temporal complexity of the order parameter at the phase transition we study a decision making model in a condition where it is equivalent to the two dimensional ising model and we show that at the onset of phase transition it generates temporal complexity namely nonstationary and nonergodic fluctuations we argue that this is a general property of criticality thereby opening the door to the application of the recently discovered phenomenon of complexity matching for an efficient transfer of information to occur a perturbing complex network must share the same temporal complexity as the perturbed complex network

comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

global solutions and blow up phenomena to a shallow water equation abstract a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated the local well posedness of solutions for the nonlinear equation in the sobolev space h s r with s 3 2 is developed provided that 1 x 2 u 0 does not change sign u 0 h s s 3 2 and u 0 l 1 r the existence and uniqueness of the global solutions to the equation are shown to be true in u t x c 0 h s r c 1 0 h s 1 r conditions that lead to the development of singularities in finite time for the solutions are also acquired

compact and stable discontinuous galerkin methods for convection diffusion problems we present a new scheme the compact discontinuous galerkin 2 cdg2 method for solving nonlinear convection diffusion problems together with a detailed comparison to other well accepted dg methods the new cdg2 method is similar to the cdg method that was recently introduced in the work of perraire and persson for elliptic problems one main feature of the cdg2 method is the compactness of the stencil which includes only neighboring elements even for higher order approximation theoretical results showing coercivity and stability of cdg2 and cdg for the poisson and the heat equation are given providing computable bounds on any free parameters in the scheme in numerical tests for an elliptic problem a scalar convection diffusion equation and for the compressible navier stokes equations we demonstrate that the cdg2 method slightly outperforms similar methods in terms of l 2 accuracy and cpu time

traces and determinants of strongly stochastic operators periodic orbit theory allows calculations of long time properties of chaotic systems from traces dynamical zeta functions and spectral determinants of deterministic evolution operators which are in turn evaluated in terms of periodic orbits for the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral techniques for evaluating such path integrals are discussed using as an example the logistic map f x lambdax 1 x with moderate to strong additive gaussian noise rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda 5 for all noise levels

noise induced escape from bifurcating attractors symplectic approach in the weak noise limit the effect of noise is studied in one dimensional maps undergoing transcritical tangent and pitchfork bifurcations the attractors of the noiseless map become metastable states in the presence of noise in the weak noise limit a symplectic two dimensional map is associated with the original one dimensional map the consequences of their noninvertibility on the phase space structures are discussed heteroclinic orbits are identified which play a key role in the determination of the escape rates from the metastable states near bifurcations the critical slowing down justifies the use of a continuous time approximation replacing maps by flows which allows the analytic calculation of the escape rates this method provides the universal scaling behavior of the escape rates at the bifurcations

the holder continuity of the solution map to the b family equation in weak topology we prove that the solution map of the b family equation is holder continuous as a map from a bounded set of h s mathbb r s frac 3 2 with h r mathbb r 0 le r 0 moreover we show that the obtained exponent of the holder continuity is optimal when s 1 r s

stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and bassi and rebay ii numerical fluxes for linear triangular elements abstract the flux reconstruction fr method has gained popularity within the research community the approach has been demonstrated to recover high order methods such as the discontinuous galerkin dg method stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction esfr methods also named vincent castonguay jameson huynh vcjh methods esfr schemes can be viewed as dg schemes with modally filtered correction fields using this class of methods the linear advection diffusion problem has been shown to be stable using the local discontinuous galerkin scheme ldg to compute the viscous numerical flux this stability proof has been extended for linear triangular and tetrahedra elements although the ldg scheme is commonly used it requires on particular meshes a wide stencil which raises the computational cost as a consequence many prefer the compact interior penalty ip or the bassi and rebay ii br2 numerical fluxes this article for the first time derives for both schemes a condition on the penalty term to ensure the stability for the linear diffusion equation the derivations are based on gauss legendre flux points and on periodic boundary conditions moreover the article proposes that for both the ip and br2 numerical fluxes the stability of the esfr scheme is independent of the auxiliary correction field a von neumann analysis is conducted to study the maximal time step of various esfr methods

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

statistics of noise driven coupled nonlinear oscillators applications to systems with kerr nonlinearity we present exact analytical results for the statistics of nonlinear coupled oscillators under the influence of additive white noise we suggest a perturbative approach for analysing the statistics of such systems under the action of a deterministic perturbation based on the exact expressions for probability density functions for noise driven oscillators using our perturbation technique we show that our results can be applied to studying the optical signal propagation in noisy fibres at nearly zero dispersion as well as to weakly nonlinear lattice models with additive noise the approach proposed can account for a wide spectrum of physically meaningful perturbations and is applicable to the case of large noise strength

blow up solutions of the general b equationa in this short paper we consider the b equation a new precise blow up scenario is described and some new results are obtained for the blow up phenomena

higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

insights on the coercivity of the esfr methods for elliptic problems abstract the flux reconstruction approach is a recent high order method which has been introduced for unsteady problems initial energy stability has been conducted for the advection problem leading to the well know energy stable flux reconstruction esfr scheme using the esfr scheme the energy stability proof has been extended for the advection diffusion using the local discontinuous galerkin ldg numerical flux recently stability conditions were derived for the compact interior penalty ip and bassi rebay ii br2 numerical fluxes here we apply esfr schemes to elliptic problems and derive the associated bilinear form for the poisson equation we show that for the compact ip and br2 numerical fluxes the bilinear form is independent of the auxiliary correction function finally we provide some insights on the coercivity of the esfr scheme

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

nearest neighbour distances of diffusing particles from a single trap we present a simple method for obtaining the typical smallest distance from an absorbing trap to the nearest particle in a system of brownian particles in terms of the overall density profile we also obtain the distribution function for the minimum distance by using a quasistatic approximation for the diffusion equation we derive new results for two dimensions the characteristic distance to the nearest particle increases asymptotically as m the technique is useful in diffusion reaction systems in the smoluchowski model for bimolecular reactions an ideal spherical trap centred at the origin is surrounded by a cloud of brownian particles which are captured upon contact with the trap the reaction rate in this formulation depends on the spatial distribution of particles about the trap recently interest has focused on finer details of this distribution such as the distance of the nearest surviving particle from the trap l 31 the density distribution function of that distance is the key to several exact results concerning some diffusion reaction systems in one dimension 4 while the spatial density of the particles surrounding the trap can be easily obtained by solving a diffusion equation with the appropriate boundary conditions the distance to the nearest particle from the trap seems to require a more sophisticated analysis 13 in this paper we suggest a simple derivation for the typical distance from the trap to the nearest particle and of the distribution of those distances based upon the more readily attainable spatial density of all particles and simple facts from the statistics of extremes 5 6 to demonstrate our method we rederive the known results for one and three dimensions the simplicity of our approach together with using a quasistatic approximation 7 for solving the diffusion equation allows us to solve the two dimensional case for the first time consider first the one dimensional problem a perfect trap is located at the origin and is initially surrounded by a homogeneous density co of brownian particles which diffuse with a diffusion constant d the density c x t of the surviving particles to the right of the trap is given by the diffusion equation

the two dimensional periodic b equation on the diffeomorphism group of the torus in this paper the two dimensional periodic b equation is discussed under geometric aspects i e as a geodesic flow on the diffeomorphism group of the torus in the framework of arnold s 1966 ann inst fourier grenoble 16 319 famous approach we achieve some well posedness results for the b equation and we perform explicit curvature computations for the 2d camassa holm equation which is obtained for b 2 finally we explain the special role of the choice b 2 by giving a rigorous proof that b 2 is the only case in which the associated geodesic flow is weakly riemannian

entropy extropy and information potential in stochastic systems far from equilibrium the relations between information entropy and energy which are well known in equilibrium thermodynamics are not clear far from equilibrium moreover the usual expression of the classical thermodynamic potentials is only valid near equilibrium in previous publications we showed for a chemical system maintained far from equilibrium that a new thermodynamic potential the information potential can be de4ned by using the stochastic formalism of the master equation here we extend this theory to a completely general discrete stochastic system for this purpose we use the concept of extropy which is de4ned in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration we show that the statistical equivalent of the thermodynamic extropy is the relative information if a coarse grained description by means of the thermodynamic extensive variables is available which is the case for many macroscopic systems the coarse grained statistical extropy allows one to de4ne the information potential in the thermodynamic limit using this potential we study the evolution of such systems towards a non equilibrium stationary state we derive a general thermodynamic inequality between energy dissipation and information dissipation which sharpens the law of the maximum available work for non equilibrium systems c 2002 published by elsevier science b v

classical nonlinear response of a chaotic system i collective resonances we develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems in contrast to behavior in integrable or almost integrable systems the nonlinear classical response in chaotic systems vanishes at long times the exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices we calculate the linear and second order response in one of the simplest chaotic systems free classical motion on a compact surface of constant negative curvature the response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories we demonstrate the relevance of the model for the interpretation of spectroscopic experiments

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

isostables isochrons and koopman spectrum for the action angle representation of stable fixed point dynamics abstract for asymptotically periodic systems a powerful phase reduction of the dynamics is obtained by computing the so called isochrons i e the sets of points that converge toward the same trajectory on the limit cycle motivated by the analysis of excitable systems a similar reduction has been attempted for non periodic systems admitting a stable fixed point in this case the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories instead the sets of interest that we call isostables are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point however it turns out that this definition of the isostables holds only for systems with slow fast dynamics also efficient methods for computing the isostables are missing the present paper provides a general framework for the definition and the computation of the isostables of stable fixed points which is based on the spectral properties of the so called koopman operator more precisely the isostables are defined as the level sets of a particular eigenfunction of the koopman operator through this approach the isostables are unique and well defined objects related to the asymptotic properties of the system also the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action angle coordinates for the dynamics in addition an efficient algorithm for computing the isostables is obtained which relies on the evaluation of laplace averages along the trajectories the method is illustrated with the excitable fitzhugh nagumo model and with the lorenz model finally we discuss how these methods based on the koopman operator framework relate to the global linearization of the system and to the derivation of special lyapunov functions

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system the finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise in the weak noise approximation the stochastic neighborhoods of deterministic periodic orbits can be computed from distributions stationary under the action of a local fokker planck operator and its adjoint we derive explicit formulas for widths of these distributions in the case of chaotic dynamics when the periodic orbits are hyperbolic the resulting neighborhoods form a basis for functions on the attractor the global stationary distribution needed for calculation of long time expectation values of observables can be expressed in this basis

first exit times of harmonically trapped particles a didactic review we revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one or multidimensional ornstein uhlenbeck process we start by recalling the main derivation steps of a propagator using langevin and fokker planck equations the mean exit time the moment generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed we also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing fokker planck operator we discuss several applications of first exit times such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single particle tracking experiments adhesion bond dissociation under mechanical stress characterization of active periods of trend following and mean reverting strategies in algorithmic trading on stock markets relation to the distribution of first crossing times of a moving boundary by brownian motion some extensions are described including diffusion under quadratic double well potential and anomalous diffusion

closure schemes in stochastic nonlinear dynamics a validation case study it is known that randomness in dynamical systems often leads to infinite hierarchies of coupled equations for relevant probabilistic quantities several closure methods for truncation of the hierarchies have been proposed in the literature in the present paper the performance of closure schemes for moment hierarchies is compared by using the well known nonlinear equation of overdamped oscillator with additive gaussian white noise in the case of bistable dynamics it is shown that the closure schemes can give incorrect results despite of their good convergence properties to overcome this deficiency new closure procedures are proposed they are based on using the hermite polynomials and their generalization

multiparticle trapping problem in the half line abstract a variation of rosenstock s trapping model in which n independent random walkers are all initially placed upon a site of a one dimensional lattice in the presence of a one sided random distribution with probability c of absorbing traps is investigated the probability survival probability n t that no random walker is trapped by time t for n 1 is calculated by using the extended rosenstock approximation this requires the evaluation of the moments of the number sn t of distinct sites visited in a given direction up to time t by n independent random walkers the rosenstock approximation improves when n increases working well in the range dt ln 2 1 c ln n d being the diffusion constant the moments of the time lifetime before any trapping event occurs are calculated asymptotically too the agreement with numerical results is excellent

on the uniqueness of discontinuous solutions to the degasperis procesi equation abstract we prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive degasperis procesi equation t u t x x 3 u 4 u x u 3 x u x x 2 u u x x x 3 u in a recent paper g m coclite k h karlsen on the well posedness of the degasperis procesi equation j funct anal 233 2006 60 91 we proved for this equation the existence and uniqueness of l 1 bv weak solutions satisfying an infinite family of kru kov type entropy inequalities the purpose of this paper is to replace the kru kov type entropy inequalities by an ole nik type estimate and to prove uniqueness via a nonlocal adjoint problem an implication is that a shock wave in an entropy weak solution to the degasperis procesi equation is admissible only if it jumps down in value like the inviscid burgers equation

spectrum of stochastic evolution operators local matrix representation approach a matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum in the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits the evaluation of perturbative corrections is easier to implement in this framework than in the standard feynman diagram perturbation theory the results are perturbative corrections to a stochastic analog of the gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum mechanical applications

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

spectral decomposition of a fokker planck equation at criticality the mean field for a complex network consisting of a large but finite number of random two state elements m has been shown to satisfy a nonlinear langevin equation the noise intensity is inversely proportional to sqrt m in the limiting case m infty the solution to the langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point when m infty the inverse power law is truncated by an exponential decay with rate vargamma the evaluation of which is the main purpose of this article an analytic numeric approach is used to obtain the lowest order eigenvalues in the spectral decomposition of the solution to the corresponding fokker planck equation and its equivalent schrodinger equation representation

fokker planck description for a linear delayed langevin equation with additive gaussian noise we construct an equivalent probability description of linear multi delay langevin equations subject to additive gaussian white noise by exploiting the time convolutionless transform and a time variable transformation we are able to write a fokker planck equation fpe for the 1 time and for the 2 time probability distributions valid irrespective of the regime of stability of the langevin equations we solve exactly the derived fpes and analyze the aging dynamics by studying analytically the conditional probability distribution we discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non markov process as both preparation and observation times have bearing on its dynamics as our analytic procedure can also be applied to linear langevin equations with memory kernels we compare the non markov dynamics of a one delay system with that of a generalized langevin equation with an exponential as well as a power law memory application to a generalization of the green kubo formula is also presented

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

closing the hierarchy of moment equations in nonlinear dynamical systems the moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems in the present paper a systematic approach for closing this hierarchy is proposed based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale the method is applied to a one dimensional vector field in the presence of noise and to two prototypes of chaotic behavior excellent agreement with numerical results is obtained special emphasis is placed on the role of symmetries and on the origin of the composite oscillations found for certain types of moments in the chaotic systems

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

equivalence between the energy stable flux reconstruction and filtered discontinuous galerkin schemes the aim of this paper is to demonstrate the equivalence between filtered discontinuous galerkin dg schemes and the energy stable flux reconstruction esfr schemes expanding on previous demonstrations in 1d 1 and for straight sided elements in 3d 2 we first derive the dg and esfr schemes in strong form and compare the respective flux penalization terms while highlighting the implications of the fundamental assumptions for stability in the esfr formulations notably that all esfr scheme correction fields can be interpreted as modally filtered dg correction fields we present the result in the general context of all higher dimensional curvilinear element formulations through a demonstration that there exists a weak form of the esfr schemes which is both discretely and analytically equivalent to the strong form we then extend the results obtained for the strong formulations to demonstrate that esfr schemes can be interpreted as a dg scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction theoretical derivations are then verified with numerical results obtained from a 2d euler testcase with curved boundaries given the current choice of high order dg type schemes and the question as to which might be best to use for a specific application the main significance of this work is the bridge that it provides between them clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use esfr schemes as opposed to the weak dg scheme when solving problems implicitly

most probable histories for nonlinear dynamics tracking climate transitions the effective action provides an appropriate cost function to determine most probable or optimal histories for nonlinear dynamics with strong noise in such strong coupling problems a nonperturbative technique is required to calculate the effective action we have proposed a rayleigh ritz variational approximation which employs simple moment closures or intuitive guesses of the statistics to calculate the effective action we consider here an application to climate dynamics within a simple bimodal langevin model similar to that proposed by c nicolis and g nicolis tellus33 225 1981 capturing climate state transitions even in this simple model is known to present a serious problem for standard methods of data assimilation in contrast it is shown that the effective action for the climate history is already well approximated by a one moment closure and that the optimal minimizing history robustly tracks climate change even with large observation errors furthermore the hessian of the effective action provides the ensemble variance as a realistic measure of confidence level in the predicted optimal history

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

asymptotic expansions for the escape rate of stochastically perturbed unimodal maps abstract the escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength in the previous work the coefficients of such an expansion for a one dimensional map were fitted to a general form containing a few parameters these parameters were found to be related to the fractal structure of the repeller of the system the parameter the noise dimension remains to be interpreted this report presents new data for showing that the relation to the dimensions is more complicated than predicted in the earlier work and oscillates as a function of the map parameter in contrast to other dimension like quantities

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

stationary probability distribution near stable limit cycles far from hopf bifurcation points we obtain analytic results for the stationary probability distribution in the vicinity of a stable limit cycle for markov systems described by a fokker planck equation or a birth death master equation the results apply best for ranges of parameters removed from hopf bifurcation points as a by product we demonstrate that there holds a liouville like theorem for the stationary probability distribution the product of the velocity along the limit cycle times the area of the cross section of the probability distribution transverse to the cycle is a constant a numerical simulation of a chemical model system with a limit cycle shows good agreement with the analytic results

nonspectral relaxation in one dimensional ornstein uhlenbeck processes the relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates which are typically considered to be independent of the initial condition the rates follow from the spectrum of a hermitian operator obtained by a similarity transformation of the initial fokker planck operator however some initial conditions are mapped by this similarity transformation to functions which growat infinity these cannot be expanded in terms of the eigenfunctions of a hermitian operator and show different relaxation patterns considering the exactly solvable examples of gaussian and generalized levy ornstein uhlenbeck processes oups we show that the relaxation rates belong to the hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise while for an ordinary oup initial conditions leading to nonspectral relaxation can be considered exotic for generalized oups driven by levy noise such initial conditions are the rule doi 10 1103 physrevlett 110 150602

limit behavior of the global solutions to the degasperis procesi type equation abstract in this paper we study the degasperis procesi equation with a physically perturbation term a linear dispersion based on the global existence result we show that the solution of the degasperis procesi equation with linear dispersion tends to the solution of the corresponding degasperis procesi equation as the dispersive parameter goes to zero moreover we prove that smooth solutions of the equation have finite propagation speed they will have compact support if its initial data has this property

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

hierarchies of landau lifshitz bloch equations for nanomagnets a functional integral framework we propose a functional integral framework for the derivation of hierarchies of landau lifshitz bloch llb equations that describe the flow toward equilibrium of the first and second moments of the magnetization the short scale description is defined by the stochastic landau lifshitz gilbert equation under both markovian or non markovian noise and takes into account interaction terms that are of practical relevance depending on the interactions different hierarchies on the moments are obtained in the corresponding llb equations two closure ans atze are discussed and tested by numerical methods that are adapted to the symmetries of the problem our formalism provides a rigorous bridge between the atomistic spin dynamics simulations at short scales and micromagnetic descriptions at larger scales

the degasperis procesi equation as a non metric euler equation in this paper we present a geometric interpretation of the degasperis procesi dp equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle we also show that for any evolution in the family of b equations there is neither gain nor loss of the spatial regularity of solutions this in turn allows us to view the dp and the camassa holm equation as an ode on the frechet space of all smooth functions on the circle

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

on the geometric approach to the motion of inertial mechanical systems according to the principle of least action the spatially periodic motions of one dimensional mechanical systems with no external forces are described in the lagrangian formalism by geodesics on a manifold configuration space the group d of smooth orientation preserving diffeomorphisms of the circle the periodic inviscid burgers equation is the geodesic equation on d with the l 2 right invariant metric however the exponential map for this right invariant metric is not a c 1 local diffeomorphism and the geometric structure is therefore deficient on the other hand the geodesic equation on d for the h 1 rightinvariant metric is also a re expression of a model in mathematical physics we show that in this case the exponential map is a c 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on d t heycan be joined by a unique length minimizing geodesic a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy we also analyse for both metrics the breakdown of the geodesic flow

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

non lorentzian spectral lineshapes near a hopf bifurcation the effects of additive white noise upon the dynamics of a system described by its hopf normal form are investigated with particular reference to the well known model of a detuned single mode laser the power spectrum corresponding to the laser amplitude is determined by finite difference solution of a partial differential equation and analytical formulas are determined in the asymptotic limits of large parameters the effect of the amplitude phase coupling parameter in generating non lorentzian lineshapes is highlighted and the regions of parameter space where accurate first eigenvalue approximations of the fokker planck equation exist are indicated

global existence of weak solutions for a shallow water equation a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated provided that initial value u email protected h s email protected email protected 32 u email protected l 1 r and email protected x 2 u 0 does not change sign it is shown that there exists a unique global weak solution to the equation

peakon solutions of the novikov equation and properties of the data to solution map abstract we consider both the periodic and the non periodic cauchy problem for the novikov equation and discuss continuity results for the data to solution map in sobolev spaces in particular we show that the data to solution map is not globally uniformly continuous in sobolev spaces with exponent less than 3 2 to accomplish this we construct sequences of peakon solutions whose distance initially goes to zero but later becomes large

spectral signature of the pitchfork bifurcation liouville equation approach the time evolution of probability densities of one dimensional nonlinear vector fields is studied using a liouville equation approach it is shown that the liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation the associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions on the other hand the spectrum of the liouville operator may become continuous at the bifurcation point a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation the relationship between the spectral decompositions of the liouville and of the fokker planck equations is discussed in particular the spectral decompositions constructed here for the liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the fokker planck equation of the associated stochastic process

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

reaction diffusion theory in the presence of an attractive harmonic potential problems involving the capture of a moving entity by a trap occur in a variety of physical situations the moving entity being an electron an excitation an atom a molecule a biological object such as a receptor cluster a cell or even an animal such as a mouse carrying an epidemic theoretical considerations have almost always assumed that the particle motion is translationally invariant we study here the case when that assumption is relaxed in that the particle is additionally subjected to a harmonic potential this tethering to a center modifies the reaction diffusion phenomenon using a smoluchowski equation to describe the system we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions interesting features emerge depending on the relative location of the trap the attractive center and the initial placement of the diffusing particle

on the model of the compressible hyperelastic rods and euler equations on the circle abstract considered herein is a geometric investigation on the one parameter equations modeled in the cylindrical compressible hyperelastic rods it is shown that the family of equations can only be realized as an euler equation on the lie group diff s 1 of all smooth and orientation preserving diffeomorphisms on the circle if the material parameter 1 which corresponds to the camassa holm equation in contrast the benjamin bona mahony bbm equation with the parameter 0 in this family of equations is not an euler equation on diff s 1 for any inertia operator

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

leading ruelle resonances of chaotic maps the leading ruelle resonances of typical chaotic maps the perturbed cat map and the standard map are calculated by variation it is found that excluding the resonance associated with the invariant density the next subleading resonances are approximately the roots of the equation z 4 ensuremath gamma where ensuremath gamma is a positive number that characterizes the amount of stochasticity of the map the results are verified by numerical computations and the implications to the form factor of the corresponding quantum maps are discussed

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

pseudospectral solution of the fokker planck equation with equilibrium bistable states the eigenvalue spectrum and the approach to equilibrium the fokker planck equation with a constant diffusion coefficient and a particular polynomial drift coefficient can exhibit a bistable equilibrium distribution such model systems have been used to study chemical reactions nucleation climate optical bistability and other phenomena in this paper we consider a particular choice for the drift coefficient of the form a x x 5 x 3 to exemplify the statistical behaviour of such systems the transformation of the fokker planck equation to a schrodinger equation leads to a potential that belongs to the class of potentials in supersymmetric susy quantum mechanics a pseudospectral method based on nonclassical polynomials is used to determine the spectrum of the fokker planck operator and of the schrodinger equation the converged numerical eigenvalues are compared with wkb and swkb approximations of the eigenvalues

global stability analysis using the eigenfunctions of the koopman operator we propose a novel operator theoretic framework to study global stability of nonlinear systems based on the spectral properties of the so called koopman operator our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems the main results establish the necessary and sufficient relationship between the existence of specific eigenfunctions of the koopman operator and the global stability property of hyperbolic fixed points and limit cycles these results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space

dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

localized partial traps in diffusion processes and random walks reaction diffusion equations in which the reaction is described by a sink term consisting of a sum of delta functions are studied it is shown that the laplace transform of the reactive green s function can be analytically expressed in terms of the green s function for diffusion in the absence of reaction moreover a simple relation between the green s functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained several applications are presented and the formalism is used to establish the relationship between the time dependent geminate recombination yield and the bimolecular reaction rate for diffusion influenced reactions finally an analogous development for lattice random walks is presented

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme

dissipation and large thermodynamic fluctuations the results of recent work of kipnis olla and varadhan on the dynamic large deviations from a hydrodynamic limit for some interacting particle models are formally extended to a general hydrodynamic situation including non equilibrium steady states as a fluctuation dissipation hypothesis the basic conjecture is that the exponent of decay in the probability of a large thermodynamic fluctuation is given by the dissipation of the force required to produce the fluctuation it is shown that this hypothesis leads to a nonlinear version of onsager machlup fluctuation theory that had previously been proposed by graham a direct consequence of the theory is a dynamic variational principle for the most probable thermodynamic history subject to imposed constraints onsager s principle of least dissipation following graham the theory leads also to a generalized potential analogous to an equilibrium free energy for the nonequilibrium steady state and an associated static variational principle finally a formulation of nonlinear fluctuating hydrodynamics is proposed in which the noise enters multiplicatively so as to reproduce the conjectured large deviations theory on a formal analogy with the results of freidlin and wentzell for finite dimensional systems