hamiltonian reformulation and pairing of lyapunov exponents for nose hoover dynamics the nose hamiltonian is adapted leading to a derivation of the nose hoover equations of motion which does not involve time transformations and in which the degree of freedom corresponding to the external reservoir is treated on the same footing as those of the rest of the system in this form it is possible to prove the conjugate pairing rule for lyapunov exponents of this system

lyapunov instabilities of lennard jones fluids recent work on many particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive lyapunov exponents les called hydrodynamic lyapunov modes until now however these modes have been found only for hard core systems here we report results on lyapunov spectra and lyapunov vectors lvs for lennard jones fluids by considering the fourier transform of the coordinate fluctuation density u ensuremath alpha x t it is found that the lvs with ensuremath lambda ensuremath approx 0 are highly dominated by a few components with low wave numbers these numerical results provide strong evidence that hydrodynamic lyapunov modes do exist in soft potential systems although the collective lyapunov modes are more vague than in hard core systems in studying the density and temperature dependence of these modes it is found that when the value of the lyapunov exponent ensuremath lambda ensuremath alpha is plotted as function of the dominant wave number k mathit max of the corresponding lv all data from simulations with different densities and temperatures collapse onto a single curve this shows that the dispersion relation ensuremath lambda ensuremath alpha vs k mathit max for hydrodynamical lyapunov modes appears to be universal for the low density cases studied here despite the wavelike character of the lvs no steplike structure exists in the lyapunov spectrum of the systems studied here in contrast to the hard core case further numerical simulations show that the finite time les fluctuate strongly we have also investigated localization features of lvs and propose a length scale to characterize the hamiltonian spatiotemporal chaotic states

a kam theory for conformally symplectic systems efficient algorithms and their validation abstract we present a kam theory for some dissipative systems geometrically these are conformally symplectic systems i e systems that transform a symplectic form into a multiple of itself for systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with a fixed n dimensional diophantine frequency by adjusting the parameters we do not assume that the system is close to integrable but we present the results in an a posteriori format our unknowns are a parameterization of the quasi periodic solution and some parameters in the system we formulate an invariance equation that expresses that the system with the parameters leaves invariant the solution given by the embedding we show that if there is a sufficiently approximate solution of the invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby the smallness assumptions above can be understood either in sobolev or in analytic norms the a posteriori format has several consequences a smooth dependence on the parameters including the singular limit of zero dissipation b estimates on the measure of parameters covered by quasi periodic solutions c convergence of perturbative expansions in dissipative analytic systems d bootstrap of regularity i e that all tori which are smooth enough are analytic if the map is analytic e a numerically efficient criterion for the breakdown of the quasi periodic solutions the proof is based on an iterative quadratically convergent method the iterative step takes advantage of some geometric identities these identities also lead to an efficient algorithm if we discretize the parameterization with n terms a modified newton step requires o n storage and o n log n operations the a posteriori theorems allow one to be confident on the numerical results even very close to breakdown the algorithm does not require that the system is close to integrable so that a continuation algorithm is guaranteed to continue the tori till the breakdown in practice only limitation being the total memory and the precision the algorithms have been implemented and run in other papers

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

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

on the validity of the conjugate pairing rule for lyapunov exponents for hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state dettmann and morriss proved a strong conjugate pairing rule scpr for pairs of lyapunov exponents in the case of isokinetic ik stationary states which have a given kinetic energy this scpr holds for all initial phases of the system all times t and all numbers of particles n this proof was generalized by wojtkowski and liverani to include hard interparticle potentials a geometrical reformulation of those results is presented the present paper proves numerically using periodic orbits for the lorentz gas that scpr cannot hold for isoenergetic ie stationary states which have a given total internal energy in that case strong evidence is obtained for cpr to hold for large n and t where it can be conjectured that the larger n the smaller t will be this suffices for statistical mechanics

aubry mather theory for conformally symplectic systems in this article we develop an analogue of aubry mather theory for a class of dissipative systems namely conformally symplectic systems and prove the existence of interesting invariant sets which in analogy to the conservative case will be called the aubry and the mather sets besides describing their structure and their dynamical significance we shall analyze their attracting repelling properties as well as their noteworthy role in driving the asymptotic dynamics of the system

time dependent mode structure for lyapunov vectors as a collective movement in quasi one dimensional systems the time dependent mode structure of the lyapunov vectors associated with the stepwise structure of the lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi one dimensional many hard disk systems we obtain the complete mode structures lyapunov modes for all components of the lyapunov vectors including the longitudinal and transverse components of both the spatial and momentum parts and their phase relations these mode structures are suggested by the form of the lyapunov vectors for the zero lyapunov exponents the spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models our main result is that the largest time oscillating period of the lyapunov modes is twice as long as the time oscillating period of the longitudinal momentum autocorrelation function this relation is satisfied irrespective of the number of particles and the boundary conditions a simple explanation for this relation is given based on the form of the time dependent lyapunov mode

the instability of naked singularities in the gravitational collapse of a scalar field one of the fundamental unanswered questions in the general theory of relativity is whether naked singularities that is singular events which are visible from inflnity may form with positive probability in the process of gravitational collapse the conjecture that the answer to this question is in the negative has been called cosmic censorship the present paper which is a continuation of the work in 1 2 3 and 4 addresses this question in the context of the spherical gravitational collapse of a scalar fleld the problem of a spherically symmetric self gravitating scalar fleld is formulated in terms of a 2 dimensional quotient space time manifold q with boundary see 1 the boundary ofq corresponds to the set of flxed points of the group action the center of symmetry which is a timelike geodesic i the manifoldq is endowed with a lorentzian metric gab an area radius function i i r

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

time oscillating lyapunov modes and the momentum autocorrelation function received 29 april 2004 published 20 april 2005 the lyapunov vectors corresponding to the steps of lyapunov spectra for many particle systems showtime oscillating behavior in two types of lyapunov modes one associated with time translationalinvariance and the other with spatial translational invariance our result is that for each coordinatedirection the longest period of the lyapunov modes is twice as long as the period of the momentumautocorrelation function a simple explanation for this relation is proposed and we argue that this result isgenerally true for many particle systems this gives the rst quantitative connection between thelyapunov modes and an experimentally accessible quantity

stability and instability of the cauchy horizon for the spherically symmetric einstein maxwell scalar field equations this paper considers a trapped characteristic initial value problem for the spherically symmetric einstein maxwell scalar field equations for an open set of initial data whose closure contains in particular reissner nordstrdata the future boundary of the maximal domain of development is found to be a light like surface along which the curvature blows up and yet the metric can be continuously extended beyond it this result is related to the strong cosmic censorship conjecture of roger penrose the principle of determinism in classical physics is expressed mathemat ically by the uniqueness of solutions to the initial value problem for certain equations of evolution indeed in the context of the einstein equations of general relativity where the unknown is the very structure of space and time uniqueness is equivalent on a fundamental level to the validity of this principle the question of uniqueness may thus be termed the issue of the predictability of the equation the present paper explores the issue of predictability in general relativity since the work of leray it has been known that for the einstein equations contrary to common experience uniqueness for the cauchy problem in the large does not generally hold even within the class of smooth solutions in other words uniqueness may fail without any loss in regularity such failure is thus a global phenomenon the central question is whether this violation of predictability may occur in solutions representing actual physical processes physical phenomena and concepts related to the general theory of relativity namely gravitational collapse black holes angular momentum etc must cer tainly come into play in the study of this problem unfortunately the math ematical analysis of this exciting problem is very difficult at present beyond reach for the vacuum einstein equations in the physical dimension conse

black holes without spacelike singularities it is shown that for small spherically symmetric perturbations of asymptotically flat two ended reissner nordstrom data for the einstein maxwell real scalar field system the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously under additional assumptions it is shown that the hawking mass blows up identically along this bifurcate null hypersurface and thus the metric cannot be extended twice differentiably in fact it cannot be extended in a weaker sense characterized at the level of the christoffel symbols the proof combines estimates obtained in previous work with an elementary cauchy stability argument there are no restrictions on the size of the support of the scalar field and the result applies to both the future and past boundary of spacetime in particular it follows that for an open set in the moduli space of solutions around reissner nordstrom there is no spacelike component of either the future or the past singularity

multidimensional measures of impulsively driven stochastic systems based on the kullback leibler distance by subjecting a dynamical system to a series of short pulses and varying several time delays we can obtain multidimensional characteristic measures of the system multidimensional kullback leibler response function klrf which are based on the kullback leibler distance between the initial and final states are defined we compare the klrf which are nonlinear in the probability density with ordinary response functions obtained from the expectation value of a dynamical variable which are linear we show that the klrf encode different level of information regarding the system s dynamics for overdamped stochastic dynamics two dimensional klrf shows a qualitatively different variation with the time delays between pulses depending on whether the system is initially in a steady state or in thermal equilibrium

multifractal characterization of stochastic resonance we use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals to characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence we use wavelet transform modulus maxima method for the singularity spectrum computations it is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic stochastic or chaotic input signals we show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization this fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also moreover our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence we use also the proposed approach to characterize the noise enhanced dynamics of a coupled stochastic neurons model

front propagation techniques to calculate the largest lyapunov exponent of dilute hard disk gases a kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point measured by the largest lyapunov exponent for a dilute system of hard disks both in equilibrium and in a uniform shear flow we derive a generalized boltzmann equation for an extended one particle distribution that includes deviations from the reference phase space point the equation is valid for very low densities n and requires an unusual expansion in powers of 1 ln n it reproduces and extends results from the earlier more heuristic clock model and may be interpreted as describing a front propagating into an unstable state the asymptotic speed of propagation of the front is proportional to the largest lyapunov exponent of the system its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand for the equilibrium case an explicit expression for the largest lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest lyapunov exponent

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

billiards correlation functions we discuss various experiments on the time decay of velocity autocorrelation functions in billiards we perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by friedman and martin fm they do not seem compatible with the stretched exponentials believed in spite of fm and more recently of chernov to describe the mixing the analysis leads to several byproducts we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timetm astm finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles

conformally symplectic dynamics and symmetry of the lyapunov spectrum a generalization of the hamiltonian formalism is studied and the symmetry of the lyapunov spectrum established for the resulting systems the formalism is applied to the gausssian isokinetic dynamics of interacting particles with hard core collisions and other systems

dynamical response of the hodgkin huxley model in the high input regime the response of the hodgkin huxley neuronal model subjected to stochastic uncorrelated spike trains originating from a large number of inhibitory and excitatory post synaptic potentials is analyzed in detail the model is examined in its three fundamental dynamical regimes silence bistability and repetitive firing its response is characterized in terms of statistical indicators interspike interval distributions and their first moments as well as of dynamical indicators autocorrelation functions and conditional entropies in the silent regime the coexistence of two different coherence resonances is revealed one occurs at quite low noise and is related to the stimulation of subthreshold oscillations around the rest state the second one at intermediate noise variance is associated with the regularization of the sequence of spikes emitted by the neuron bistability in the low noise limit can be interpreted in terms of jumping processes across barriers activated by stochastic fluctuations in the repetitive firing regime a maximization of incoherence is observed at finite noise variance finally the mechanisms responsible for the different features appearing in the interspike interval distributions like multimodality and exponential tails are clearly identified in the various regimes

evaluation of stochastic resonance based detectors of weak harmonic signals in additive white gaussian noise a thorough evaluation of stochastic resonance in the framework of statistical detection theory is presented both as a nonlinear signal preprocessor and as a detector the pertinent receiver operating characteristics are compared with those of the known statistically optimum detector using extensive monte carlo simulations parameter optimization and computational budget aspects are discussed

late time asymptotics for the wave equation on spherically symmetric stationary spacetimes abstract we derive precise late time asymptotics for solutions to the wave equation on spherically symmetric stationary and asymptotically flat spacetimes including as special cases the schwarzschild and reissner nordstrom families of black holes we also obtain late time asymptotics for the time derivatives of all orders and for the radiation field along null infinity we show that the leading order term in the asymptotic expansion is related to the existence of the conserved newman penrose quantities on null infinity as a corollary we obtain a characterization of all solutions which satisfy price s polynomial law 3 as a lower bound our analysis relies on physical space techniques and uses the vector field approach for almost sharp decay estimates introduced in our companion paper in the black hole case our estimates hold in the domain of outer communications up to and including the event horizon our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors

weak null singularities in general relativity we construct a class of spacetimes without symmetry assumptions satisfying the vacuum einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2 sphere the metric of these spacetimes extends continuously beyond the singularities while the christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries the construction shows moreover that the singularities are stable in a suitable sense these singularities are stronger than the impulsive gravitational spacetimes considered by luk rodnianski and are conjecturally the generic type of singularity in the interior of black holes arising from gravitational collapse

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

dynamical entropies applied to stochastic resonance we will show that sr is accompanied by an extremal behavior of these quantities when sweeping noise intensity over the resonance region we will prove that an increase of applied noise at first leads to an increase of temporal order in the output sequence simultaneously the predictability of the output sequence will improve in the resonance region from the kullback entropy analysis we will learn that for a certain value of noise intensity the input and output distributions of binary subsequences maximally match moreover input and output subsequences registered at equal times exhibit maximum correlation finally we relate these results to the syn

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

local propagation of impulsive gravitationalwaves in this paper we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves we construct such spacetimes as solutions to the characteristic initial value problem of the einstein vacuum equations with a data curvature delta singularity we show that in the resulting spacetime the delta singularity propagates along a characteristic hypersurface while away from that hypersurface the spacetime remains smooth unlike the known explicit examples of impulsive gravitational spacetimes this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions the arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data 2015 wiley periodicals inc

on the global uniqueness for the einstein maxwell scalar field system with a cosmological constant part 2 structure of the solutions and stability of the cauchy horizon this paper is the second part of a trilogy dedicated to the following problem given spherically symmetric characteristic initial data for the einstein maxwell scalar field system with a cosmological constant lambda with the data on the outgoing initial null hypersurface given by a subextremal reissner nordstrom black hole event horizon study the future extendibility of the corresponding maximal globally hyperbolic development as a suitably regular lorentzian manifold in the first paper of this sequence costa et al class quantum gravity 32 015017 2015 we established well posedness of the characteristic problem with general initial data in this second paper we generalize the results of dafermos ann math 158 875 928 2003 on the stability of the radius function at the cauchy horizon by including a cosmological constant this requires a considerable deviation from the strategy followed in dafermos ann math 158 875 928 2003 focusing on the level sets of the radius function instead of the red shift and blue shift regions we also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin in the third and final paper costa et al on the global uniqueness for the einstein maxwell scalar field system with a cosmological constant part 3 mass inflation and extendibility of the solutions arxiv 1406 7261 2015 we will consider the issue of mass inflation and extendibility of solutions beyond the cauchy horizon

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

ergodicity of a single particle confined in a nanopore we analyze the dynamics of a gas particle moving through a nanopore of adjustable width with particular emphasis on ergodicity we give a measure of the portion of phase space that is characterized by quasiperiodic trajectories which break ergodicity the interactions between particle and wall atoms are mediated by a lennard jones potential so that an analytical treatment of the dynamics is not feasible but making the system more physically realistic in view of recent studies which proved non ergodicity for systems with scatterers interacting via smooth potentials we find that the non ergodic component of the phase space for energy levels typical of experiments is surprisingly small i e we conclude that the ergodic hypothesis is a reasonable approximation even for a single particle trapped in a nanopore due to the numerical scope of this work our focus will be the onset of ergodic behavior which is evident on time scales accessible to simulations and experimental observations rather than ergodicity in the infinite time limit

recent results for the thermostatted lorentz gas abstract we review the behaviour of a driven thermostatted lorentz gas the two variable poincare section confirms that the attractor fills the whole phase space and that the associated stationary measure is ergodic and multifractal for fields below 2 2 such a property of the attractor ends with either a crisis or the emergence of elliptical regions followed eventually by stable orbits we present accurate periodic orbit expansion calculations for the diffusion coefficient pressure and lyapunov exponent the periodic orbit approach suggests the definition of a partition function and gives a simple explanation for a positive conductivity and diffusion coefficient

the maximum number of highly localized lyapunov vectors at low density abstract the localization spectra which describes the magnitude of the localization of the lyapunov vectors in many particle systems exhibit a characteristic bending behavior at low density it is shown that this behavior is due to a restriction on the maximum number of the most localized lyapunov vectors determined by the system configuration and mutual orthogonality for a quasi one dimensional system using a randomly distributed brick model this leads to a predicted bending point at n c 0 432 n for an n particle system numerical evidence is presented that confirms this predicted bending point as a function of the number of particles n

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

instability results for the wave equation in the interior of kerr black holes abstract we prove that a large class of smooth solutions to the linear wave equation g 0 on subextremal rotating kerr spacetimes which are regular and decaying along the event horizon become singular at the cauchy horizon more precisely we show that assuming appropriate upper and lower bounds on the energy along the event horizon the solution has infinite non degenerate energy on any spacelike hypersurfaces intersecting the cauchy horizon transversally extrapolating from known results in the reissner nordstrom case the assumed upper and lower bounds required for our theorem are conjectured to hold for solutions arising from generic smooth and compactly supported initial data on a cauchy hypersurface this result is motivated by the strong cosmic censorship conjecture in general relativity

the number dependence of the maximum lyapunov exponent we examine the number dependence of the largest lyapunov exponent for nonlinear dynamical systems in one two and three cartesian dimensions our results suggest that the largest lyapunov exponent diverges logarithmically with system size independently of the number of cartesian dimensions and the interaction potential

approach to block entropy modeling and optimization abstract this paper introduces an approach to the block entropy modeling of stationary signals the block entropies constructed according to the tsallis generalized formalism are optimized with respect to the block length and the partition of the signal value domain to appropriately measure the complexity of the signal the optimal partition is known to be addressed by the euclidean mean expressing the signal optimized level based on the least squares fitting method however this is valid only for random signal values following a symmetric distribution alternatively and within the framework of fitting methods based on non euclidean metrics we implement the q th order means to consistently describe the optimal signal level and to clearly present the difference between the optimal signal level and the optimal partition the signal optimization is utilized for detecting the distribution modes developing a technique being resistant to the noise corruption that can be useful for detecting the optimal partition moreover the mechanisms that affect the optimal partition are identified and thoroughly investigated we first consider random signals following an arbitrary distribution where the block entropy modeling reveals that the optimal partition is located at the median thereafter we consider persistent signals specifying a degree of determinism where the optimal partition is found to be driven far from the median and close to the persistent mode the existence of persistent modes of small hitting time is the key point of this dissertation highlighting their implications on the block entropy modeling finally efforts towards block entropy modeling of non stationary signals are discussed

exit times and entropy for dynamical systems stochastic processes and turbulence abstract we present an investigation of entropy h in dynamical systems stochastic processes and turbulence this tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion particular emphasis is put on a recently proposed approach to the calculation of the entropy based on the exit time statistics the advantages of this method are demonstrated in examples of deterministic diffusive maps intermittent maps stochastic self and multi affine signals and experimental turbulent data concerning turbulence the multifractal formalism applied to the exit time statistics allows us to predict that h 3 for velocity time measurement this power law is independent of the presence of intermittency and has been confirmed by the experimental data analysis moreover we show that the entropy density of a three dimensional velocity field is affected by the correlations induced by the sweeping of large scales

statistical analysis of stochastic resonance in a simple setting a subthreshold signal may be detected if noise is added to the data we study a simple model consisting of a constant signal to which at uniformly spaced times independent and identically distributed noise variables with known distribution are added a detector records the times at which the noisy signal exceeds a threshold there is an optimal noise level called stochastic resonance we explore the detectability of the signal in a system with one or more detectors with different thresholds we use a statistical detectability measure the asymptotic variance of the best estimator of the signal from the thresholded data or equivalently the fisher information in the data in particular we determine optimal configurations of detectors varying the distances between the thresholds and the signal as well as the noise level the approach generalizes to nonconstant signals

asymptotics for the wave equation on differential forms on kerr de sitter space we study asymptotics for solutions of maxwell s equations in fact of the hodge de rham equation d u 0 without restriction on the form degree on a geometric class of stationary spacetimes with a warped product type structure without any symmetry assumptions which in particular include schwarzschild de sitter spaces of all spacetime dimensions n 4 we prove that solutions decay exponentially to 0 or to stationary states in every form degree and give an interpretation of the stationary states in terms of cohomological information of the spacetime we also study the wave equation on differential forms and in particular prove analogous results on schwarzschildde sitter spacetimes we demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of maxwell s equations the hodge de rham equation and the wave equation on differential forms on kerr de sitter spacetimes with small angular momentum

the interior of charged black holes and the problem of uniqueness in general relativity we consider a spherically symmetric double characteristic initial value problem for the real einstein maxwell scalar field equations on the initial outgoing characteristic the data is assumed to satisfy the price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat cauchy surface we establish that the heuristic mass inflation scenario put forth by israel and poisson is mathematically correct in the context of this initial value problem in particular the maximal future development has a future boundary over which the space time is extendible as a c0 metric but along which the hawking mass blows up identically thus the space time is inextendible as a c1 metric in view of recent results of the author in collaboration with i rodnianski which rigorously establish the validity of price s law as an upper bound for the decay of scalar field hair the c0 extendibility result applies to the collapse of complete asymptotically flat spacelike initial data where the scalar field is compactly supported this shows that under christodoulou s c0 formulation the strong cosmic censorship conjecture is false for this system 2005 wiley periodicals inc

late time asymptotics for the wave equation on extremal reissner nordstrom backgrounds abstract we derive the precise late time asymptotics for solutions to the wave equation on extremal reissner nordstrom black holes and explicitly express the leading order coefficients in terms of the initial data our method is based on purely physical space techniques we derive novel weighted energy hierarchies and develop a singular time inversion theory which allow us to uncover the subtle contribution of both the near horizon and near infinity regions to the precise asymptotics we introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes our work confirms and in some cases extends the numerical and heuristic analysis of lucietti murata reall tanahashi ori sela and blaksley burko

from lyapunov modes to their exponents for hard disk systems we demonstrate the preservation of the lyapunov modes in a system of hard disks by the underlying tangent space dynamics this result is exact for the zero modes and correct to order for the transverse and longitudinal momentum modes where is linear in the mode number for sufficiently large mode numbers the terms become significant and the dynamics no longer preserves the mode structure we propose a modified gram schmidt procedure based on orthogonality with respect to the center zero space that produces the exact numerical mode this gram schmidt procedure can also exploit the orthogonality between conjugate modes and their symplectic structure in order to find a simple relation that determines the lyapunov exponent from the lyapunov mode this involves a reclassification of the modes into either direction preserving or form preserving these analytic methods assume a knowledge of the ordering of the modes within the lyapunov spectrum but gives both predictive power for the values of the exponents from the modes and describes the modes in greater detail than was previously achievable thus the modes and the exponents contain the same information