telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

numerical semi global analysis of a 1 2 resonant hopf hopf bifurcation abstract in this paper a numerical semi global analysis of the dynamics near a 1 2 resonant hopf hopf bifurcation on a four dimensional mathematical model of a simple nonlinear oscillator is performed the 1 2 resonant hopf hopf bifurcation is a codimension three singularity denoted by two pairs of purely imaginary eigenvalues with frequency ratio 1 2 a structure involving 1 1 and 1 2 resonant neimark sacker bifurcations is clearly identified both resonances are coupled by lower codimension singularities such as generalized hopf and period doubling cusp points and cyclic fold curves a three parameter semi global analysis is performed and some of the codimension two singularities unfolded by the 1 2 resonant hopf hopf bifurcation are identified several codimension three points are also detected the obtained results can be useful for further theoretical analysis of the corresponding normal form

evidence of dispersion relations for the nonlinear response of the lorenz 63 system along the lines of the nonlinear response theory developed by ruelle in a previous paper we have proved under rather general conditions that kramers kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of axiom a non equilibrium steady state systems to weak monochromatic forcings we present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed lorenz 63 system by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions some new theoretical results showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables are also presented our findings confirm the conceptual validity of the nonlinear response theory suggest that the theory can be extended for more general non equilibrium steady state systems and shed new light on the applicability of very general tools based only upon the principle of causality for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals in situations where the fluctuation dissipation relation is not of great help

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

parareal multiscale methods for highly oscillatory dynamical systems we introduce a new strategy for coupling the parallel in time parareal iterative methodology with multiscale integrators following the parareal framework the algorithm computes a low cost approximation of all slow variables in the system using an appropriate multiscale integrator which is refined using parallel fine scale integrations convergence is obtained using an alignment algorithm for fast phase like variables the method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables or to resolve all the state variables the numerical scheme does not require that the system is split into slow and fast coordinates moreover the dynamics may involve hidden slow variables for example due to resonances we propose an alignment algorithm for almost periodic solutions in which case convergence of the parareal iterations is proved the applicability of the method is demonstrated in numerical examples

travelling waves and their bifurcations in the lorenz 96 model abstract in this paper we study the dynamics of the monoscale lorenz 96 model using both analytical and numerical means the bifurcations for positive forcing parameter f are investigated the main analytical result is the existence of hopf or hopf hopf bifurcations in any dimension n 4 exploiting the circulant structure of the jacobian matrix enables us to reduce the first lyapunov coefficient to an explicit formula from which it can be determined when the hopf bifurcation is sub or supercritical the first hopf bifurcation for f 0 is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave furthermore by unfolding the codimension two hopf hopf bifurcation it is shown to act as an organising centre explaining dynamics such as quasi periodic attractors and multistability which are observed in the original lorenz 96 model finally the region of parameter values beyond the first hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and lyapunov exponents the observed routes to chaos are various but without clear pattern as n

algebraic analysis of fuzzy systems in this paper we have developed an algebraic theory suitable for the analysis of fuzzy systems we have used the notions of semiring and semimodule introduced the notion of semilinear space have given numerous examples of them and defined also the notions of linear dependence and independence we have shown that the composition operation which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference can be interpreted as a homomorphism between special semimodules consequently this operation is in a certain sense a linear operation this property formally explains why fuzzy systems are attractive for applications

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

complicated quasiperiodic oscillations and chaos from driven piecewise constant circuit chenciner bubbles do not necessarily occur via simple phase locking abstract we analyze the complex quasiperiodic oscillations and chaos generated by two coupled piecewise constant hysteresis oscillators driven by a rectangular wave force oscillations generate arnol d resonance webs wherein lower dimensional resonance tongues extend such as that of a web in numerous directions we provide the fundamental tools for bifurcation analysis of nonautonomous piecewise constant oscillators to optimize the outstanding simplicity of piecewise constant circuits we formulate a generalized procedure for calculating the lyapunov exponents in nonautonomous piecewise constant dynamics the lyapunov exponents in these dynamics can be calculated with a precision approximately similar to that of maps we observe two parameter lyapunov diagrams near the fundamental resonance region called chenciner bubbles wherein the oscillation frequencies of the two oscillators and the force are synchronized with a ratio of 1 1 1 inevitably the hysteresis considerably distorts the chenciner bubbles this result suggests that the chenciner bubbles do not necessarily occur due to simple phase locking of two dimensional tori that can be explained by homeomorphism on the circle furthermore we observe the farey sequence in the experimental measurements

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

coexistence of chaotic and non chaotic states in the two dimensional gauss navier stokes dynamics recently gallavotti proposed an equivalence conjecture in hydrodynamics which states that forced damped fluids can be equally well represented by means of the navier stokes equations ns and by means of time reversible modifications of ns called gauss navier stokes equations gns this equivalence conjecture received numerical support in several recent papers concerning two dimensional fluid mechanics the corresponding results rely on the fact that the ns and gns systems only have one attracting set performing similar two dimensional simulations we find that there are conditions to be met by the gns system for this to be the case in particular increasing the reynolds number while keeping fixed the number of fourier modes leads to the coexistence of different attractors this makes difficult a test of the equivalence conjecture but constitutes a spurious effect due to the insufficient spectral resolution with sufficiently fine spectral resolution the steady states are unique and the equivalence conjecture can be conveniently established

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

persistent chaos in high dimensions an extensive statistical survey of universal approximators shows that as the dimension of a typical dissipative dynamical system is increased the number of positive lyapunov exponents increases monotonically and the number of parameter windows with periodic behavior decreases a subset of parameter space remains where noncatastrophic topological change induced by a small parameter variation becomes inevitable a geometric mechanism depending on dimension and an associated conjecture depict why topological change is expected but not catastrophic thus providing an explanation of how and why deterministic chaos persists in high dimensions

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

numerical and experimental observation of arnol d resonance webs in an electrical circuit abstract an extensive bifurcation analysis of partial and complete synchronizations of three frequency quasi periodic oscillations generated in an electric circuit is presented our model uses two coupled hysteresis oscillators and a rectangular wave forcing term the governing equation of the circuit is represented by a piecewise constant dynamics generating a three dimensional torus the lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation by analyzing this extremely simple circuit we clearly demonstrate that it generates an extremely complex bifurcation structure called arnol d resonance web inevitably chaos is observed in the neighborhood of chenciner bubbles around which regions generating three dimensional tori emanate furthermore the numerical results are experimentally verified

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

the local structure of a chaotic attractor in four dimensions abstract a simple system of four ordinary differential equations exhibiting chaotic behavior is introduced the lyapunov exponents are approximately 3 34 0 00 1 79 and 5 55 points on the attractor where two variables simultaneously assume prechosen values are located by a numerical integration procedure in which suitable antecedent conditions are obtained by successive approximations the points are organized into curves which appear more numerous under higher resolution a backward integration procedure indicates that a line transverse to the curves intersects each curve rather than passing through gaps while the set of intersections forms a cantor set thus the attractor appears to be locally the product of three continua and one cantor set the continua are found to extend in the expanding neutral and less rapidly contracting local directions while the cantor set extends in the more rapidly contracting direction

logarithmic bred vectors in spatiotemporal chaos structure and growth bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos we study the structure spatial correlations and the growth rates of logarithmic bred vectors which are constructed by using a given norm we find that after a suitable transformation logarithmic bred vectors are roughly piecewise copies of the leading lyapunov vector this fact allows us to deduce a scaling law for the bred vector growth rate as a function of its amplitude in addition we relate growth rates with the spectrum of lyapunov exponents corresponding to the most expanding directions we illustrate our results with simulations of the lorenz 1996 model

the gbq algorithm for constructing start systems of homotopies for polynomial systems homotopy methods have become a standard tool for the computation of all solutions of a polynomial system this paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications the gbq algorithm presented consists of two parts the computation of a generalized bezout number gb and the construction of a multi homogeneous start system q the approach generalizes m homogenization into multihomogenization it can also be regarded as a generalization towards the random product homotopy however without making assumptions on the coefficients of the polynomials in the system as is illustrated in the examples symmetric polynomial systems also can be solved more efficiently

models for stochastic climate prediction there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory this analysis demonstrates the potential need for more sophisticated models beyond stable linear langevin equations the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances the strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta plane with topography and a mean flow in this example the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom

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

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

towards a general theory of extremes for observables of chaotic dynamical systems in this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values we show that the extremes of so called physical observables are distributed according to the classical generalised pareto distribution and derive explicit expressions for the scaling and the shape parameter in particular we derive that the shape parameter does not depend on the chosen observables but only on the partial dimensions of the invariant measure on the stable unstable and neutral manifolds the shape parameter is negative and is close to zero when high dimensional systems are considered this result agrees with what was derived recently using the generalized extreme value approach combining the results obtained using such physical observables and the properties of the extremes of distance observables it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow moreover by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments of the partial dimensions and of the kaplan yorke dimension of the attractor preliminary numerical investigations provide encouraging results on the applicability of the theory presented here the results presented here do not apply for all combinations of axiom a systems and observables but the breakdown seems to be related to very special geometrical configurations

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

a computational strategy for multiscale systems with applications to lorenz 96 model numerical schemes for systems with multiple spatio temporal scales are investigated the multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales the multiscale schemes are analyzed in general then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to lorenz issues like consistency accuracy and efficiency are discussed in detail the role of possible hidden slow variables as well as additional effects arising on the diffusive time scale are also investigated as a byproduct we obtain a rather complete characterization of the effective dynamics in lorenz model

stochastic modelling replacing fast degrees of freedom by noise in systems with timescale separation where the fast degrees of freedom exhibit chaotic motion the latter are replaced by suitable stochastic processes a projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones the resulting equations can be approximated in a controlled way by fokker planck equations or equivalently by stochastic differential equations for slow degrees of freedom we discuss some model situations and explore the accuracy of the approximations by numerical simulations

discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics many physical systems are described by nonlinear differential equations that are too complicated to solve in full a natural way to proceed is to divide the variables into those that are of direct interest and those that are not formulate solvable approximate equations for the variables of greater interest and use data and statistical methods to account for the impact of the other variables in the present paper we consider time dependent problems and introduce a fully discrete solution method which simplifies both the analysis of the data and the numerical algorithms the resulting time series are identified by a narmax nonlinear autoregression moving average with exogenous input representation familiar from engineering practice the connections with the mori zwanzig formalism of statistical physics are discussed as well as an application to the lorenz 96 system

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

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

computing singularly perturbed differential equations abstract a computational tool for coarse graining nonlinear systems of ordinary differential equations in time is discussed three illustrative model examples are worked out that demonstrate the range of capability of the method this includes the averaging of hamiltonian as well as dissipative microscopic dynamics whose slow variables defined in a precise sense can often display mixed slow fast response as in relaxation oscillations and dependence on initial conditions of the fast variables also covered is the case where the quasi static assumption in solid mechanics is violated the computational tool is demonstrated to capture all of these behaviors in an accurate and robust manner with significant savings in time a practically useful strategy for accurately initializing short bursts of microscopic runs for the evolution of slow variables is integral to our scheme without the requirement that the slow variables determine a unique invariant measure of the microscopic dynamics

1 soliton solution of the generalized kdv equation with generalized evolution this paper obtains the 1 soliton solution of three variants of the generalized kdv equation with generalized evolution the solitary wave ansatz is used to carry out the integration of such equation the parameter domain is also identified in the process a couple of conserved quantities are also calculated for each of these variants the numerical simulations are also carried out

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

contributions to max min convex geometry i segments abstract we give some contributions to the theory of max min convex geometry that is convex geometry in the semimodule r max min n over the max min semiring r max min r we introduce elementary segments that generalize from n 2 the horizontal vertical or oblique segments contained in the main bisector of r max min 2 we show that every segment in r max min n is a concatenation of a finite number of elementary subsegments at most 2 n 1 respectively at most 2 n 2 in the case of comparable respectively incomparable endpoints x y in this first part we study max min segments and in the subsequent second part submitted we study max min semispaces and some of their relations to max min convex sets

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

bifurcation boundaries of three frequency quasi periodic oscillations in discrete time dynamical system abstract this report presents an extensive investigation of bifurcations of quasi periodic oscillations based on an analysis of a coupled delayed logistic map this map generates an invariant two torus it 2 that corresponds to a three torus in vector fields we illustrate detailed lyapunov diagrams and by observing attractors derive a quasi periodic saddle node qsn bifurcation boundary with a precision of 1 0 9 we derive a stable invariant one torus it 1 and a saddle it 1 which correspond to a stable two torus and a saddle two torus in vector fields respectively we confirmed that the qsn bifurcation boundary coincides with a saddle node bifurcation point of a stable it 1 and a saddle it 1 our major concern in this study is whether the qualitative transition from an it 1 to an it 2 via qsn bifurcations includes phase locking we prove with a precision of 1 0 9 that there is no resonance at the bifurcation point

eigenvalues and eigen functionals of diagonally dominant endomorphisms in min max analysis the so called min analysis may be viewed as an extension to the continuous case and to functional spaces of shortest path algebras in graphs we investigate here min max analysis which extends in some similar way minimum spanning tree problems and maximum capacity path problems in graphs an endomorphisms a of the functional min max semi module acts on any functional to produce a where a x miny max a x y y we present here a complete characterization of eigenvalues and eigen functionals of diagonally dominant endomorphisms i e such that x y a x x 0a a x y 0a it is shown in particular that any real value 0a is an eigenvalue and that the associated eigen semi module has a unique minimal generator

hopf bifurcation and strange attractors in chebyshev spectral solutions of the burgers equation abstract in this paper we apply the chebyshev pseudospectral collocation and spectral methods to the solution of the burgers equation the results provide evidence for the existence of strange attractors it is shown that there is a critical viscosity v c above which the solution is stable and below which the motion becomes highly unstable in favor of motion on a strange attractor

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

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

proof of lyapunov exponent pairing for systems at constant kinetic energy we present a proof that a system consisting of any finite number of particles that move under the action of a scalar potential at constant kinetic energy exhibits conjugate pairing of lyapunov exponents that is the lyapunov exponents come in pairs which sum to the same constant this result generalizes previous results because it is independent of the size of the system s1063 651x 96 51206 7

eigenvectors in bottleneck algebra abstract let b be a nonempty linearly ordered set without maximum and minimum and max min a vector x is said to be an eigenvector of a square matrix a if a x x the aim of the present paper is to characterize the eigenvectors by means of the associated graph of the matrix and to give bounds for the set of all eigenvectors we define the lower and the upper basic eigenvectors and derive efficient methods for computing them

linear theory for filtering nonlinear multiscale systems with model error in this paper we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes in particular the analysis assumes the availability of continuous time noisy observations of all components of the slow variables mathematically this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure in particular we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables given noisy observation of the slow variables alone from the mathematical analysis we learn that for a continuous time linear model with gaussian noise there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed moreover these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system and as a consequence they can be estimated offline from the equilibrium statistics of the true signal by examining a nonlinear test model we show that the linear theory extends in this non gaussian nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model however when the stochastic parametrization model is inappropriate parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa this finding is based on analytical and numerical results on our nonlinear test model and the two layer lorenz 96 model finally even when the correct stochastic ansatz is given it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates in numerical experiments on the two layer lorenz 96 model we find that the parameters estimated online as part of a filtering procedure simultaneously produce accurate filtering and equilibrium statistical prediction in contrast an offline estimation technique based on a linear regression which fits the parameters to a training dataset without using the filter yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed this finding does not imply that all offline methods are inherently inferior to the online method for nonlinear estimation problems it only suggests that an ideal estimation technique should estimate all parameters simultaneously whether it is online or offline

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

finite difference schemes for three dimensional time dependent convection diffusion equation using full global discretization abstract the three dimensional time dependent convection diffusion equation cde is considered an exponential transformation is used to collectively transform the cde the idea of global discretization is used where attention is paid to the whole transformed cde but not to the individual spatial and temporal derivatives in the equation four finite difference schemes for both cde and transformed cde are established the modified partial differential equations of these schemes are obtained which indicate that the trunction errors of the schemes can be of second and fourth order depending on the prescription of the time step length some characteristic physical parameters i e local reynolds number local strouhal number and viscous diffusive length are introduced into the schemes and the viscous diffusive length is found to be a significant parameter in relating temporal discretisation with spatial discretisation a series of benchmark analytical solutions of navier stokes and burgers equations as well as the numerical solutions using the well known discretisation schemes are used to investigate the properties of the derived schemes the high order schemes achieve higher resolutions over the conventional schemes without decreasing much the sparsity of the matrix structures grid refinement studies reveal that the inverse exponential transformation of the finite difference schemes tends to destroy some resolution of the schemes especially for large local reynolds number

dynamics and synchronization of numerical solutions of the burgers equation for chebyshev spectral solutions of the forced burgers equation with low values of the viscosity coefficient several bifurcations and stable attractors can be observed periodic orbits quasiperiodic and strange ones may arise bistability can also be observed necessary conditions for these attractors to appear are discussed and justification for the non emerging of bistability for an example of a system symmetry break is presented as an application for the dynamical behavior of spectral solutions of burgers equation the dynamics and synchronization of unidirectionally coupling of chebyshev spectral solutions of burgers equations by means of a linear coupling are described and discussed also a nonlinear coupling is proposed and discussed

a galerkin finite element approach to burgers equation a galerkin finite element method is presented for the numerical solution of burgers equation a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found via a crank nicolson approach involving a product approximation it is shown that this method is capable of solving burgers equation accurately for a wide range of viscosity values the results show that the new method performs better than the most of the methods available in the literature

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

equivalence of non equilibrium ensembles and representation of friction in turbulent flows the lorenz 96 model we construct different equivalent non equilibrium statistical ensembles in a simple yet instructive n degrees of freedom model of atmospheric turbulence introduced by lorenz in 1996 the vector field can be decomposed into an energy conserving time reversible part plus a non time reversible part including forcing and dissipation we construct a modified version of the model where viscosity varies with time in such a way that energy is conserved and the resulting dynamics is fully time reversible for each value of the forcing the statistical properties of the irreversible and reversible model are in excellent agreement if in the latter the energy is kept constant at a value equal to the time average realized with the irreversible model in particular the average contraction rate of the phase space of the time reversible model agrees with that of the irreversible model where instead it is constant by construction we also show that the phase space contraction rate obeys the fluctuation relation and we relate its finite time corrections to the characteristic time scales of the system a local version of the fluctuation relation is explored and successfully checked the equivalence between the two non equilibrium ensembles extends to dynamical properties such as the lyapunov exponents which are shown to obey to a good degree of approximation a pairing rule these results have relevance in motivating the importance of the chaotic hypothesis in explaining that we have the freedom to model non equilibrium systems using different but equivalent approaches and in particular that using a model of a fluid where viscosity is kept constant is just one option and not necessarily the only option for describing accurately its statistical and dynamical properties

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

multiple time scales in nonlinear flight mechanics diagnosis and modeling there are often disparate time scales in the dynamics of flight creating the potential for reduced order modeling to simplify simulation analysis and design there have been notable successes in developing reduced order models however in the case of nonlinear dynamics which one must typically deal with in guidance problems there has not been a systematic reliable means of diagnosing disparate time scales and developing reduced order models focusing on two time scale behavior in nonlinear dynamical systems we recall fenichel s characterization of the geometric structure of the flow and his theorem establishing the existence and properties of coordinates adapted to this structure adapted coordinates are difficult to construct directly without an appropriate singularly perturbed model of the dynamics we discuss the use of lyapunov exponents and vectors to diagnose two time scale behavior and to determine the corresponding tangent space structure for the linearized dynamics the structure of the linearized flow can then be translated into the manifold structure of the nonlinear flow we briefly mention the use of lyapunov vectors to locate a slow manifold and contrast this approach with two existing approaches the minimum time to climb problem provides an example of two time scale behavior and motivates the discussion

a multiscale method for highly oscillatory ordinary differential equations with resonance a multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations odes is presented the oscillations may be in resonance with on

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

global structure of periodicity hubs in lyapunov phase diagrams of dissipative flows infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow namely a r ossler s oscillator we show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal like sheaves of shilnikov homoclinic bifurcations of a common saddle focus equilibrium the specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor

characterisation of intermittency in chaotic systems the authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory a set of exponents is introduced which generalise the maximum lyapunov characteristic exponent the link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems they perform some analytical computations in simple cases and give a few numerical examples

parametric smoothness and self scaling of the statistical properties of a minimal climate model what beyond the mean field theories abstract a quasi geostrophic intermediate complexity model of the mid latitude atmospheric circulation is considered featuring simplified baroclinic conversion and barotropic convergence processes the model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator to pole temperature difference t e as t e increases a transition takes place from a stationary regime hadley equilibrium to a periodic regime and eventually to a chaotic regime where evolution takes place on a strange attractor the attractor dimension metric entropy and bounding box volume in phase space have a smooth dependence on t e which results in power law scaling properties power law scalings with respect to t e are detected also for the statistical properties of global physical observables the total energy of the system and the averaged zonal wind the scaling laws which constitute the main novel result of the present work can be thought to result from the presence of a statistical process of baroclinic adjustment which tends to decrease the equator to pole temperature difference and determines the properties of the attractor of the system the self similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding on a heuristic basis both data analysis and realistic simulations going beyond the unsatisfactory mean field theories and brute force approaches a leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters

numerical methods for two parameter local bifurcation analysis of maps we discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits cycles of maps and their implementation in matcont a matlab toolbox for continuation and bifurcation analysis of dynamical systems this includes the numerical continuation of fixed points of iterates of the map with one control parameter detecting and locating their bifurcation points i e limit point period doubling and neimark sacker and their continuation in two control parameters as well as detection and location of all codimension 2 bifurcation points on the corresponding curves for all bifurcations of codim 1 and 2 the critical normal form coefficients are computed both numerically with finite directional differences and using symbolic derivatives of the original map using a parameter dependent center manifold reduction explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period these asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations we provide two examples illustrating the developed techniques a generalized henon map and a juvenile adult competition model from mathematical biology

lyapunov spectra and nonequilibrium ensembles equivalence in 2d fluid mechanics we perform numerical experiments to study the lyapunov spectra of dynamical systems associated with the navier stokes ns equation in two spatial dimensions truncated over the fourier basis recently new equations called gns equations have been introduced and conjectured to be equivalent to the ns equations at large reynolds numbers the lyapunov spectra of the ns and of the corresponding gns systems overlap adding evidence in favor of the conjectured equivalence already studied and partially extended in previous papers we make use of the lyapunov spectra to study a fluctuation relation which had been proposed to extend the fluctuation theorem to strongly dissipative systems preliminary results towards the formulation of a local version of the fluctuation formula are also presented

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

a sparse mode spectral method for the simulation of turbulent flows we propose a new algorithm belonging to the family of the sparse mode spectral method to simulate turbulent flows in this method the number of fourier modeskincreases withkmore slowly thankd 1in dimensiond while retaining the advantage of the fast fourier transform examples of applications of the algorithm are given for the one dimensional burgers equation and two dimensional incompressible mhd flows

dioids and semirings links to fuzzy sets and other applications besides the classical algebraic structures of groups rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling other algebraic structures such as dioids and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non classical problems e g in decision analysis fuzzy set theory operations research automatic control and mathematical physics the present paper aims at providing an overview of applications of dioid and semiring structures stressing links with fuzzy sets and emphasizing linear algebraic problems solving linear systems computing eigenvalues and eigenvectors non classical path finding problems using algebras of endomorphisms and connections between dioid structure and nonlinear analysis in view of solving problems in mathematical physics

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

bifurcations de points fixes elliptiques ii orbites periodiques et ensembles de cantor invariants formes normales orbites bien ordonnees estimation a priori sur c estimations lipschitziennes sur les orbites bien ordonnees elimination des problemes de bord existence d ensembles invariants d aubry mather ensembles translates d aubry mather

applied smoothing techniques for data analysis the kernel approach with s plus illustrations 1 density estimation for exploring data 2 density estimation for inference 3 nonparametric regression for exploring data 4 inference with nonparametric regression 5 checking parametric regression models 6 comparing regression curves and surfaces 7 time series data 8 an introduction to semiparametric and additive models references

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

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

dynamical ensembles equivalence in fluid mechanics abstract an attempt to put together various theoretical mathematical or experimental results recently developed in apparently unrelated subjects namely ruelle s 1980 approach to turbulence the body of nose hoover type of molecular dynamics experiments evans and morriss 1990 evans et al 1993 bonetto et al 1996 mathematical results on lyapunov exponents the pairing rule dressler 1988 and dettman and morriss 1995 and experimental results on them sarman et al 1992 bohr et al 1996 theoretical as well as mathematical results on fluctuations frisch and parisi 1984 multifractality and gallavoti and cohen 1995 chaotic hypothesis the key idea that we try to clarify is that of dynamical ensembles as a generalization of the classical equilibrium ensembles arguing that they should be identified with the srb distributions and that they share several properties with the classical ensembles most of the results invoked here did not deal directly with the navier stokes equations and yet they seem to have a lot to do with them as we shall argue here the discussion will focus on the navier stokes and dissipative euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis the ideas developed to a great extent from the efforts put in interpreting the experimental results in bonetto et al 1996

dynamics in spectral solutions of burgers equation for low values of the viscosity coefficient burgers equation can develop sharp discontinuities which are difficult to simulate in a computer oscillations can occur by discretization through spectral collocation methods due to gibbs phenomena under a dynamic point of view these instabilities are related to bifurcations arising to the discretized equation for different values of discretized points herein a study is performed of the dynamics and bifurcations occurring in the spectral solutions of burgers equation with symmetry several phenomena are observed from limit cycles strange attractors to the presence of bistability with two periodic attractors with a periodic attractor and a strange attractor and with two strange attractors also other stable equilibrium points can occur diverse from the ones corresponding to the solution of burgers equation

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems