of smaller dimension

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

robust estimation of tangent maps and liapunov spectra abstract in this paper we describe three algorithms for estimating the spectrum of liapunov exponents which we have found to be robust and reasonably accurate when used in practical circumstances attention is focussed on least squares and total least squares methods for estimating tangent maps from time series data the properties of the algorithms are demonstrated by application to a range of data sets we consider numerical and experimental data and discuss the effect of additive noise and system noise on the proposed methods

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

exponential decay of power spectra at high frequency and positive lyapunov exponents abstract it is generally accepted that power spectra of time series taken from continuous time dynamical systems exhibiting deterministic chaos will decay exponentially at high frequency the exponential decay constant is an inverse time scale that is an invariant of the dynamics of the system and that is unique to deterministic chaos theoretical arguments are presented that suggest that should be related to the positive lyapunov exponents numerical results are presented from four models with attractor dimensions ranging from 2 009 to 20 28 which suggest that is roughly proportional to the sum of the positive lyapunov exponents i for these models the ratio of to i lies between the values 0 639 and 2 46 and shows no systematic change with attractor dimension

complex amplitude estimation in the known steering matrix and generalized waveform case we consider a special growth curve sgc model with a known steering matrix and generalized waveform in the presence of unknown interference and noise several estimators of the complex amplitude based on this model are derived including the methods of approximate maximum likelihood aml minimum variance distortionless response mvdr and amplitude and phase estimation apes we analyze the statistical properties of these estimators and show that in the presence of temporally white but spatially correlated noise and interference aml is asymptotically statistically efficient for a large snapshot number while mvdr and apes are asymptotically equivalent but not statistically efficient via several numerical examples we also show that when the noise and interference are both spatially and temporally correlated the apes estimator can achieve better estimation accuracy and exhibit greater robustness than the other methods

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

numerical estimates of hausdorff dimension numerical methods for estimating hausdorff dimension useful in the analysis of turbulence are explained and applied to a specific example in particular methods involving rescaling and approximation by cantor sets are discussed

array signal processing in the known waveform and steering vector case the amplitude estimation of a signal that is known only up to an unknown scaling factor with interference and noise present is of interest in several applications including using the emerging quadrupole resonance qr technology for explosive detection in such applications a sensor array is often deployed for interference suppression this paper considers the complex amplitude estimation of a known waveform signal whose array response is also known a priori two approaches viz the capon and the maximum likelihood ml methods are considered for the signal amplitude estimation in the presence of temporally white but spatially colored interference and noise we derive closed form expressions for the expected values and mean squared errors mses of the two estimators a comparative study shows that the ml estimate is unbiased whereas the capon estimate is biased downwards for finite data sample lengths we show that both methods are asymptotically statistically efficient when the number of data samples is large but not when the signal to noise ratio snr is high furthermore we consider a more general scenario where the interference and noise are both spatially and temporally correlated we model the interference and noise vector as a multichannel autoregressive ar random process an alternating least squares als method for parameter estimation is presented we show that in most cases the als method is superior to the model mismatched ml m sup 3 l method which ignores the temporal correlation of the interference and noise

ising type and other transitions in one dimensional coupled map lattices with sign symmetry we consider a one dimensional lattice of expanding antisymmetric maps 1 1 1 1 with nearest neighbor diffusive coupling for such systems it is known that if the coupling parameter e is small there is unique stationary in time state which is chaotic in space time a disputed question is whether such systems can exhibit ising type phase transitions as e grows beyond some critical value ec we present results from computer experiments which give definite indication that such a transition takes place the mean square magnetization appears to diverge as e approaches some critical value with a critical exponent around 0 9 we also study other properties of the coupled map system

adaptation and its impact on the dynamics of a system of three competing populations abstract we investigate how the adaptation of the competition coefficients of the competing populations for the same limited resource influences the system dynamics in the regions of the parameter space where chaotic motion of shilnikov kind exists we present results for two characteristic values of the competition coefficient adaptation factor the first value 0 05 belongs to the small interval of possible negative values of for this a transition to chaos by period doubling bifurcations occurs and a window of periodic motion exists between the two regions of chaotic motion with increasing the system becomes more dissipative and the number of the windows of periodic motion increases when 1 0 a region of transient chaos is observed after the last window of periodic motion we verify the picture of the system dynamics by power spectra histograms and autocorrelations and calculate the lyapunov exponents and kaplan yorke dimension finally we discuss the eligibility of the investigated system for a topological analysis

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

regular and chaotic dynamic analysis and control of chaos of an elliptical pendulum on a vibrating basement the dynamic behavior of an elliptical pendulum subjected to external disturbance is studied in this paper the lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system by applying numerical results phase diagrams power spectrum period t maps and lyapunov exponents are presented to observe periodic and chaotic motions the effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams for global analysis the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping micm method finally several methods the delayed feedback control the addition of constant torque the addition of periodic force adaptive control algorithm and bang bang control are used to control chaos effectively

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

a semi implicit spectral method for compressible convection of rotating and density stratified flows in cartesian geometry in this paper we have described a stratified semi implicit spectral method to study compressible convection in cartesian geometry the full set of compressible hydrodynamic equations are solved in conservative forms the numerical scheme is accurate and efficient based on fast fourier sin cos spectral transforms in the horizontal directions chebyshev spectral transform or second order finite difference scheme in the vertical direction and second order semi implicit scheme in time marching of linear terms we have checked the validity of both the fully pseudo spectral scheme and the mixed finite difference pseudo spectral scheme by studying the onset of compressible convection the difference of the critical rayleigh number between our numerical result and the linear stability analysis is within two percent besides we have computed the mach numbers with different rayleigh numbers in compressible convection it shows good agreement with the numerical results of finite difference methods and finite volume method this model has wide application in studying laminar and turbulent flow illustrative examples of application on horizontal convection gravity waves and long lived vortex are given in this paper

joint estimation of time delays and directions of arrival of multiple reflections of a known signal we present an efficient algorithm for estimating the time delays and the directions of arrival doas of multiple reflections of a known signal the algorithm is based on an iterative scheme that transforms the multidimensional maximum likelihood criterion into two sets of simple one dimensional 1 d maximization problems simulation results illustrating the performance of the algorithm in comparison with the cramer rao bound are included

spectral methods and a maximum principle etude de diverses approximations spectrales de tchebychev d un probleme de couche limite pour un operateur d helmholtz et d advection diffusion comportement des solutions spectrales dans l espace des frequences et dans l espace physique estimations d erreur

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

modal interaction in chaotic vibrations of a shallow double curved shell panel abstract experimental results and analytical results are presented on chaotic vibrations of a shallow double curved shell panel subjected to gravity and periodic excitation modal interactions in the chaotic responses are discussed the shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained in the experiment time histories of the chaotic responses at the spatial multiple positions of the shell panel are measured for the inspection of modal interaction in the analysis the shallow shell panel is assumed to have constant curvatures along to orthogonal directions and geometric initial imperfection the donnell mushtari vlasov type equation is used as governing equation with lateral inertia force assuming deflection with multiple modes of vibration the governing equation is reduced to a set of nonlinear ordinary differential equations by the bubnov galerkin procedure chaotic responses are integrated numerically the chaotic responses which are obtained by the experiment and the analysis are inspected with the fourier spectra the poincare projections the maximum lyapunov exponents and the lyapunov dimension contribution of modes of vibration to the chaotic responses is analyzed by the principal component analysis i e karhunen loeve transformation

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

quantisation of the kaplan yorke model a classically chaotic model with a strange attractor introduced by kaplan and yorke 1979 is quantised and analysed in the heisenberg picture the equivalence is shown to quantisation in the schrodinger picture given earlier by one of the authors as an advantage of the approach presented here the expectation values of observables can be calculated it is shown that in expectation values the limits to 0 and t to infinity commute

driven chaotic motion in single and double well potentials abstract the stochastic motions in stable and bistable potentials are compared on the basis of a langevin like equation with a deterministic chaotic process replacing gaussian white noise ergodic and mixing properties are obtained under rather general circumstances in the case of the symmetric double well potential the system exhibits a behavior quite different from what one observes for stable potentials as the control parameters are varied the system undergoes a phase transition from an ergodic to a nonergodic phase lyapunov exponents and fractal dimensions are studied for more detailed insight

the mechanism of the increase of the generalized dimension of a filtered chaotic time series the determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes while this is expected from the kaplan yorke conjecture we show that for one dimensional maps a weak filter can induce a self similarity which is responsible for the increase of the hausdorff dimension we are able to calculate the increase of the generalized dimensiond q for the filtered time series of the logistic mapx i 1 rx i 1 x i atr 4 analytically

detection of synchronization between chaotic signals an adaptive similarity based approach we present an adaptive similarity based approach to detect generalized synchronization gs with n m phase synchronization ps where n and m are integers and one of them is 1 this approach is based on the similarity index si and gaussian mixture model with the minimum description length criterion the clustering method which is shown to be superior to the closeness and connectivity of a continuous function is employed in this study to detect the existence of gs with n m ps we conducted a computer simulation and a finger lifting experiment to illustrate the effectiveness of the proposed method in the simulation of a rossler lorenz system our method outperformed the conventional si and gs with 2 1 ps within the coupled system was found in the experiment of self paced finger lifting movement cortico muscular gs with 1 2 and 1 3 ps was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area the gs with n m ps n or m 1 has been simultaneously resolved from both simulation and experiment the proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science

chaotic attractors in the four dimensional leslie gower competition model abstract we study the occurrence of the chaotic attractor in the four dimensional classical leslie gower competition model we find that chaos can be generated by a cascade of quasiperiod doubling bifurcations starting from a supercritical neimark sacker bifurcation of the positive fixed point in this model the chaotic attractor is contained in the three dimensional carrying simplex that is a globally attracting invariant manifold biologically the result implies that the invasion attempts by an invader into a trimorphic population under the leslie gower dynamics can lead to chaos

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

chaos chaos control and synchronization of a gyrostat system abstract the dynamic behavior of a gyrostat system subjected to external disturbance is studied in this paper by applying numerical results phase diagrams power spectrum period t maps and lyapunov exponents are presented to observe periodic and choatic motions the effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams for global analysis the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping micm method several methods the delayed feedback control the addition of constant torque the addition of periodic force the addition of periodic impulse torque injection of dither signal control adaptive control algorithm aca control and bang bang control are used to control chaos effectively finally synchronization of chaos in the gyrostat system is studied

a computation of the limit capacity of the lorenz attractor abstract attempts to compute directly the limit capacity or fractal dimension of the lorenz attractor using the box counting algorithm suggested by takens encounter a convergence problem this convergence problem is discussed leading to an extrapolation procedure that resolves it and yields a fractal dimension that is accurate to the resolution of the smallest boxes used

maximum likelihood angle estimation for signals with known waveforms maximum likelihood estimation mle of the arrival angles of narrowband plane waves when one or all signals have known waveforms is considered computationally efficient and rapidly converging algorithms that iteratively maximize the likelihood functions are presented cramer rao bounds for these estimators are obtained the conditions under which incorporating knowledge of one or all of the signal waveforms in the estimators improves the accuracy of the angle estimates are described

rigorous statistical procedures for data from dynamical systems various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations this includes symmetry tests and the determination of dimension coefficients the paper contains the theoretical results and several simulations explain the methods

performance analysis of multivariate complex amplitude estimators we consider multivariate complex amplitude estimation in the presence of unknown interference and noise two multivariate approaches maximum likelihood ml and capon are provided we derive the closed form expression of the crame spl acute r rao bound crb for the unknown complex amplitudes we also analyze the bias properties and mean squared errors mse of the two estimators a comparative study shows that the multivariate ml estimator is unbiased whereas the multivariate capon estimator is biased downward for finite snapshots both estimators are asymptotically statistically efficient when the number of snapshots is large

turbulence in multistep methods for initial value problems solving a scalar initial value problem with a numerical k step scheme amounts to iterating a k dimensional discrete dynamical system in this paper we investigate regular and chaotic behaviour of some dynamical systems associated with adams bashforth formulas applied to a simple nonlinear test equation

chaotic motion of a parametrically forced simple pendulum abstract this paper describes the behavior of a simple pendulum with support in a steady and rather slow circular motion in a vertical plane the governing equation of this model is nonlinear and its motion is about the direction of gravity in the equation of motion there are three parameters which are related to the radius and angular velocity of the circular motion the length of the pendulum and the damping constant for some combinations of these parameters this model is found to have chaotic motion by numerical integration this paper not only draws the time series and poincare map to show the characteristics of regular and chaotic motions but also explores the feasibility of using the amplitude probability distribution by numerical method to diagnose whether the motion is chaotic the effect of geometry and damping constant on the motion of the system is investigated furthermore whether the poincare map and amplitude probability distribution depend sensitively on the initial conditions is also studied

some applications of hausdorff dimension inequalities for ordinary differential equations upper bounds are obtained for the hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable from these are derived sufficient conditions for dissipative analytic n dimensional periodic differential equations to have only a finite number of periodic solutions for autonomous equations the same conditions ensure that each bounded semi orbit converges to a critical point these results yield some information about the lorenz equation and the forced duffing equation

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

mathematical theory of lyapunov exponents this paper reviews some basic mathematical results on lyapunov exponents one of the most fundamental concepts in dynamical systems the first few sectionscontainsomeverygeneralresultsinnonuniformhyperbolictheory we consider f where f is an arbitrary dynamical system and is an arbitrary invariant measure and discuss relations between lyapunov exponents and several dynamical quantities of interest including entropy fractal dimension and rates of escape the second half of this review focuses on observable chaos characterized by positive lyapunov exponents on positive lebesgue measure sets much attention is given to srb measures a very special kind of invariantmeasuresthatofferawaytounderstandobservablechaosindissipative systems paradoxical as it may seem given a concrete system it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied case studies will be discussed the final section is on noisy or stochastically perturbed systems for which we present a dynamical picture simpler than that for purely deterministic systems in this short review we have elected to limit ourselves to finite dimensional systems and to discrete time the phase space which is assumed to be r d or a riemannian manifold is denoted by m throughout the lebesgue or the riemannian measure on m is denoted by m and the dynamics are generated by iterating a self map of m written f m for flows the reviewed results

non linear dynamics and control of chaos for a tachometer abstract the dynamic behaviors of a rotational tachometer with vibrating support are studied in the paper both analytical and computational results are used to obtain the characteristics of the system the lyapunov direct method is applied to obtain the conditions of stability of the equilibrium position of the system the center manifold theorem determines the conditions of stability for the system in a critical case by applying various numerical analyses such as phase plane poincare map and power spectrum analysis a variety of periodic solutions and phenomena of the chaotic motion are observed the effects of the changes of parameters in the system can be found in the bifurcation diagrams and parametric diagrams by using lyapunov exponents and lyapunov dimensions the periodic and chaotic behaviors are verified finally various methods such as the addition of a constant torque the addition of a periodic torque delayed feedback control adaptive control bang bang control optimal control and the addition of a periodic impulse are used to control chaos effectively

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

time delay and spatial signature estimation using known asynchronous signals this paper considers the problem of estimating the parameters of known signals received asynchronously by an array of antennas the parameters of primary interest are the time delays of the signals and their spatial signatures at the array estimates of the signal directions of arrival are also considered but are of secondary importance in this work maximum likelihood algorithms and more computationally efficient approximations are developed for both the case where all received signals are identical the channel estimation overlapping echo problem and where they are all distinct conditions are also derived under which the standard matched filter approach yields consistent and statistically efficient parameter estimates the issue of solution uniqueness is addressed and in particular an upper bound on the number of signals whose parameters may be uniquely identified is derived for a number of different cases typically the bound is far greater than the number of sensors and is limited only by the number of data samples collected some representative simulation examples are also included to illustrate the algorithms performance relative to the cramer rao bound

nonlinear dynamical behavior of the limited explodator in a cstr under square wave perturbation of the flow rate abstract results of studies of the limited explodator model in a continuous flow stirred tank reactor cstr under square wave perturbation of the flow rate are reported the perturbation is applied in such a way that the system is alternately attracted to two different periodic attractors in the parameter region close the hopf bifurcation point the system is shown to display a variety of entrainment bands birhythmicity quasiperiodicity resonance like phenomenon period doubling and intermittency routes to chaos and a complicated window structure of the chaotic region in addition a novel phenomenon intermittent alternative laminar oscillations was observed in a chaotic regime sandwiched between two entrainment bands transient chaos occurs in one of the entrainment bands which intimates chaos in the adjacent regime positive lyapunov exponents were found to be associated with the chaotic behavior the folding and stretching property of the chaotic attractors was analyzed through stroboscopic representations the deterministic nature of the chaotic behavior was confirmed by the quadratic like curve formed in the one dimensional map

a three dimensional multidimensional gas kinetic scheme for the navier stokes equations under gravitational fields this paper extends the gas kinetic scheme for one dimensional inviscid shallow water equations k xu a well balanced gas kinetic scheme for the shallow water equations with source terms j comput phys 178 2002 533 562 to multidimensional gas dynamic equations under gravitational fields four important issues in the construction of a well balanced scheme for gas dynamic equations are addressed first the inclusion of the gravitational source term into the flux function is necessary second to achieve second order accuracy of a well balanced scheme the chapman enskog expansion of the boltzmann equation with the inclusion of the external force term is used third to avoid artificial heating in an isolated system under a gravitational field the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface fourth the multidimensional approach with the inclusion of tangential gradients in two dimensional and three dimensional cases becomes important in order to maintain the accuracy of the scheme many numerical examples are used to validate the above issues which include the comparison between the solutions from the current scheme and the strang splitting method the methodology developed in this paper can also be applied to other systems such as semi conductor device simulations under electric fields

information flow and maximum entropy measures for 1 d maps abstract the invariant measures of maximal metric entropy are constructed explicitly for some maps of the interval by iterating the maps backward the construction illustrates in a particularly clear way the information flow in simple systems as well as recently conjectured relationships between dimensions of invariant measures lyapunov exponents and entropies maps it is conjectured that the natural measure is the invariant measure with strongest mixing

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

a p ge semiconductor experiment showing chaos and hyperchaos abstract a p ge semiconductor experiment is investigated by the help of both probabilistic and dynamical characterization methods dimensions lyapunov exponents and the corresponding scaling functions are calculated two exemplary files of data from the p ge semiconductor experiment exhibiting spontaneous i e undriven resistance oscillations in the low temperature avalanche breakdown are shown to be chaotic and hyperchaotic respectively for the first file we obtained a fractal dimension between two and three and one positive lyapunov exponent whereas for the second file we found a fractal dimension between three and four and two positive lyapunov exponents adopting the terminology introduced by rossler the behavior corresponding to the latter file is called hyperchaotic furthermore using the language of the thermodynamical formalism the probabilistic scaling function evaluated for the hyperchaotic state indicates a phase transition like behavior

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

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

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

the dimension spectrum of axiom a attractors some rigorous results on the dimension spectrum of expanding markov maps of the interval are extended to axiom ac2 diffeomorphisms of a compact two dimensional manifold

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

approach to equilibrium for locally expanding maps in ℝ k by using a well known technique from classical statistical mechanics of one dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapst of the unit cube in k we generalize herewith in a certain sense the results of lasota and yorke on piecewise expanding maps of the unit interval to higher dimensions we show a kuzmin type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow