networks of theta neurons with time varying excitability macroscopic chaos multistability and final state uncertainty abstract using recently developed analytical techniques we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons excitability parameter varies in time we demonstrate that such periodic variation can lead to the emergence of macroscopic chaos multistability and final state uncertainty in the collective behavior of the network finite size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated

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

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

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

the pim simplex method an extension of the pim triple method to saddles with an arbitrary number of expanding directions abstract the growing interest in non attracting chaotic sets of high dimensional dynamical systems requires the development of numerical techniques for their study the pim triple method h e nusse j a yorke physica d 36 1989 137 is a very good method to obtain trajectories on saddles with one positive lyapunov exponent in this paper we combine the same ideas with an algorithm for finding local extrema of multi variable functions to develop an extension of the method the pim simplex method that is suitable for the study of sets with an arbitrary number of expanding directions

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

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

multi dimensioned intertwined basin boundaries basin structure of the kicked double rotor abstract using numerical computations on a map which describes the time evolution of a particular mechanical system in a four dimensional phase space the kicked double rotor we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale in particular for the double rotor map if one chooses a restricted region of the phase space and examines the basin boundary in that region then either one observes that the boundary is a smooth three dimensional surface or one observes that the boundary is fractal with dimension d 3 9 and which of these two possibilities applies depends on the particular phase space region chosen for examination furthermore for any region no matter how small for which d 3 9 one can choose subregions within it for which d 3 hence d 3 9 region and d 3 region are intertwined on arbitrarily fine scale other examples will also be presented and analyzed to show how this situation can arise these include one dimensional map cases a map of the plane and the lorenz equations in one of our one dimensional map cases the boundary will be fractal everywhere but the dimension can take on either of two different values both of which lie between 0 and 1 these examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values more specifically let these values be denoted d 1 d 2 d n choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region then our conjecture is that for all typical volume choices the evaluated dimension within the chosen volume will be one of the values d 1 d 2 d n for example in our double rotor map it appears that n 2 and d 1 3 0 and d 2 3 9

chaotic principle an experimental test abstract the chaotic hypothesis discussed in gallavotti and cohen 1995 is tested experimentally in a simple conduction model besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which as the forcing strength grows the attractor ceases to be an anosov system and becomes an axiom a attractor a first text of new predictions is also attempted

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

chaos and crises in more than two dimensions instituto de astronomi ayfi sica del espacio conicet casilla de correo 67 socursal 28 1428 buenos aires argentina received 23 december 1996 noisy chaotic trajectories with nite time lyapunov exponents that uctuate about zero are basicallyunshadowable s dawson c grebogi t sauer and j a yorke phys rev lett 73 1927 1994 this canoccur when periodic orbits with different numbers of unstable directions coexist inside the attractor thepresence of a he non type chaotic saddle i e a nonattracting chaotic set with a structure similar to that of thehe non attractor guarantees such coexistence in a persistent manner s p dawson phys rev lett 76 4348 1996 in this paper we describe how these sets appear naturally in maps of more than two dimensions howthey can be found and what crises they produce s1063 651x 97 13005 7 pacs number s 05 45 1b 05 40 1ji introduction

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

rotation numbers of periodic orbits in the henon map for invertible area contracting maps of the plane it is common for a basin of attraction to have a fractal basin boundary certain periodic orbits on the basin boundary are distinguished by being accessible by a path from the interior of the basin a numerical study is made of the accessible periodic orbits for the henon family of maps theoretical results on rotary homoclinic tangencies are given which describe the appearance of the accessible saddles and organize them in a natural way according to the continued fractions expansions of their rotation numbers

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

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

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

permutation and grouping methods for sharpening gaussian process approximations abstractvecchia s approximate likelihood for gaussian process parameters depends on how the observations are ordered which has been cited as a deficiency this article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations indeed the first part of the article includes a systematic study of how ordering affects the accuracy of vecchia s approximation we demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate based orderings additional ordering schemes are described and analyzed numerically including orderings capable of improving on random orderings the second contribution of this article is a new automatic method for grouping calculations of components of the approximation the grouping methods simultaneously improve approximation accuracy and reduce computational burden in common settings reordering combined with grouping reduces kullback leibler divergence from the target model by more th

on the fluctuation relation for nose hoover boundary thermostated systems we discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures the system is a modified lorentz gas in which the fixed scatterers exchange energy with the gas of particles and the thermostats are modelled by two nose hoover thermostats applied at the boundaries of the system the transient fluctuation relation which holds only for a precise choice of the initial ensemble is verified at all times as expected times longer than the mesoscopic scale needed for local equilibrium to be settled are required if a different initial ensemble is considered this shows how the transient fluctuation relation asymptotically leads to the steady state relation when as explicitly checked in our systems the condition found in d j searles et al j stat phys 128 1337 2007 for the validity of the steady state fluctuation relation is verified for the steady state fluctuations of the phase space contraction rate and of the dissipation function a similar relaxation regime at shorter averaging times is found the quantity satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale the quantity appears to begin a monotonic convergence after such times this is consistent with the fact that and differ by a total time derivative and that the tails of the probability distribution function of are gaussian

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

characterization in bi parameter space of a non ideal oscillator abstract we investigate the dynamical behavior of a non ideal duffing oscillator a system composed of a mass spring pendulum driven by a dc motor with limited power supply to identify new features on duffing oscillator parameter space due to the limited power supply we provide an extensive numerical characterization in the bi parameter space by using lyapunov exponents following this procedure we identify remarkable new organized distribution of periodic windows the ones known as arnold tongues and also shrimp shaped structures in addition we also identify intertwined basins of attraction for coexisting multiple attractors connected with tongues

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

note on two theorems in nonequilibrium statistical mechanics an attempt is made to clarify the difference between a theorem derived by evans and searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state

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

universal and nonuniversal features in shadowing dynamics of nonhyperbolic chaotic systems with unstable dimension variability an important quantity characterizing the shadowability of computer generated trajectories in nonhyperbolic chaotic system is the shadowing time which measures for how long a numerical trajectory remains valid this time depends sensitively on an initial condition here we show that for nonhyperbolic systems with unstable dimension variability the probability distribution of the shadowing time contains two distinct scaling behaviors an algebraic scaling for short times and an exponential scaling for long times the exponential behavior depends on the system details but the small time algebraic behavior appears to be universal

periodicity and decomposability of basin boundaries with irrational maps on prime ends planar basin boundaries of iterated homeomorphisms induce homeomorphisms on prime ends when the basin is connected simply connected and has a compact connected boundary the space of prime ends is a topological circle if the induced homeomorphism on prime ends has rational rotation number the basin boundary contains periodic orbits several questions as to basin boundary periodics decomposability and minimality when the induced map on prime ends has irrational rotation number are answered by construction of both homeomorphisms and diffeomorphisms examples in the literature of basin boundaries with interesting prime end dynamics have been sparse prime end dynamics has drawn recent interest as a natural tool for the study of strange attractors in their 1951 manuscript entitled some fixed point theorems m l cartwright and j e littlewood proved that if the induced map on caratheodory s prime ends has rational rotation number then a connected basin boundary of an orientation preserving planar homeomorphism must contain a periodic orbit cl 1 c in this note we explore the relationship between aperiodic maps induced on the circle of prime ends and the periodicity minimality and decomposability of basin boundaries attractors as frequently defined serve as connected basin boundaries and one usually requires them to contain dense orbits the existence of a dense orbit would distinguish an attractor from a union of periodic orbits or other attracting sets can the map on prime ends induced from the basin itself help detect these dense orbits we construct a basin separating attractor of a c1 diffeomorphism containing no dense orbits which induces pure irrational rotation of both circles of prime ends example a but some minimal attractors all orbits dense can induce two denjoy maps on prime ends on both sides which have no dense orbits den example e conceptually one imagines the map on the circle of prime ends as the action of the planar homeomorphism on the basin boundary as viewed from the basin precise definitions of key terminology follow an excellent reference is j n mather s exposition m received by the editors october 28 1988 and in revised form march 14 1989 1980 mathematics subject classification 1985 revision primary 54h20 secondary 58f12 58f22 1991 american mathematical society 0002 9947 91 1 00 25 per page

dynamics from multivariate time series abstract multivariate time series data are common in experimental and industrial systems if the generating system has nonlinear dynamics we may be able to construct a model that reproduces the dynamics and can be used for control and other purposes in principle multivariate time series are not necessary for recovering dynamics according to the embedding theorem only one time series should be needed however for real data there may be large gains in using all of the measurements in this paper we examine the issues of how to use multiple data streams most effectively for modeling and prediction for example perhaps the data are redundant in that only a subset of the data streams is useful and how should we embed the data if indeed embedding is required at all we show how these questions can be answered and describe some numerical experiments which show that using multivariate time series can significantly improve predictability we also demonstrate a somewhat surprising synchronization between different reconstructions

a procedure for finding numerical trajectories on chaotic saddles abstract examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors if almost every trajectory eventually leaves some region but the region contains a chaotic set then typical trajectories will behave chaotically for a while and then will leave the region and so we will observe chaotic transients the main objective that will be addressed is the dynamic restraint problem given a region that contains a chaotic set but does not contain an attractor find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time systems with horseshoes have such regions as do systems with fractal basin boundaries as does the henon map for suitably chosen parameters we present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory the method is called the pim triple procedure since it is based on so called pim triples a pim proper interior maximum triple is three points a c b in a straight line segment such that the interior point c i e c is between a and b has the maximum escape time that is its escape time from the region is greater than the escape time of both a and b proper means the segment from a to b is smaller than a previously obtained segment we show rigorously that the pim triple procedure works in ideal situations we find it works well even in less than ideal cases this procedure can also be used for the computation of lyapunov exponents furthermore the accessible pim triple procedure a refined pim triple procedure for finding accessible trajectories on the chaotic saddle will also be discussed

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

approximating likelihoods for large spatial data sets summary likelihood methods are often difficult to use with large irregularly sited spatial data sets owing to the computational burden even for gaussian models exact calculations of the likelihood for n observations require o n3 operations since any joint density can be written as a product of conditional densities based on some ordering of the observations one way to lessen the computations is to condition on only some of the past observations when computing the conditional densities we show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating through theoretical numerical and practical examples we show that there can often be considerable benefit in conditioning on some distant observations as well

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

prediction of reservoir variables based on seismic data and well observations offshore petroleum reservoirs are usually explored by acquisition of vast amounts of seismic reflection data and observations along a small number of wells drilled through the reservoirs the seismic data are indirect measurements of reservoir characteristics and have good spatial coverage but low precision the well observations are sparse but precise information from these sources of data are integrated to characterize the reservoir variables of interest reservoir porosity in the current study critical parameters in the acquisition procedure are unknown and are simultaneously estimated this process termed seismic inversion constitutes a spatial multivariate ill posed inverse problem the problem is cast in a bayesian framework with a focus on sampling from the posterior model high dimensionality nonlinear components and unknown parameters in the likelihood model along with a complex design of the conditioning data complicate the problem a fast sequential sampling algorithm containing severa

fluctuation relation beyond linear response theory the fluctuation relation fr is an asymptotic result onthe distribution of certain observables averaged over timeintervals as and it is a generalization of thefluctuation dissipation theorem to far from equilibrium systemsin a steady state which reduces to the usual green kubo gk relation in the limit of small external non conservative forces fr is a theorem for smooth uniformly hyperbolic systems and it isassumed to be true in all dissipative chaotic enough systemsin a steady state in this paper we develop a theory of finitetime corrections to fr needed to compare the asymptoticprediction of fr with numerical observations which necessarilyinvolve fluctuations of observables averaged over finite timeintervals we perform a numerical test of fr in two cases inwhich non gaussian fluctuations are observable while gk does notapply and we get a non trivial verification of fr that is independent of and different from linear response theory our results are compatible with the theory of finite timecorrections to fr while fr would be observably violated well within the precision of our experiments if such correctionswere neglected

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

joseph geology and seismic deconvolution common practice in seismic deconvolution is to assume that the reflection sequence is uncorrelated that is that the sequence has a white power spectrum and a delta function autocorrelation a white spectrum implies that the acoustic impedance function has a power spectrum proportional to 1 f2 which is characteristic of a nonstationary brownian process f is frequency however the maximum power spectrum permissible for the acoustic impedance function is 1 f we call a spectrum of this kind a joseph spectrum a joseph spectrum corresponds to a reflection sequence with a power spectrum proportional to f and a negative autocorrelation at small lags joseph spectrum behavior for reflection sequences has been seen before and we show it again in a well off newfoundland and in two wells from quebec if the power spectrum is proportional to f then the first term of the discretized autocorrelation function is 0 405 of the zero lag term and higher terms are negligible we construct a joseph filter analogous t

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

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

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

jumping stable manifolds for dissipative maps of the plane abstract one of the more drastic changes in the nature of a chaotic dynamical system is a transformation of the basins of competing attractors in a two dimensional system the basin boundary is typically the stable manifold of a saddle point the transformation of the basin boundary from smooth to fractal is presaged by a homoclinic tangency for the saddle point as a varying parameter in a dissipative map of the plane forces a homoclinic tangency we show that the location of the stable manifold changes discontinuously this forces a basin boundary to change in a dramatic way which is often an immediate precursor of the collision between a strange attractor and its basin boundary finally we describe the application of our results to the behavior of the ikeda map from nonlinear optics

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

plotting stable manifolds error estimates and noninvertible maps a numerical procedure is described that can accurately compute the stable manifold of a saddle fixed point for a map of r2 even if the map has no inverse conventional algorithms use the inverse map to compute an approximation of the unstable manifold of the fixed point we rigorously analyze the errors that arise in the computation and guarantee that they are small we also argue that a simpler nonrigorous algorithm nevertheless produces highly accurate representations of the stable manifold

periodic orbit theory of the blowout bifurcation this paper presents a theory for characterization of the blowout bifurcation by periodic orbits blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric invariant subspace becomes transversely unstable we present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits

extended heat fluctuation theorems for a system with deterministic and stochastic forces heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components a brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity using a langevin equation we find the exact fourier transform of the distribution of these fluctuations for all tau by a saddle point method we obtain analytical results for the inverse fourier transform which for not too small tau agree very well with numerical results from a sampling method as well as from the fast fourier transform algorithm due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid the conventional heat fluctuation theorem is for infinite and for finite tau replaced by an extended fluctuation theorem that differs noticeably and measurably from it in particular for large fluctuations the ratio of the probability for absorption of heat by the particle from the fluid to the probability to supply heat by the particle to the fluid is much larger here than in the conventional fluctuation theorem

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

statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability severe obstruction to shadowing of computer generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability that is when the dimension of the unstable eigenspace changes along a trajectory in the invariant set no true trajectory of reasonable length can be found to exist near any numerically generated trajectory an important quantity characterizing the shadowability of numerical trajectories is the shadowing time which measures for how long a trajectory remains valid this time depends sensitively on initial condition here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors an algebraic scaling for short times and an exponential scaling for long times the exponential behavior depends on system details but the small time algebraic behavior appears to be universal we describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

optimal escape from potential wells patterns of regular and chaotic bifurcation abstract the patterns of bifurcation governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier are studied by numerical simulation both the generic asymmetric single well cubic potential and the symmetric twin well potential duffing oscillator are surveyed by varying three parameters forcing frequency forcing amplitude and damping coefficient the close relationship between optimal escape and nonlinear resonance within the well is confirmed over a wide range of damping subtle but significant differences are observed at higher damping ratios the possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is cmppletely suppressed above a critical level of the damping ratio about 0 12 for the asymmetric single well oscillator coincidentally at almost the same level of damping the optimal escape condition becomes distinct from the apex in the f plane of the bistable regime this corresponds to the appearance of chaotic attractors which subsume both resonant and non resonant motions within one well at higher damping levels further changes occur involving conversions from chaotic saddle to regular saddle bifurcations these changes in optimal escape phenomena correspond to codimension three bifurcations at exceptional points in the space of three parameters these bifurcations are described in terms of homoclinic and heteroclinic structures of invariant manifolds and changes in accessible boundary orbits the same sequence of codimension three bifurcations is observed in both the twin well duffing oscillator and the asymmetric single well escape equation within the codimension three bifurcation patterns governing escape one particular codimension two global bifurcation involves a chaotic attractor explosion or interior crisis compounded with a blue sky catastrophe or boundary crisis of the exploded attractor this codimension two bifurcation has structure containing a form of predictive power knowledge of attractor bifurcations in part of the codimension two pattern permits inference of the attractor and basin bifurcations in the remainder this predictive power is applicable beyond the context of escape from potential wells quantitative correlation of bifurcation patterns between the two equations according to simple scaling laws is tested the unstable periodic orbits which figure most prominently in the major attractor basin bifurcations are of periods one and three their linking is conveniently interpreted by a three layer spiral horseshoe structure for the folding action in phase space within a well the structure of this 3 shoe implies a partial ordering among order three subharmonic saddle node bifurcations this helps explain the sequence of codimension three bifurcations near optimal escape some bifurcational precedence relations are known to follow from the linking of periodic orbits in a braid on a 3 shoe additional bifurcational precedence relations follow from a quantitative property of generic potential wells the dynamic hilltop saddle has a very large expanding multiplier over one cycle of forcing near fundamental resonance this quantitative property explains the close coincidence of codimension three bifurcations near the suppression of indeterminate outcomes an experimentalist s approach to identifying the three layer template structure from time series data is discussed including a consistency check involving poincare indices the bifurcation patterns emerging at higher damling values create favorable conditions for realizing experimental strategies to recognize optimal escape and locate it in parameter space strategies based solely on observations of quasi steady behavior while remaining always within one well are discussed

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

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

the steady state fluctuation relation for the dissipation function we give a proof of transient fluctuation relations for the entropy production dissipation function in nonequilibrium systems which is valid for most time reversible dynamics we then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax producing a unique nonequilibrium steady state although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown if such a steady state exists the generation of the steady state fluctuation relation from the transient relation is shown to be very general it is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation which is needed also for e g the existence of transport coefficients because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities the steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor whose convergence is problematic for systems close to equilibrium we examine some model dynamics that have been considered previously and show how they are described in the context of this work

fluctuation theorems for entropy production and heat dissipation in periodically driven markov chains asymptotic fluctuation theorems are statements of a gallavotti cohen symmetry in the rate function of either the time averaged entropy production or heat dissipation of a process such theorems have been proved for various general classes of continuous time deterministic and stochastic processes but always under the assumption that the forces driving the system are time independent and often relying on the existence of a limiting ergodic distribution in this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous time processes without a stationary distribution considering specifically a finite state markov chain driven by periodic transition rates we find that for both entropy production and heat dissipation the usual gallavotti cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process in which the trajectory and the driving protocol have been time reversed the effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process and vice versa revealing that the distributions of fluctuations in universes in which time moves forward and backward are related as an additional result the asymptotic time averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics

bayesian linearized avo inversion a new linearized avo inversion technique is developed in a bayesian framework the objective is to obtain posterior distributions for p wave velocity s wave velocity and density distributions for other elastic parameters can also be assessed for example acoustic impedance shear impedance and p wave to s wave velocity ratio the inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the zoeppritz equation the solution is represented by a gaussian posterior distribution with explicit expressions for the posterior expectation and covariance hence exact prediction intervals for the inverted parameters can be computed under the specified model the explicit analytical form of the posterior distribution provides a computationally fast inversion method tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero with realistic noise levels acoustic impedance was the best determined parameter

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

application of the gallavotti cohen fluctuation relation to thermostated steady states near equilibrium the fluctuation relation of the gallavotti cohen fluctuation theorem gcft concerns fluctuations in the phase space compression rate of dissipative reversible dynamical systems it has been proven for anosov systems but it is expected to apply more generally this raises the question of which non anosov systems satisfy the fluctuation relation we analyze time dependent fluctuations in the phase space compression rate of a class of n particle systems that are at equilibrium or in near equilibrium steady states this class does not include anosov systems or isoenergetic systems however it includes most steady state systems considered in molecular dynamics simulations of realistic systems we argue that the fluctuations of the phase space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the gcft although the discrepancies become somewhat smaller as the systems move further from equilibrium in contrast similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium

zon cohen singularity and negative inverse temperature in a trapped particle limit we study a brownian particle on a moving periodic potential we focus on the statistical properties of the work done by the potential and the heat dissipated by the particle when the period and the depth of the potential are both large by using a boundary layer analysis we calculate a cumulant generating function and a biased distribution function the result allows us to understand a zon cohen singularity for an extended fluctuation theorem from a viewpoint of rare trajectories characterized by a negative inverse temperature of the biased distribution function

the equality of fractal dimension and uncertainty dimension for certain dynamical systems mgoy introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system in mgoy and p it was conjectured that the box counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems in this paper our main result is that the box counting dimension the uncertainty dimension and the hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems which are uniformly hyperbolic on their basin boundary when the box counting dimension of the basin boundary is large that is near the dimension of the phase space this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in

chaotic hypothesis fluctuation theorem and singularities the chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them however its application to physics problems requires attention and can lead to apparent inconsistencies in particular there are several cases that have been considered in the literature in which singularities are built in the models for instance when among the forces there are lennard jones potentials which are infinite in the origin and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded the situation is well understood in certain special cases in which the system is subject to gaussian noise here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived the main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider ie deterministic systems with thermostat forces acting according to gauss principle for the constraint of constant total kinetic energy isokinetic gaussian thermostats close and far from equilibrium near equilibrium it even predicts a fluctuation relation which in deterministic cases with more general thermostat forces ie not necessarily of gaussian isokinetic nature extends recent relations obtained in situations in which the thermostatting forces satisfy gauss principle this relation agrees where expected with the fluctuation theorem for perfectly chaotic systems the results are compared with some recent works in the literature

test of the fluctuation relation in lagrangian turbulence on a free surface the statistics of velocity divergence are studied for an assembly of particles that float on a closed turbulent fluid under an appropriate definition of entropy the two dimensional lagrangian velocity divergence of a particle trajectory represents the local entropy rate dot s a random variable in time the statistics of this rate measured in the lagrangian frame are collected over a wide range of values this permits a severe test of the fluctuation relation fr over a range that exceeds prior experiments out to a regime beyond which the fr no longer holds notably the probability density functions pdf of the dimensionless divergence are strongly non gaussian

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

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

a bayesian approach to stochastic capture zone delineation incorporating tracer arrival times conductivity measurements and hydraulic head observations this paper presents a methodology to invert tracer arrival times and to incorporate travel time data in the delineation of well capture zones within a bayesian framework the observed arrival times are used to obtain probability based weights for each realization of the hydraulic conductivity field realizations that closely reproduce the observed arrival times are more likely to represent the true or real conductivity field than realizations yielding worse predictions and are consequently characterized by a higher conditional probability in the prediction of the capture zones the realizations are weighted by their respective probability we combine the arrival time data with conductivity measurements and head observations to delineate stochastic capture zones the conductivity measurements update the prior distributions specified for the unknown structural parameters of the conductivity field and are used as conditioning data in the generation of conditional conductivity fields the parameter values used to generate the conductivity realizations are sampled by monte carlo from the updated parameter distributions the head and travel time observations are subsequently used to assign probability based weights to the conductivity realizations by applying bayes theorem the hyperparameters of the error model are assumed unknown and their effect is accounted for by marginalization a synthetic flow setup consisting of a single extraction well in a fully confined aquifer under a regional gradient is used to illustrate the method we evaluate the relative worth of the different data types by introducing them sequentially in the inverse methodology results indicate that the different data types are complementary and that the travel time data are essential to improve the predictions of the capture zones this is confirmed by the results for the case where uncertainty in the homogeneous porosity is accounted for

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

method for measuring unstable dimension variability from time series many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity one of the possible violations of this condition is the presence of unstable dimension variability udv i e the existence in a chaotic attractor of sets of unstable periodic orbits each with a different number of expanding directions it has been shown that the presence of udv poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems the shadowing time in this work we propose a method to detect the presence of udv in real systems from time series measurements variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor we show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

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

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3