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

a manifold independent approach to understanding transport in stochastic dynamical systems abstract we develop a new collection of tools aimed at studying stochastically perturbed dynamical systems specifically in the setting of bi stability that is a two attractor system it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero noise case barriers in the phase space pseudo barriers thus creating a pre heteroclinic tangency chaos like behavior the stochastic dynamical system has a corresponding frobenius perron operator with a stochastic kernel which describes how densities of initial conditions move under the noisy map thus in studying the action of the frobenius perron operator we learn about the transport of the map we have employed a galerkin ulam like method to project the frobenius perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space graph theoretic methods allow us to re order the resulting finite dimensional markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo barriers of the stochastic dynamics our toolbox allows us to find 1 regions of high activity of transport 2 flux across pseudo barriers and also 3 expected time of escape from pseudo basins some of these quantities are also possible via the manifold dependent stochastic melnikov method but melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available our methods are unique in that they can essentially be considered as a black box of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures we use here a model of childhood diseases to showcase our methods our tools will allow us to make specific observations of 1 loss of reducibility between basins with increasing noise 2 identification in the phase space of active regions of stochastic transport 3 stochastic flux which essentially completes the heteroclinic tangle

modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps abstract we model a two dimensional open fluid flow that has temporally irregular time dependence by a random map n 1 m n n where on each iterate n the map m n is chosen from an ensemble we show that a tracer distribution advected through a chaotic region can be entrained on a set that becomes fractal as time increases theoretical and numerical results on the multifractal dimension spectrum are presented

stochastic climate dynamics random attractors and time dependent invariant measures a b s t r a c t this article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s the nonlinear deterministic and the linear stochastic one this unification via the theory of random dynamical systems rds allows one to consider the detailed geometric structure of the random attractors associated with nonlinear stochastically perturbed systems we report on high resolution numerical studies of two idealized models of fundamental interest for climate dynamics the first of the two is a stochastically forced version of the classical lorenz model the second one is a low dimensional nonlinear stochastic model of the el nino southern oscillation enso these studies provide a good approximation of the two models global random attractors as well as of the time dependent invariant measures supported by these attractors the latter are shown to have an intuitive physical interpretation as random versions of sinai ruelle bowen srb measures

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

bi instability and the global role of unstable resonant orbits in a driven laser driven class b lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior we describe the onset of global heteroclinic connections which give rise to chaotic saddles these form the precursor topology which creates both localized homoclinic chaos as well as global mixed mode heteroclinic chaos to locate the relevant periodic orbits creating the precursor topology approximate maps are derived using matched asymptotic expansions and subharmonic melnikov theory locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser 2000 published by elsevier science b v

proof of certain diophantine conjectures and identification of remarkable classes of orthogonal polynomials certain diophantine conjectures are proven and to do so certain remarkable classes of orthogonal polynomials are identified yielding additional diophantine findings

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

tridiagonal matrices orthogonal polynomials and diophantine relations i it is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three term recursion relations and being therefore members of an orthogonal set a class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers in the process certain neat formulae are also obtained which perhaps deserve to be included in standard compilations since they involve classical polynomials such as the jacobi polynomials and other named polynomials

characterization of nonstationary chaotic systems nonstationary dynamical systems arise in applications but little has been done in terms of the characterization of such systems as most standard notions in nonlinear dynamics such as the lyapunov exponents and fractal dimensions are developed for stationary dynamical systems we propose a framework to characterize nonstationary dynamical systems a natural way is to generate and examine ensemble snapshots using a large number of trajectories which are capable of revealing the underlying fractal properties of the system by defining the lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots we show that the kaplan yorke formula which is fundamental in nonlinear dynamics remains valid most of the time even for nonstationary dynamical systems

escape from attracting sets in randomly perturbed systems the dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science previous studies of escape in chaotic systems have rather focused on the case of unbounded noise usually assumed to have gaussian distribution in this paper we address the problem of escape induced by bounded noise we show that the dynamics of escape from an attractor s basin is equivalent to that of a closed system with an appropriately chosen hole using this equivalence we show that there is a minimum noise amplitude above which escape takes place and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition we verify our analytical predictions through numerical simulations of two well known two dimensional maps with noise

on isochronous bruschi ragnisco ruijsenaars toda lattices equilibrium configurations behaviour in their neighbourhood diophantine relations and conjectures isochronous versions of the bruschi ragnisco ruijsenaars toda lattice and of some of its also integrable variants are introduced their equilibrium configurations are found when they exist and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known

driving a conceptual model climate by different processes snapshot attractors and extreme events in a low order chaotic model of global atmospheric circulation the effects of driving i e time dependent periodic chaotic and noisy forcing are investigated with particular interest in extremal behavior an approach based on snapshot attractors formed by a trajectory ensemble is applied to represent the time dependent likelihood of extreme events in terms of a physical observable a single trajectory based framework on the other hand is used to determine the maximal value and the kurtosis of the distribution of the same observable we find the most significant effect of the driving on the magnitude relative frequency and variability of extreme events when its characteristic time scale becomes comparable to that of the model climate extreme value statistics is pursued by the method of block maxima and found to follow weibull distributions deterministic drivings result in shape parameters larger in modulus than stochastic drivings but otherwise strongly dependent on the particular type of driving the maximal effects of deterministic drivings are found to be more pronounced both in magnitude and variability of the extremes than white noise and the latter has a stronger effect than red noise

analysis of transient electromagnetic scattering from overfilled cavities in this paper we consider the time domain scattering problem of a two dimensional overfilled cavity embedded in the infinite ground plane the problem is first discretized in time by the beta gamma newmark time marching scheme at each time step the variational formulation of the semidiscrete problem is derived via a nonlocal boundary condition to truncate the infinite problem domain existence and uniqueness of the variational solutions are established error analysis of the fully discrete problem is performed stability criteria of the time stepping scheme are also obtained

fractal snapshot components in chaos induced by strong noise in systems exhibiting transient chaos in coexistence with periodic attractors the inclusion of weak noise might give rise to noise induced chaotic attractors when the noise amplitude exceeds a critical value an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set a further increase of noise leads to a fuzzy nonfractal pattern by means of the concept of snapshot attractors and random maps we point out that the fuzzy pattern can be decomposed into well defined fractal components the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories the pattern of the snapshot attractor and its characteristic numbers such as the finite time lyapunov exponents and numerically evaluated fractal dimensions change continuously in time we find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size the validity of the kaplan yorke formula is also investigated a superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise induced attractor at the same noise strength

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

noise induced unstable dimension variability and transition to chaos in random dynamical systems results are reported concerning the transition to chaos in random dynamical systems in particular situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle which can be expected in any periodic window of a nonlinear dynamical system under noise the asymptotic attractor of the system can become chaotic as characterized by the appearance of a positive lyapunov exponent generic features of the transition include the following 1 the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions unstable dimension variability and this nonhyperbolicity develops as soon as the attractor becomes chaotic 2 for systems described by differential equations the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero lyapunov exponent after the noisy attractor becomes chaotic and 3 the largest lyapunov exponent becomes positive from zero in a continuous manner and its scaling with the variation of the noise amplitude is algebraic formulas for the scaling exponent are derived in all dimensions numerical support using both low and high dimensional systems is provided

finite dimensional representations of difference operators and the identification of remarkable matrices two square matrices of arbitrary order n are introduced they are defined in terms of n arbitrary numbers zn and of an arbitrary additional parameter a respectively q and provide finite dimensional representations of the two operators acting on a function f z as follows f z a f z a respectively f qz f z q 1 z these representations are exact in a sense explained in the paper when the function f z is a polynomial in z of degree less than n this formalism allows to transform difference equations valid in the space of polynomials of degree less than n into corresponding matrix vector equations as an application of this technique several remarkable square matrices of order n are identified which feature explicitly n arbitrary numbers zn or the n zeros of polynomials belonging to the askey and q askey schemes several of these findings have a diophantine character

dynamics of leaking hamiltonian systems in order to understand the dynamics in more detail in particular for visualizing the space filling unstable foliation of closed chaotic hamiltonian systems we propose to leak them up the cutting out of a finite region of their phase space the leak through which escape is possible leads to transient chaotic behavior of nearly all the trajectories the never escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically it is an approximant of the full hamiltonian foliation the better the smaller the leak is the escape rate depends sensitively on the orientation of the leak even if its area is fixed the applications for chaotic advection for chemical reactions superimposed on hydrodynamical flows and in other branches of physics are discussed

advection in chaotically time dependent open flows the passive advection of tracer panicles is considered in open two dimensional incompressible flows with chaotic time dependence as illustrative examples we investigate flows produced by chaotically moving ideal point vortices the advection problem can be seen as a chaotic scattering process in a chaotically driven hamiltonian system studying the motion of tracer ensembles we present numerical evidence for the existence of a bounded chaotic set containing infinitely many aperiodic trajectories never leaving the mixing region of the flow these ensembles converge to filamental patterns which however do not follow self similar scaling nevertheless they possess a fractal dimension after averaging over several finite time realizations of the flow we propose random maps as simple models of the phenomenon

stochastic perturbations in open chaotic systems random versus noisy maps we investigate the effects of random perturbations on fully chaotic open systems perturbations can be applied to each trajectory independently white noise or simultaneously to all trajectories random map we compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time dependent conditionally map invariant measure for the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map in random maps we show that the escape rate ensuremath kappa and dimensions d of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation we discuss the accuracy bias and precision variance of finite size estimators of ensuremath kappa and d and show that the improvement of the precision of the estimations with the number of trajectories n is extremely slow ensuremath propto 1 mathrm ln n we also argue that the finite size d estimators are typically biased general theoretical results are combined with analytical calculations and numerical simulations in area preserving baker maps