singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

finite approximations of frobenius perron operators a solution of ulam s conjecture to multi dimensional transformations abstract we prove that ulam s piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise c2 expanding transformation or a jablonski transformation s 0 1 n rn 0 1 n this solves an extension of ulam s conjecture to multi dimensions and generalizes the convergence result given by t y li for one dimensional transformations

precursors of extreme increments we investigate precursors and the predictability of extreme increments in a time series the events we are focusing on consist in large increments within successive time steps we are especially interested in understanding how the quality of the predictions depends on the strategy to choose precursors on the size of the event and on the correlation strength we study the prediction of extreme increments analytically in an autoregressive process of order 1 and numerically in wind speed recordings and long range correlated autoregressive moving average processes data we evaluate the success of predictions via receiver operator characteristics roc curves furthermore we observe an increase of the quality of predictions with increasing event size and with decreasing correlation in all examples both effects can be understood by using the likelihood ratio as a summary index for smooth roc curves

influence of noise on statistical properties of nonhyperbolic attractors we analyze effects of bounded white and colored noise on nonhyperbolic chaotic attractors in two dimensional invertible maps it is shown that first the nonhyperbolic nature is kept even in the presence of strong noise but secondly already due to weak noise some properties of nonhyperbolic chaos can become similar to those of hyperbolic and almost hyperbolic chaos we also estimate the stationary probability measure of noisy nonhyperbolic attractors for this purpose two different methods for calculating the probability density are applied and the obtained results are compared in detail

extreme sensitivity and climate tipping points a climate state close to a tipping point will have a degenerate linear response to perturbations which can be associated with extreme values of the equilibrium climate sensitivity ecs in this paper we contrast linearized instantaneous with fully nonlinear geometric two point notions of ecs in both presence and absence of tipping points for a stochastic energy balance model of the global mean surface temperature with two stable regimes we confirm that tipping events cause the appearance of extremes in both notions of ecs moreover multiple regimes with different mean sensitivities are visible in the two point ecs we confirm some of our findings in a physics based multi box model of the climate system

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

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

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

random periodic solutions of spdes via integral equations and wiener sobolev compact embedding abstract in this paper we study the existence of random periodic solutions for semilinear spdes on a bounded domain with a smooth boundary we identify them as the solutions of coupled forward backward infinite horizon stochastic integral equations on l 2 d in general cases for this we use mercer s theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 d and generalized schauder s fixed point theorem to prove the existence of a solution of the integral equations this is the first paper in literature to study random periodic solutions of spdes our result is also new in finding semi stable stationary solution for non dissipative spdes while in literature the classical method is to use the pull back technique so researchers were only able to find stable stationary solutions for dissipative systems

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

influence of the event magnitude on the predictability of an extreme event we investigate the predictability of extreme events in time series the focus of this work is to understand under which circumstances large events are better predictable than smaller events therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event size is closely linked to the probability distribution function of the underlying stochastic process we evaluate this dependence on the probability distribution function analytically and numerically if we assume that the optimal precursory structures are used to make the predictions we find that large increments are better predictable if the underlying stochastic process has a gaussian probability distribution function whereas larger increments are harder to predict if the underlying probability distribution function has a power law tail in the case of an exponential distribution function we find no significant dependence on the event size furthermore we compare these results with predictions of increments in correlated data namely velocity increments of a free jet flow the velocity increments in the free jet flow are in dependence on the time scale either asymptotically gaussian or asymptotically exponential distributed the numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers

probabilistic view of explosion in an inelastic kac model let mu cdot t t ge 0 be the family of probability measures corresponding to the solution of the inelastic kac model introduced in pulvirenti and toscani j stat phys 114 1453 1480 2004 it has been proved by gabetta and regazzini j stat phys 147 1007 1019 2012 that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law in the present paper it is shown that for initial data which are heavier tailed than the aforementioned ones the limiting distribution is improper in the sense that it has probability 1 2 adherent to infty and probability 1 2 adherent to infty it is explained in which sense this phenomenon is amenable to a sort of explosion and the main result consists in an explicit expression of the rate of such an explosion the presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved this gives the chance to make an adjustment to a portion of a proof contained in the above mentioned paper by gabetta and regazzini

stochastic euler bernoulli beam driven by additive white noise global random attractors and global dynamics abstract this paper concerns with the long time dynamical behavior of a stochastic euler bernoulli beam driven by additive white noise by verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam the existence of global random attractors that attracts all bounded sets in phase space is proved furthermore the finite hausdorff dimension for the global random attractors is attained in light of the relationship between global random attractor and random invariant probability measure the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

parameter estimation for energy balance models with memory we study parameter estimation for one dimensional energy balance models with memory ebmms given localized and noisy temperature measurements our results apply to a wide range of nonlinear parab

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

a non uniform berry esseen bound via stein s method this paper is part of our efforts to develop stein s method beyond uniform bounds in normal approximation our main result is a proof for a non uniform berry esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments it is proved by combining truncation with stein s method and by taking the concentration inequality approach improved and adapted for non uniform bounds to illustrate the technique we give a proof for a uniform berry esseen bound without assuming the existence of third moments

response theory for equilibrium and non equilibrium statistical mechanics causality and generalized kramers kronig relations we consider the general response theory recently proposed by ruelle for describing the impact of small perturbations to the non equilibrium steady states resulting from axiom a dynamical systems we show that the causality of the response functions entails the possibility of writing a set of kramers kronig k k relations for the corresponding susceptibilities at all orders of nonlinearity nonetheless only a special class of directly observable susceptibilities obey k k relations specific results are provided for the case of arbitrary order harmonic response which allows for a very comprehensive k k analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short time response of the perturbed system these results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models and shed light on the very general impact of considering the principle of causality for testing self consistency the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey the theory exposed in the present paper is dual to the time dependent theory of perturbations to equilibrium states and to non equilibrium steady states and has in principle similar range of applicability and limitations in order to connect the equilibrium and the non equilibrium steady state case we show how to rewrite the classical response theory by kubo so that response functions formally identical to those proposed by ruelle apart from the measure involved in the phase space integration are obtained these results taking into account the chaotic hypothesis by gallavotti and cohen might be relevant in several fields including climate research in particular whereas the fluctuation dissipation theorem does not work for non equilibrium systems because of the non equivalence between internal and external fluctuations k k relations might be robust tools for the definition of a self consistent theory of climate change

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

stochastic perturbations to dynamical systems a response theory approach using the formalism of the ruelle response theory we study how the invariant measure of an axiom a dynamical system changes as a result of adding noise and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics we first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system both in the additive and in the multiplicative case we also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing this provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings using kramers kronig theory it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the green function of the system for any desired observable we then extend our results to rather general patterns of random forcing from the case of several white noise forcings to noise terms with memory up to the case of a space time random field explicit formulas are provided for each relevant case analysed as a general result we find using an argument of positive definiteness that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system we provide an example of application of our results by considering the spatially extended chaotic lorenz 96 model these results clarify the property of stochastic stability of srb measures in axiom a flows provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies and introduce new ways to study the response of a system to external perturbations taking into account the chaotic hypothesis we expect that our results have practical relevance for a more general class of system than those belonging to axiom a

some new results for mckean s graphs with applications to kac s equation the main goal of the present paper is to sharpen some results about the error made when the wild sums used to represent the solution of the kac analog of boltzmann s equation are truncated at the n th stage more precisely in carlen carvalho and gabetta j funct anal 220 362 387 2005 one finds a bound for the above mentioned error which depends on an e on the one hand it is shown that the least negative eigenvalue of the linearized collision operator is the best possible exponent on the other hand e is an extra strictly positive number and a a positive coefficient which depends on e too thus it is interesting to check whether e can be removed from the above bound according to the aforesaid reference this problem is studied here by means of the probability distribution of the depth of a leaf in a mckean random tree in fact an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with a n

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

predictability of critical transitions critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages from bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations the consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition however especially in the presence of noise it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions in this contribution we investigate the predictability of critical transitions in conceptual models we study the quadratic integrate and fire model and the van der pol model under the influence of external noise we focus especially on the statistical analysis of the success of predictions and the overall predictability of the system the performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it furthermore we study the influence of the magnitude of transitions on the predictive performance

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

probabilistic study of the speed of approach to equilibrium for an inelastic kac model this paper deals with a one dimensional model for granular materials which boils down to an inelastic version of the kac kinetic equation with inelasticity parameter p 0 in particular the paper provides bounds for certain distances such as specific weighted distances and the kolmogorov distance between the solution of that equation and the limit it is assumed that the even part of the initial datum which determines the asymptotic properties of the solution belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent 2 1 p with such initial data it turns out that the limit exists and is just the aforementioned stable distribution a necessary condition for the relaxation to equilibrium is also proved some bounds are obtained without introducing any extra condition sharper bounds of an exponential type are exhibited in the presence of additional assumptions concerning either the behaviour close to the origin of the initial characteristic function or the behaviour at infinity of the initial probability distribution function

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

probability distribution for the number of cycles between successive regime transitions for the lorenz model abstract the lorenz model has been widely used for exploring many real world problems in this paper we obtain with the help of an invariant manifold technique the return map for the maximum value of the variable x of the model and use this return map to derive the simple empirically obtained regime transition rules for forecasting regime changes and length in the new regime for the model the probability distribution for number of cycles between successive regime transitions of the lorenz model may be of interest in many disciplines we apply the perron frobenius algorithm over the return map to estimate the probability distribution for the number of cycles between successive regime transitions these probabilities are also estimated for the forced lorenz model which is a conceptual model to explore the effects of sea surface temperature on seasonal rainfall

a stochastic model of indopacific sea surface temperature anomalies abstract it is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system a review of this approximation is presented the white noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations the properties of which are summarized as an application of stochastic differential equations we review the evidence that indopacific sea surface temperature anomalies sstas can be represented as a stable linear process driven by spatially coherent stochastic forcing an inversemodeling approach is used that is the relevant parameters of the best fit stable linear process are obtained from observations and given these parameters the assumptions of stability and linearity are subsequently tested an optimal initial structure for growth predicted by the model is found to occur approximately seven months before most of the major warm and cold el nino la nina events in the data record the optimal structure preferentially occurs in the boreal spring with the mature phase of the extreme event occurring in the subsequent late fall winter the best model to fit the observations including the dependence of the dynamical system on the annual cycle is driven by stochastic forcing with periodic statistics these statistics are inferred using a time dependent fluctuation dissipation relation the variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure yet when a linear numerical model is driven with the derived stochastic forcing the optimal structure including its variation with the annual cycle is recovered this implies that deterministic dynamics play an important role in setting up the optimal structure and therefore a mature phase of el nino may be predicted before the optimal structure actually appears

random periodic solutions of random dynamical systems abstract in this paper we give the definition of the random periodic solutions of random dynamical systems we prove the existence of such periodic solutions for a c 1 perfect cocycle on a cylinder using a random invariant set the lyapunov exponents and the pullback of the cocycle

longtime robustness of pullback random attractors for stochastic magneto hydrodynamics equations abstract we establish backward compactness of a pullback random attractor pra from a backward limit set compact cocycle we prove that backward compactness of a pra is a necessary and sufficient condition such that its time fibers are upper semi continuous at negative infinity and obtain the minimal compact limit set also we obtain the maximal limit set in the lower semi continuity sense these results greatly reinforce the upper semi continuity result at the finite time given by cui et al 2018 in the application part we show existence backward compactness and longtime robustness of a pra for the non autonomous stochastic magneto hydrodynamics equations perturbed by a backward tempered force moreover we prove that the pra does not change on different attracted universes this fact allows us to establish the finite time robustness of the pra if the force is tempered only

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

on the relation between rates of relaxation and convergence of wild sums for solutions of the kac equation abstract in the case of maxwellian molecules the wild summation formula gives an expression for the solution of the spatially homogeneous boltzmann equation in terms of its initial data f as a sum f v t n 1 e t 1 e t n 1 q n f v here q n f is an average over n fold iterated wild convolutions of f if m denotes the maxwellian equilibrium corresponding to f then it is of interest to determine bounds on the rate at which q n f m l 1 r tends to zero in the case of the kac model we prove that for every e 0 if f has moments of every order and finite fisher information there is a constant c so that for all n q n f m l 1 r cn e where is the least negative eigenvalue for the linearized collision operator we show that is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f t to m a key role in the analysis is played by a decomposition of q n f into a smooth part and a small part this depends in an essential way on a probabilistic construction of mckean it allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data

anticipating random periodic solutions i sdes with multiplicative linear noise in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify them as solutions of coupled forward backward infinite horizon stochastic integral equations ihsies using the substitution theorem of stochastic differential equations with anticipating initial conditions in general random periodic solutions and the solutions of ihsies are anticipating for the linear noise case with the help of the exponential dichotomy given in the multiplicative ergodic theorem we can identify them as the solutions of infinite horizon random integral equations ihsies we then solve a localised forward backward ihrie in c r lloc2 using an argument of truncations the malliavin calculus the relative compactness of wiener sobolev spaces in c 0 t l2 c 0 t l2 and schauder s fixed point theorem we finally measurably glue the local solutions together to obtain a global solution in c r l2 c r l2 thus we obtain the existence of a random periodic solution and a periodic measure

improved risk estimation in multifractal records application to the value at risk in finance we suggest a risk estimation method for financial records that is based on the statistics of return intervals between events above below a certain threshold q and is particularly suited for multifractal records the method is based on the knowledge of the probability w q t ensuremath delta t that within the next ensuremath delta t units of time at least one event above q occurs if the last event occurred t time units ago we propose an analytical estimate of w q and show explicitly that the proposed method is superior to the conventional precursory pattern recognition technique widely used in signal analysis which requires considerable fine tuning and is difficult to implement we also show that the estimation of the value at risk which is a standard tool in finances can be improved considerably by the method

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

open loop sustained chaos and control a manifold approach we present a general method for preserving chaos in nonchaotic parameter regimes as well as preserving periodic behavior in chaotic regimes using a multifrequency phase control the systems considered are nonlinear systems driven at a base frequency multifrequency phase control is defined as the addition of small subharmonic amplitude modulation coupled with a phase shift by implementing multifrequency control stable and unstable manifold intersections in postcrisis regimes may be manipulated to sustain chaos as well as to sustain periodic behavior the theory and a preliminary experiment are demonstrated for a co 2 driven laser

data driven non markovian closure models abstract this paper has two interrelated foci i obtaining stable and efficient data driven closure models by using a multivariate time series of partial observations from a large dimensional system and ii comparing these closure models with the optimal closures predicted by the mori zwanzig mz formalism of statistical physics multilayer stochastic models msms are introduced as both a generalization and a time continuous limit of existing multilevel regression based approaches to closure in a data driven setting these approaches include empirical model reduction emr as well as more recent multi layer modeling it is shown that the multilayer structure of msms can provide a natural markov approximation to the generalized langevin equation gle of the mz formalism a simple correlation based stopping criterion for an emr msm model is derived to assess how well it approximates the gle solution sufficient conditions are derived on the structure of the nonlinear cross interactions between the constitutive layers of a given msm to guarantee the existence of a global random attractor this existence ensures that no blow up can occur for a broad class of msm applications a class that includes non polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants the emr msm methodology is first applied to a conceptual nonlinear stochastic climate model of coupled slow and fast variables in which only slow variables are observed it is shown that the resulting closure model with energy conserving nonlinearities efficiently captures the main statistical features of the slow variables even when there is no formal scale separation and the fast variables are quite energetic second an msm is shown to successfully reproduce the statistics of a partially observed generalized lotka volterra model of population dynamics in its chaotic regime the challenges here include the rarity of strange attractors in the model s parameter space and the existence of multiple attractor basins with fractal boundaries the positivity constraint on the solutions components replaces here the quadratic energy preserving constraint of fluid flow problems and it successfully prevents blow up

central limit theorem for a class of one dimensional kinetic equations we introduce a class of kinetic type equations on the real line which constitute extensions of the classical kac caricature the collisional gain operators are defined by smoothing transformations with rather general properties by establishing a connection to the central limit problem we are able to prove long time convergence of the equation s solutions toward a limit distribution for example we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law then the limit is a scale mixture of under some additional assumptions explicit exponential rates for the convergence to equilibrium in wasserstein metrics are calculated and strong convergence of the probability densities is shown

characterization of weak convergence of probability valued solutions of general one dimensional kinetic equations for a general inelastic kac like equation recently proposed this paper studies the long time behaviour of its probability valued solution in particular the paper provides necessary and sufficient conditions for the initial datum in order that the corresponding solution converges to equilibrium the proofs rest on the general clt for independent summands applied to a suitable skorokhod representation of the original solution evaluated at an increasing and divergent sequence of times it turns out that roughly speaking the initial datum must belong to the standard domain of attraction of a stable law while the equilibrium is presentable as a mixture of stable laws

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

controlling chaos with weak periodic signals optimized by a genetic algorithm in the present study we develop a relatively novel and effective chaos control approach with a multimode periodic disturbance applied as a control signal and perform an in depth analysis on this nonfeedback chaos control strategy different from previous chaos control schemes the present method is of two characteristic features 1 the parameters of the controlling signal are optimized by a genetic algorithm ga with the largest lyapunov exponent used as an index of the stability and 2 the optimization is justified by a fitness function defined with the target lyapunov exponent and the controlling power this novel method is then tested on the noted rossler and lorenz systems with and without the presence of noise the results disclosed that compared to the existing chaos control methods the present ga based control needs only significantly reduced signal power and a shorter transient stage to achieve the preset control goal the switching control ability and the robustness of the proposed method for cases with sudden change in a system parameter and or with the presence of noise environment are also demonstrated

central limit theorems for solutions of the kac equation speed of approach to equilibrium in weak metrics this paper is part of our efforts to show how direct application of probabilistic methods pertaining to central limit general theory can enlighten us about the convergence to equilibrium of the solutions of the kac equation here we consider convergence with respect to the following metrics kolmogorov s uniform metric 1 and 2 gini s dissimilarity indices widely known as 1 and 2 wasserstein metrics weighted metrics our main results provide new bounds or improvements on already well known ones for the corresponding distances between the solution of the kac equation and the limiting gaussian maxwellian distribution the study is conducted both under the necessary assumption that initial data have finite energy without assuming existence of moments of order greater than 2 and under the condition that the 2 moment of the initial distribution is finite for some 0

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

lyapunov exponents of two stochastic lorenz 63 systems two different types of perturbations of the lorenz 63 dynamical system for rayleigh benard convection by multiplicative noise called stochastic advection by lie transport salt noise and fluctuation dissipation fd noise are found to produce qualitatively different effects possibly because the total phase space volume contraction rates are different in the process of making this comparison between effects of salt and fd noise on the lorenz 63 system a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical lyapunov exponents was developed with this stochastic version of the algorithm the value of the sum of the lyapunov exponents for the fd noise was found to differ significantly from the value of the deterministic lorenz 63 system whereas the salt noise preserves the lorenz 63 value with high accuracy the lagrangian averaged version of the salt equations la salt is found to yield a closed deterministic subsystem for the expected solutions which is isomorphic to the original lorenz 63 dynamical system the solutions of the closed chaotic subsystem in turn drive a linear stochastic system for the fluctuations of the la salt solutions around their expected values

attractors of a randomly forced electronic oscillator abstract this paper examines an electronic oscillator forced by a pseudo random noise signal we give evidence of the existence of one or more random attractors for the system depending on noise amplitude and system parameters these random attractors may appear to be random fixed points or random chaotic attractors in the latter case we observe a form of intermittent synchronization of the response of the system to the noise signal we show how this can be understood as on off intermittency in an extended system

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

waves vs particles in the atmosphere s phase space a pathway to long range forecasting thirty years ago e n lorenz provided some approximate limits to atmospheric predictability the details in space and time of atmospheric flow fields are lost after about 10 days certain gross flow features recur however after times of the order of 10 50 days giving hope for their prediction over the last two decades numerous attempts have been made to predict these recurrent features the attempts have involved on the one hand systematic improvements in numerical weather prediction by increasing the spatial resolution and physical faithfulness in the detailed models used for this prediction on the other hand theoretical attempts motivated by the same goal have involved the study of the large scale atmospheric motions9 phase space and the inhomogeneities therein these coarse graining studies have addressed observed as well as simulated atmospheric data sets two distinct approaches have been used in these studies the episodic or intermittent and the oscillatory or periodic the intermittency approach describes multiple flow or weather regimes their persistence and recurrence and the markov chain of transitions among them the periodicity approach studies intraseasonal oscillations with periods of 15 70 days and their predictability we review these two approaches particles vs waves in the quantum physics analogy alluded to in the title of this article discuss their complementarity and outline unsolved problems

triggering and enhancing chaos with a prescribed target lyapunov exponent using optimized perturbations of minimum power the objective of the present work is to propose a method for nonfeedback anticontrol of chaos with perturbations of minimum power for a preset control goal the noted lorenz system is employed as the test model for chaotification with the target state specified by a prescribed positive value of the largest lyapunov exponent lle lambda over 0 periodic and quasiperiodic perturbations are used as control signals and the signals parameters are optimized using a genetic algorithm under restriction of minimum power performance of the optimized signals in triggering chaos at an ordered state fixed point or periodic as well as further enhancing chaoticity at a chaotic state is explored the present numerical experiments reveal the following interesting physics about chaotification in general the power for chaotification increases with the preset value of lambda and quasiperiodic signals can achieve the control goal with a lower power than periodic ones given the same increment of lle from that of the uncontrolled state lambda1 0 i e deltalambda lambda lambda1 0 the further enhancement of chaoticity in a chaotic state needs a higher control power than the triggering of chaos from an ordered state the minimum power required for chaotification of an ordered state increases relatively slowly for lower lambda over but increases drastically as the preset target lle reaches a certain critical value most strikingly the numerical experiments demonstrate that this critical value of lambda corresponds to lle of the nearest chaotic state in the neighborhood of the uncontrolled state robustness of applying the present method in the presence of external noise is also demonstrated

stochastic stability of the classical lorenz flow under impulsive type forcing we introduce a novel type of random perturbation for the classical lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow the perturbation acts on the system in an impulsive way hence is not of diffusive type as those already discussed in keller attractors and bifurcations of the stochastic lorenz system report 389 institut fur dynamische systeme universitat bremen 1996 kifer random perturbations of dynamical systems birkhauser basel 1988 and metzger commun math phys 212 277 296 2000 namely given a cross section mathcal m for the unperturbed flow each time the trajectory of the system crosses mathcal m the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one the resulting random evolution is therefore described by a piecewise deterministic markov process the proof of the stochastic stability for the umperturbed flow is then carried on working either in the framework of the random dynamical systems or in that of semi markov processes

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

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

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