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

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

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

intertwined basins of attraction of dynamical systems in this paper we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results

adaptive noise smoothing filter for images with signal dependent noise in this paper we consider the restoration of images with signal dependent noise the filter is noise smoothing and adapts to local changes in image statistics based on a nonstationary mean nonstationary variance nmnv image model for images degraded by a class of uncorrelated signal dependent noise without blur the adaptive noise smoothing filter becomes a point processor and is similar to lee s local statistics algorithm 16 the filter is able to adapt itself to the nonstationary local image statistics in the presence of different types of signal dependent noise for multiplicative noise the adaptive noise smoothing filter is a systematic derivation of lee s algorithm with some extensions that allow different estimators for the local image variance the advantage of the derivation is its easy extension to deal with various types of signal dependent noise film grain and poisson signal dependent restoration problems are also considered as examples all the nonstationary image statistical parameters needed for the filter can be estimated from the noisy image and no a priori information about the original image is required

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

pressure and fractal indices for the gibbs measures of hyperbolic julia sets the authors give a simple algorithm to compute the topological pressure for the julia sets j of hyperbolic rational endomorphism this allows one to compute the spectra of the generalised dimensions and entropies with respect to the whole class of gibbs measures on j

states that are far from being stabilizer states stabilizer states are eigenvectors of maximal commuting sets of operators in a finite heisenberg group states that are far from being stabilizer states include magic states in quantum computation mub balanced states and sic vectors in prime dimensions the latter two fall under the umbrella of minimum uncertainty states muss in the sense of wootters and sussman we study the correlation between two ways in which the notion of far from being a stabilizer state can be quantified two theorems valid for all prime dimensions are given as well as detailed results for low dimensions in dimension 7 we identify the mub balanced states as being antipodal to the sic vectors within the set of mus in a sense that we make definite in dimension 4 we show that the states that come closest to being mus with respect to all of the six stabilizer mubs are the fiducial vectors for alltop mubs

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

sparse geometric image representations with bandelets this paper introduces a new class of bases called bandelet bases which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow this geometric flow indicates directions in which the image gray levels have regular variations the image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm bandelet bases lead to optimal approximation rates for geometrically regular images for image compression and noise removal applications the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion comparisons are made with wavelet image compression and noise removal algorithms

chaos and indecomposability abstract we use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space earlier mouron proved that if x is an arc like continuum which admits a homeomorphism f with positive topological entropy then x contains an indecomposable subcontinuum barge and diamond proved that if g is a finite graph and f g g is any map with positive topological entropy then the inverse limit space lim g f contains an indecomposable continuum in this paper we show that if x is a g like continuum for some finite graph g and f x x is any map with positive topological entropy then lim x f contains an indecomposable continuum as a corollary we obtain that in the case that f is a homeomorphism x contains an indecomposable continuum moreover if f has uniformly positive upper entropy then x is an indecomposable continuum our results answer some questions raised by mouron and generalize the above mentioned work of mouron and also that of barge and diamond we also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

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

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

a simple feedback control for a chaotic oscillator with limited power supply we proposed a simple feedback control method to suppress chaotic behavior in oscillators with limited power supply the small amplitude controlling signal is applied directly to the power supply system so as to alter the characteristic curve of the driving motor numerical results are presented showing the method efficiency for a wide range of control parameters moreover we have found that for some parameters this kind of control may introduce coexisting periodic attractors with complex basins of attraction and therefore serious problems with predictability of the final state the system will asymptote to

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

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

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

wada dye boundaries in open hydrodynamical flows abstract dyes of different colors advected by two dimensional flows which are asymptotically simple can form a fractal boundary that coincides with a chaotic saddle s unstable manifold we show that such dye boundaries can have the wada property every boundary point of a given color on this fractal set is on the boundary of at least two other colors the condition for this is the nonempty intersection of the saddle s stable manifold with at least three differently colored domains in the asymptotic inflow region

some properties of mixing repellers applying the thermodynamic formalism to mixing repellers for regular maps the author obtains some rigorous relations between the dimensions of the repellers and the dynamic properties of the transformation on them for a class of one dimensional expanding maps these quantities can be approximated to any desired accuracy by means of the corresponding variables for linear expanding maps which can be computed exactly finally a mathematical foundation and some properties of the generalised dimensions are given

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

unpredictable behavior in the duffing oscillator wada basins this paper describes some numerical experiments giving evidence of wada basin boundaries for the duffing oscillator we suggest some mechanisms by which this fractal property of the boundaries appears and discuss the difficulties that the wada property presents for predicting the behavior of dynamical systems

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

indeterminate bifurcational phenomena in hardening systems indeterminate bifurcations are emerging as an important ingredient of nonlinear dissipating dynamical systems in this paper we show that indeterminate bifurcations with an unpredictable outcome are a typical ingredient of nonlinear hardening systems such phenomena are clearly important concepts in the theory of nonlinear resonance and their discovery complements the earlier work carried out on softening systems as an illustrative example we examine the dynamics of a parametrically excited hardening system and show that when the trivial solution is located on a highly intertwined basin boundary its loss of stability gives a dynamic jump whose outcome is indeterminate in the sense that we cannot predict on to which coexisting attractor the system will settle

the topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

the dynamics of continuous maps of finite graphs through inverse limits suppose that f g g is a continuous piecewise monotone function on a finite graph g then the following are equivalent i f has positive topological entropy ii there are disjoint intervals i1 and i2 and a positive integer n with i1 u i2 c fn ji n fn i2 iii the inverse limit space constructed by using f on g as a single bonding map contains an indecomposable subcontinuum this result generalizes known results for the interval and circle

topological conjugacy and transitivity for a class of piecewise monotone maps of the interval on dit qu une application continue f d un intervalle compact dans lui meme est markov lineaire s il est lineaire par morceaux et l ensemble de tous les f k x ou k 0 et x une extremite d un morceau lineaire est fini on donne une classification effective pour des applications de markov lineaires et une procedure effective pour determiner quand une telle application est transitive

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

origin of chaotic transients in excitatory pulse coupled networks we develop an approach to understanding long chaotic transients in networks of excitatory pulse coupled oscillators our idea is to identify a class of attractors sequentially active firing saf attractors in terms of the temporal event structure of firing and receipt of pulses then all attractors can be classified into two groups saf attractors and non saf attractors we establish that long transients typically arise in the transitional region of the parameter space where the saf attractors are collectively destabilized bifurcation behavior of the saf attractors is analyzed to provide a detailed understanding of the long irregular transients although demonstrated using pulse coupled oscillator networks our general methodology may be useful in understanding the origin of transient chaos in other types of networked systems an extremely challenging problem in nonlinear dynamics and complex systems

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

wavelet based texture retrieval using generalized gaussian density and kullback leibler distance we present a statistical view of the texture retrieval problem by combining the two related tasks namely feature extraction fe and similarity measurement sm into a joint modeling and classification scheme we show that using a consistent estimator of texture model parameters for the fe step followed by computing the kullback leibler distance kld between estimated models for the sm step is asymptotically optimal in term of retrieval error probability the statistical scheme leads to a new wavelet based texture retrieval method that is based on the accurate modeling of the marginal distribution of wavelet coefficients using generalized gaussian density ggd and on the existence a closed form for the kld between ggds the proposed method provides greater accuracy and flexibility in capturing texture information while its simplified form has a close resemblance with the existing methods which uses energy distribution in the frequency domain to identify textures experimental results on a database of 640 texture images indicate that the new method significantly improves retrieval rates e g from 65 to 77 compared with traditional approaches while it retains comparable levels of computational complexity

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

on the application of normal forms near attracting fixed points of dynamical systems abstract it is shown on an integrable example in the plane that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem however as is demonstrated on a nonintegrable example with 3 dimensional phase space the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point whose invariant manifolds embrace the attracting fixed point forming a complicated basin boundary thus in such problems normal forms are shown to be useful in practice as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t

order 3 symmetry in the clifford hierarchy we investigate the action of the first three levels of the clifford hierarchy on sets of mutually unbiased bases comprising the ivanovic mutually unbiased base mub and the alltop mubs vectors in the alltop mubs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all alltop vectors splits into three clifford orbits these vectors form configurations with so called zauner subspaces eigenspaces of order 3 elements of the clifford group highly relevant to the sic problem we identify alltop vectors as the magic states that appear in the context of fault tolerant universal quantum computing wherein the appearance of distinct clifford orbits implies a surprising inequivalence between some magic states

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

a dynamical meaning of fractal dimension when two atlractors of a dynamical system have a common basin boundary b small changes in initial conditions which lie near b can result in radically different long term behavior of the trajectory a quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary b

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

on the intertwined basins of attraction for planar flows in this paper we discuss the intertwined basins of attraction one of the simplest cases of obstruction to predictability some sufficient conditions to guarantee the existence of intertwined basins of attraction are given

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

on basin boundaries in this paper we make a study of a class of basin boundaries we show that an efficient characterization of a given element of this class can be constructed with segments of differentiable curves

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

multiresolution map despeckling of sar images based on locally adaptive generalized gaussian pdf modeling in this paper a new despeckling method based on undecimated wavelet decomposition and maximum a posteriori map estimation is proposed such a method relies on the assumption that the probability density function pdf of each wavelet coefficient is generalized gaussian gg the major novelty of the proposed approach is that the parameters of the gg pdf are taken to be space varying within each wavelet frame thus they may be adjusted to spatial image context not only to scale and orientation since the map equation to be solved is a function of the parameters of the assumed pdf model the variance and shape factor of the gg function are derived from the theoretical moments which depend on the moments and joint moments of the observed noisy signal and on the statistics of speckle the solution of the map equation yields the map estimate of the wavelet coefficients of the noise free image the restored sar image is synthesized from such coefficients experimental results carried out on both synthetic speckled images and true sar images demonstrate that map filtering can be successfully applied to sar images represented in the shift invariant wavelet domain without resorting to a logarithmic transformation

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

sensitivity of set valued discrete systems abstract consider the surjective continuous map f x x and the continuous map f of k x into itself induced by f where x is a compact metric space and k x is the space of all non empty compact subsets of x endowed with a hausdorff metric in this paper we give examples showing that sensitivity of f does not imply sensitivity of f furthermore we prove that if f is a surjective continuous interval map then f is sensitive if and only if f is sensitive

a transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map a powerful theorem and construction of wayne lewis are used to build two homeomorphisms on the pseudoarc each of which is semiconjugate to the tent map on the unit interval the first homeomorphism is transitive thus answering a question of marcy barge as to whether such homeomorphisms exist the second homeomorphism admits wandering points also it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points a continuum is a compact connected metric space a continuum x is indecomposable if every proper subcontinuum of x is nowhere dense in x and x is hereditarily indecomposable if each subcontinuum of x is indecomposable a continuum x is chainable if for each e 0 there is a chain c c0 cn of open sets of diameter less than e that covers x c is a chain means that ci n ej 0 iff ii ii 2 or in the hilbert cube are pseudoarcs here we will show that this continuum admits a transitive homeomorphism thus answering a question of marcy barge in addition this homeomorphism has the property that it is semiconjugate to the tent map on the unit interval for more information on pseudoarcs and indecomposable continua see bil bi4 l1 l2 k km kr and ot exotic continua have been making their appearance in dynamical systems ever since the early part of this century with the work of c caratheodory c received by the editors may 20 1988 and in revised form july 25 1989 presented at the 1988 spring topology conference university of florida gainesville florida 32611 1980 mathematics subject classification 1985 revision primary 54f20 secondary 54h20

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

wavelet based sar image despeckling and information extraction using particle filter this paper proposes a new wavelet based synthetic aperture radar sar image despeckling algorithm using the sequential monte carlo method a model based bayesian approach is proposed this paper presents two methods for sar image despeckling the first method called wggpf models a prior with generalized gaussian gg probability density function pdf and the second method called wgmpf models prior with a generalized gaussian markov random field ggmrf the likelihood pdf is modeled using a gaussian pdf the ggmrf model is used because it enables texture parameter estimation the prior is modeled using gg pdf when texture parameters are not needed a particle filter is used for drawing particles from the prior for different shape parameters of gg pdf when the ggmrf prior is used the particles are drawn from prior in order to estimate noise free wavelet coefficients and for those coefficients the texture parameter is changed in order to obtain the best textural parameters the texture parameters are changed for a predefined set of shape parameters of ggmrf the particles with the highest weights represents the final noise free estimate with corresponding textural parameters the despeckling algorithms are compared with the state of the art methods using synthetic and real sar data the experimental results show that the proposed despeckling algorithms efficiently remove noise and proposed methods are comparable with the state of the art methods regarding objective measurements the proposed wgmpf preserves textures of the real high resolution sar images well

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

fractals and dynamical chaos in a random 2d lorentz gas with sinks abstract two dimensional random lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap in systems of finite spatial extension the asymptotic decay of the survival probability is exponential and characterized by an escape rate which can be related to the average positive lyapunov exponent and to the dimension of the fractal repeller of the system for infinite systems the survival probability obeys a stretched exponential law of the form p c t exp ct1 2 the transition between the two regimes is studied and we show that for a given trap density the non integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space nevertheless the repeller remains fractal we determine the special scaling properties of this fractal

sic povms and clifford groups in prime dimensions we show that in prime dimensions not equal to 3 each group covariant symmetric informationally complete positive operator valued measure sic povm is covariant with respect to a unique heisenberg weyl hw group moreover the symmetry group of the sic povm is a subgroup of the clifford group hence two sic povms covariant with respect to the hw group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended clifford group in dimension 3 each group covariant sic povm may be covariant with respect to three or nine hw groups and the symmetry group of the sic povm is a subgroup of at least one of the clifford groups of these hw groups respectively there may exist two or three orbits of equivalent sic povms for each group covariant sic povm depending on the order of its symmetry group we then establish a complete equivalence relation among group covariant sic povms in dimension 3 and classify inequivalent ones according to the geometric phases associated with fiducial vectors finally we uncover additional sic povms by regrouping of the fiducial vectors from different sic povms which may or may not be on the same orbit of the extended clifford group

analysis of chaotic saddles in low dimensional dynamical systems the derivative nonlinear schrodinger equation abstract in this paper we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low dimensional dynamical system describing stationary solutions of the derivative nonlinear schrodinger equation a driven dissipative model for alfven waves these chaotic saddles have gaps which are filled at chaotic transitions such as a saddle node bifurcation and an interior crisis we give a detailed explanation of how to numerically determine the chaotic saddles and describe how a chaotic attractor after an interior crisis point can be decomposed into two chaotic saddles dynamically connected by a set of coupling unstable periodic orbits created by a gap filling explosion after the crisis this coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

the nonsubsampled contourlet transform theory design and applications in this paper we develop the nonsubsampled contourlet transform nsct and study its applications the construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks the result is a flexible multiscale multidirection and shift invariant image decomposition that can be efficiently implemented via the a trous algorithm at the core of the proposed scheme is the nonseparable two channel nonsubsampled filter bank nsfb we exploit the less stringent design condition of the nsfb to design filters that lead to a nsct with better frequency selectivity and regularity when compared to the contourlet transform we propose a design framework based on the mapping approach that allows for a fast implementation based on a lifting or ladder structure and only uses one dimensional filtering in some cases in addition our design ensures that the corresponding frame elements are regular symmetric and the frame is close to a tight one we assess the performance of the nsct in image denoising and enhancement applications in both applications the nsct compares favorably to other existing methods in the literature

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

orbits of mutually unbiased bases we express alltop s construction of mutually unbiased bases as orbits under the weyl heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4 we reproduce alltop s mutually unbiased bases using abelian subgroups of the clifford group in prime dimensions in direct analogy to the well known construction of mutually unbiased bases using abelian subgroups of the weyl heisenberg group finally we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases

self similar constructions in smooth dynamics rigidity smoothness and dimension in the first part of this paper for eachd 2 we construct diffeomorphisms of thed dimensional ball which have zero entropy one periodic orbit with period 2n for eachn 0 no other periodic orbits and a single invariant cantor set which has a continuum of possible but in any case simple geometric structures these diffeomorphisms arecr d smooth wherer d is a strictly increasing function ofd which goes to infinity withd the second part contains a more general result about smooth maps obtained by an infinite sequence of surgeries and further particular cases

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

unpredictability of the asymptotic attractors in phase coupled oscillators an array of phase coupled oscillators may exhibit multiple coexisting chaotic and nonchaotic attractors the system of coupled circle maps is such an example we demonstrate that it is common for this type of system to exhibit an extreme type of final state sensitivity in both parameter and phase space numerical computations reveal that there exist substantial regions of the parameter space where arbitrarily small perturbations in parameters or initial conditions can alter the asymptotic attractor of the system completely consequently asymptotic attractors of the system cannot be predicted reliably for specific parameter values and initial conditions

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

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

on the creation of wada basins in interval maps through fixed point tangent bifurcation basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology chemistry economics engineering and physics one of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins a basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins is called a wada basin and its boundary is called a wada basin boundary here we consider maps on the interval we present a sufficient and necessary condition guaranteeing that three wada basins are emerging from a tangent bifurcation for certain one dimensional maps having negative schwarzian derivative two fixed point attractors on one side of the tangent bifurcation and three fixed point attractors on the other side of the tangent bifurcation all the conditions involved are numerically verifiable

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

a splitting theorem for transitive maps in this paper we prove a splitting theorem for continuous transitive maps on locally connected compact metric spaces which generalizes the results of barge and martin 4 for transitive maps on the interval and of blokh 7 for transitive graph maps as a consequence lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings are obtained also we show that for every connected graph which is not a tree the infimum of the topological entropy of the transitive maps having a periodic point is zero since the topological entropy of transitive maps without periodic points is zero 7 and positive lower bounds of the entropy were given for the transitive maps on trees 2 this paper completes the problem of obtaining lower bounds of the topological entropy for transitive maps of any connected graph

classical codes in quantum state space we present a construction of hermitian operators and quantum states labelled by strings from a finite field the distance between these operators or states is then simply related typically proportional to the hamming distance between their corresponding strings this allows a straightforward application of classical coding theory to find arrangements of operators or states with a given distance distribution using the simplex or extended reed solomon code in our construction recovers the discrete wigner function which has important applications in quantum information theory

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

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

dynamic bifurcations in non stationary systems transitions with an unpredictable outcome in nonlinear non stationary systems dynamic bifurcations result in a transition to a qualitatively new state in this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction we delineate the phenomenon of indeterminate dynamic bifurcations where it is shown that the response after the system passes through critical parameter values may be extremely sensitive to the choice of initial conditions or parameter states this new form of unpredictability in systems whose parameters vary with time is clearly an important concept to be assimilated in the theory of non stationary dynamics