a kam theorem for one dimensional schrodinger equation with periodic boundary conditions abstract in this paper one dimensional 1 d nonlinear schrodinger equation iu t u xx mu g u u u 0 with periodic boundary conditions is considered m 1 12 z is a real parameter and the nonlinearity g u u j l j l 4 a jl u j u l a jl a lj r a 22 0 is a real analytic function in a neighborhood of the origin the kam machinery is adapted to fit the above equation so as to construct small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

a kam theorem for hamiltonian partial differential equations in higher dimensional spaces in this paper we give a kam theorem for a class of infinite dimensional nearly integrable hamiltonian systems the theorem can be applied to some hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi periodic solutions and its local birkhoff normal form the applications to the higher dimensional beam equations and the higher dimensional schrodinger equations with nonlocal smooth nonlinearity are also given in this paper

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

analysis of dynamics in magnetotelluric data by using the fisher shannon method abstract the fisher shannon information fs plane defined by the fisher information measure and the shannon entropy power is used to investigate the complex dynamics of magnetotelluric data of three stations in taiwan in the fs plane the electric and magnetic components are significantly separated characterized by different degrees of order further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated

degenerate lower dimensional tori in reversible systems abstract in this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems the normal variable is two dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure

wavelet analysis and nonlinear dynamics in a nonextensive setting abstract we undertake the study of signals originated in time dependent nonlinear systems by recourse to a wavelet based multiresolution analysis as adapted to a nonextensive tsallis scenario diverse applications are discussed it is shown that the tsallis environment provides one with more detailed information than the conventional shannon one

construction of response solutions for two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point under small perturbations abstract by developing two kam theorems in this paper we show that two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point a reversible system and a non conservative system admit a response solution under small perturbations for the degenerate reversible system applying special structure of unperturbed nonlinear term and herman method we successfully control the shift of equilibrium point which is difficult in view of the degenerate linear term for the degenerate non conservative system kam method is brought into force even in completely degenerate case because of the restrictions on the smallness and average of perturbation moreover arithmetic condition on the frequency is assumed to satisfy the brjuno russmann s non resonant condition by the poschel russmann kam method we prove that these two kinds of perturbed systems can be reduced to a suitable normal form with zero as equilibrium point by a quasi periodic transformation

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

nonlinear wave and schrodinger equations on compact lie groups and homogeneous spaces we develop linear and nonlinear harmonic analysis on compact lie groups and homogeneous spaces relevant for the theory of evolutionary hamiltonian pdes a basic tool is the theory of the highest weight for irreducible representations of compact lie groups this theory provides an accurate description of the eigenvalues of the laplace beltrami operator as well as the multiplication rules of its eigenfunctions as an application we prove the existence of cantor families of small amplitude time periodic solutions for wave and schrodinger equations with differentiable nonlinearities we apply an abstract nash moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the laplace operator we provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on banach scales of sobolev functions

fisher based thermodynamics its legendre transform and concavity properties it is shown that the legendre transform structure of thermodynamics can be replicated without any change if one replaces the entropy s by fisher s information measure i also the important thermodynamic property of concavity is shown to be obeyed by i by this use of the fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one

elliptic type degenerate invariant tori for quasi periodically forced four dimensional non conservative systems abstract in this paper we will focus on the persistence of elliptic type degenerate invariant tori with the same frequency vector r d as the forcing in a class of quasi periodically forced four dimensional non conservative systems it is shown that under suitable hypothesis of smoothness with respect to and the brjuno russmann s non resonant condition instead of ordinary diophantine condition with respect to the frequency vector r d the invariant tori persist under small perturbations for most of the sufficiently small parameters in the sense of the lebesgue measure hence the system has a quasi periodic solution with the same frequency vector as the forcing the proof is based on the poschel russmann kam method which is a kam type method in which one uses the polynomial structure of function to truncate introduce a parameter q and make the steps of kam iteration infinitely small in the speed of function q n 0 q 1 rather than super exponential function like traditional kam method

a birkhoff lewis type theorem for the nonlinear wave equation we give an extension of the celebrated birkhoff lewis theorem to the nonlinear wave equation accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin which is an elliptic equilibrium of the associated infinite dimensional hamiltonian system

almost periodic solutions for a class of linear schrodinger equations with almost periodic forcing in this paper the existence of almost periodic solutions corresponding to full dimensional invariant tori is shown for a class of linear schrodinger equations with almost periodic forcing subject to periodic boundary conditions it is proved via an improved kolmogorov arnold moser reducibility

fisher information measure of geoelectrical signals abstract we analyzed the time fluctuations in the dynamics of geoelectrical data recorded in a seismic area of southern italy by means of fisher information measure which is a powerful tool to investigate complex and nonstationary signals the time evolution of the fisher information measure calculated for our signal reveals some links with the earthquakes which occurred in the investigated area

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

quasi periodic solutions in a nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 4 u 0 with the periodic boundary condition is considered it is proved that for each given constant potential m and each prescribed integer n 1 the equation admits a whitney smooth family of small amplitude time quasi periodic solutions with n diophantine frequencies the proof is based on a partial birkhoff normal form reduction and an improved kam method

localized breather like solution in a discrete klein gordon model and application to dna abstract non propagating localized oscillating solutions are studied in the case of a discrete one dimensional system with the 2 3 power onsite potential a multiple scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider starting from an exact pulse like solution in the continuum we derive an approximate oscillating solution our numerical simulations reveal that whereas the excitations are strongly damped in the strong intersite coupling case they are long lived in the discrete cases an additional modulation phenomenon of the oscillation seen in the simulation is explained by combination of the breather oscillations with the zero group velocity phonons modes then applying the green s functions formalism in analogy with a previous work s takeno et al prog theor phys suppl 94 1988 we derive a very accurate expression for the discrete breather with a quasi infinite lifetime lastly we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of dna

computer methodologies for comparison of computational efficiency of simultaneous methods for finding polynomial zeros abstract the study and analysis of iterative methods for the determination of zeros of algebraic polynomials the subject of this paper is one of the most actual tasks in the theory and practice our main goal is to examine and compare the quality of most frequently used methods for simultaneously finding all zeros of a polynomial by combining numerical experiments theoretical models and empirical cpu time methodology special attention is devoted to empirical average case model assuming that average cpu times are calculated performing the experiment i on several different computers ii running each method through many loops for a fixed example iii testing reasonably large number of polynomials of different structure body and iv the use of almost arbitrary initial approximations the empirical model ranks the tested methods on the base of the consumed cpu times needed for the fulfillment of the required accuracy of approximations under the same initial conditions our aim is to select a group of the most powerful simultaneous methods with globally convergent performance convergence behavior of two most powerful methods is illustrated by trajectories in the complex plane that consist of the point approximations starting with initial points and going towards the endpoints where the iterative process stops due to the termination criterion

a family of simultaneous zero finding methods abstract applying hansen patrick s formula for solving the single equation f z 0 to a suitable function appearing in the classical weierstrass method two one parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived it is shown that all the methods of these families have fourth order of convergence some computational aspects of the proposed methods and numerical examples are given

kam for reversible derivative wave equations we prove the existence of cantor families of small amplitude analytic linearly stable quasi periodic solutions of reversible derivative wave equations

persistence of lagrange invariant tori at tangent degeneracy abstract in the present paper without russmann s nondegenderate condition we show the persistence of kam invariant tori using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three body problems especially for lagrange equilibrium position l2 with certain degeneracy

a renormalization approach to lower dimensional tori with brjuno frequency vectors abstract we extend the renormalization scheme for vector fields on t d r m in order to construct lower dimensional invariant tori with brjuno frequency vectors for near integrable hamiltonian flows for every brjuno frequency vector r d and every vector r d satisfying a diophantine condition with respect to there exists an analytic manifold w of infinitely renormalizable hamiltonian vector fields each vector field on w is shown to have an analytic invariant torus with frequency vector

on the guaranteed convergence of the fourth order simultaneous method for polynomial zeros one of the most important and challenging problems in solving nonlinear equations is the construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of the considered numerical method a suitable convergence procedure based partially on smale s point estimation theory from 1981 is applied in this paper to the fourth order iterative method for the simultaneous approximation of simple zeros of polynomials proposed by zheng and sun in 1999 we have stated initial conditions which ensure the guaranteed convergence of this method these conditions are of significant practical importance since they depend only on available data the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

kam for quasi linear and fully nonlinear forced perturbations of airy equation we prove the existence of small amplitude quasi periodic solutions for quasi linear and fully nonlinear forced perturbations of the linear airy equation for hamiltonian or reversible nonlinearities we also prove their linear stability the key analysis concerns the reducibility of the linearized operator at an approximate solution which provides a sharp asymptotic expansion of its eigenvalues for quasi linear perturbations this cannot be directly obtained by a kam iteration hence we first perform a regularization procedure which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder these transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo differential operators at this point we implement a nash moser iteration with second order melnikov non resonance conditions which completes the reduction to constant coefficients

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

a quasi periodic poincare s theorem we study the persistence of invariant tori on resonant surfaces of a nearly integrable hamiltonian system under the usual kolmogorov non degenerate condition by introducing a quasi linear iterative scheme to deal with small divisors we generalize the poincare theorem on the maximal resonance case i e the periodic case to the general resonance case i e the quasi periodic case by showing the persistence of majority of invariant tori associated to non degenerate relative equilibria on any resonant surface

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

a kam theorem for the defocusing nls equation in this paper we prove a kam theorem for the defocusing nls equation in one space dimension with periodic boundary conditions the novelty of our result is that it is valid not only near the zero solution but on the entire sobolev space h n t c with n is an element of z 1 in particular the invariant tori which persist under small hamiltonian perturbations might be far away from the zero potential c 2011 elsevier inc all rights reserved

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

quasi periodic breathers in hamiltonian networks of long range coupling this work is concerned with hamiltonian networks of weakly and long range coupled oscillators with either variable or constant on site frequencies we derive an infinite dimensional kam like theorem by which we establish that given any n sites of the lattice there is a positive measure set of small amplitude quasi periodic breathers solutions of the hamiltonian network that are quasi periodic in time and exponentially localized in space having n frequencies which are only slightly deformed from the on site frequencies

discrete breathers localization and transfer of energy in discrete hamiltonian nonlinear systems we present a partial review of results concerning the theory of discrete breathers in nonlinear hamiltonian and discrete systems these special time periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer we first review the principles which govern their existence and which are used in the existence proofs available up to now next we discuss their linear stability and the interaction of discrete breathers with small amplitude waves showing also how they could grow or decay we also briefly discuss the existence of intraband dbs in systems with a linear discrete spectrum which are not spatially periodic random or otherwise and where linear waves cannot propagate we also show that nonlinearity may restore the existence of solutions that propagate energy we review some results concerning energy transportation by dbs and especially the main features associated with their mobility we briefly discuss new perspectives opened up by the theory for applications that in particular look especially interesting for biophysics

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

on the reducibility of linear differential equations with quasiperiodic coefficients the system x a eq t x in rd is considered where a is a constant matrix and q a quasiperiodic analytic matrix with r basic frequencies the eigenvalues of a are arbitrary including the purely imaginary case suppose that the set formed by the eigenvalues of a and the basic frequencies of q satisfies a nonresonant condition then there is a positive measure cantorian set e such that for e e the system is reducible to constant coefficients by means of a quasiperiodic change of variables provided a nondegeneracy condition holds this condition prevents locking at resonance

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

fisher information and nonlinear dynamics abstract the past decade has witnessed an impressive growth in the scope and variety of physical applications of fisher s information measure fim as a consequence fim is widely regarded today as providing one with a unifying principle for physics in the present paper we investigate its usefulness in connection with the description of important aspects of the behavior of dynamical systems and find that the unification referred to above can be extended to the field of these systems

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

quasi periodic solutions of nonlinear beam equation with prescribed frequencies consider the one dimensional nonlinear beam equation utt uxxxx mu u3 0 under dirichlet boundary conditions we show that for any m 0 but a set of small lebesgue measure the above equation admits a family of small amplitude quasi periodic solutions with n dimensional diophantine frequencies these diophantine frequencies are the small dilation of a prescribed diophantine vector the proofs are based on an infinite dimensional kolmogorov arnold moser iteration procedure and a partial birkhoff normal form

a parallel algorithm for simple roots for polynomials abstract a method for finding simple roots of arbitrary polynomials based on divided differences is discussed theoretical background is presented for the case of simple roots numerical results are presented which show the algorithm finds simple and multiple zeros to an accuracy usually limited by the accuracy of polynomial evaluation the method is designed for a simd parallel computer the algorithm is compared to two other frequently used polynomial root finders the jenkins traub algorithm and laguerre s method

multiscale kam theorem for hamiltonian systems abstract in this paper we study the persistence of invariant tori in nearly integrable multiscale hamiltonian systems with highorder degeneracy in the integrable part such hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems we introduce multiscale nondegenerate condition and multiscale diophantine condition comparable to the usual diophantine condition using quasilinear kam method we prove a multiscale kam theorem

persistence of hyperbolic tori in hamiltonian systems abstract we generalize the well known result of graff and zehnder on the persistence of hyperbolic invariant tori in hamiltonian systems by considering non floquet frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies the preservation of part or full frequency components associated to the degree of non degeneracy is considered as applications we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non integrable hamiltonian system

kam for the derivative nonlinear schrodinger equation with periodic boundary conditions abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f u 2 u x 0 x t r 2 z where f is real analytic in some neighborhood of the origin in c f 0 0 and f 0 0 we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

on a simultaneous method of newton weierstrass type for finding all zeros of a polynomial combining a suitable two point iterative method for solving nonlinear equations and weierstrass correction a new iterative method for simultaneous finding all zeros of a polynomial is derived it is proved that the proposed method possesses a cubic convergence locally numerical examples demonstrate a good convergence behavior of this method in a global sense it is shown that its computational efficiency is higher than the existing derivative free methods

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

construction of zero finding methods by weierstrass functions abstract in this paper we present a simple and elegant procedure for the construction of iterative methods for the simultaneous determination of simple or multiple zeros of an algebraic polynomial this procedure is based on the application of a special type of functions called weierstrass functions to suitable zero finding methods for a single zero for demonstration using this approach we derive many known iterative methods in a simpler way compared with original derivations as well as some new methods aside from the presented methodology in developing zero finding methods the paper offers a short review of simultaneous methods including some historical notes

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems

oscillatory wave fronts in chains of coupled nonlinear oscillators wave front pinning and propagation in damped chains of coupled oscillators are studied there are two important thresholds for an applied constant stress f for f f cs static peierls stress there are only stable moving wave fronts for piecewise linear models extending an exact method of atkinson and cabrera s to chains with damped dynamics corroborates this description for smooth nonlinearities an approximate analytical description is found by means of the active point theory generically for small or zero damping stable wave front profiles are nonmonotone and become wavy oscillatory in one of their tails

global density of reducible quasi periodic cocycles on bf t 1 times roman su 2 we prove that given a in a set of total haar measure in t 1 r z the set of a c t 1 su 2 for which the skew product system a t 1 su 2 t 1 su 2 a y a y is reducible that is a b a 0 b 1 for some a 0 su 2 b c t 1 su 2 is dense for the c topology

analysis of electromagnetic pre seismic emissions using fisher information and tsallis entropy abstract in this work we investigate the dynamics of electromagnetic precursors recorded prior to significant earthquakes in greece the analysis of these signals is performed using fisher information which is a powerful tool for investigating complex and non stationary signals our results point to a decrease of the precursor complexity as the main tectonic event is approaching the results are compatible to those derived using tsallis entropy the sensitivity of fisher information and tsallis entropy are compared on the basis of the features of the underlying fracture process which they reveal

the fisher information measure and shannon entropy for particulate matter measurements abstract we investigated the dynamics of particulate matter data recorded in tito a small industrial area of southern italy the analysis of these signals was performed using the fisher information measure fim which is a powerful tool for investigating complex and nonstationary signals and the shannon entropy which is a well known tool for investigating the degree of disorder in dynamical systems our results point to an increase of disorder and complexity from fine to coarse particulates

symmetries of the fokker planck equation and the fisher frieden arrow of time we give here a concrete illustration of frieden s arrow of time connected with fisher s information measure it is shown that this arrow of time is related to symmetries of the fokker planck equation textcopyright 1996 the american physical society

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

a posteriori error bound methods for the inclusion of polynomial zeros using carstensen s results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4 these methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial we establish computationally verifiable initial conditions that guarantee the convergence of these methods some computational aspects and the possibility of implementation on parallel computers are considered including two numerical examples a comparison of a posteriori error bound methods with the corresponding circular interval methods regarding the computational costs and sizes of produced inclusion disks were given

invariant tori of nonlinear schrodinger equation abstract in this paper we consider one dimensional nonlinear schrodinger equation i u t u x x v x u f u 2 u 0 on 0 r under the boundary conditions a 1 u t 0 b 1 u x t 0 0 a 2 u t b 2 u x t 0 a i 2 b i 2 0 for i 1 2 it is proved that for a prescribed and analytic positive potential v x the above equation admits small amplitude quasi periodic solutions corresponding to d dimensional invariant tori of the associated infinite dimensional dynamical system

point estimation of a family of simultaneous zero finding methods abstract the construction of initial conditions which provide the safe convergence of iterative processes is one of the most important problems in solving equations in this paper we give initial conditions for a one parameter family of iterative methods for the simultaneous determination of all simple zeros of a polynomial zeros this family has the order of convergence four and generates some new methods with good convergence properties the presented initial conditions are of practical importance since depend only of available data coefficients of a polynomial and initial approximations to the zeros

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

singular spectrum analysis and fisher shannon analysis of spring flow time series an application to anjar spring lebanon abstract in this study the time dynamics of water flow from anjar spring was investigated which is one of the major issuing springs in the central part of lebanon likewise many water sources in lebanon this spring has no continuous records for the discharge and this would prevent the application of standard time series analysis tools furthermore the highly nonstationary character of the series implies that suited methodologies can be employed to get insight into its dynamical features therefore the singular spectrum analysis ssa and fisher shannon fs method which are useful methods to disclose dynamical features in noisy nonstationary time series with gaps are jointly applied to analyze the anjar spring water flow series the ssa revealed that the series can be considered as the superposition of meteo climatic periodic components low frequency trend and noise like high frequency fluctuations the fs method allowed to extract and to identify among all the ssa reconstructed components the long term trend of the series the long term trend is characterized by higher fisher information measure fim and lower shannon entropy and thus represents the main informative component of the whole series generally water discharge time series presents very complex time structure therefore the joint application of the ssa and the fs method would be very useful in disclosing the main informative part of such kind of data series in the view of existing climatic variability and or anthropogenic challenges

informational analysis of seismic sequences by applying the fisher information measure and the shannon entropy an application to the 2004 2010 seismicity of aswan area egypt abstract the interevent time iet and interevent distance ied series of seismic events occurred at aswan area egypt from 2004 to 2010 were investigated by means of the fisher information measure and the shannon entropy the analysis was performed varying the depth and the magnitude thresholds the results point out to an increase of level of organization and order with the decrease of magnitude threshold and the increase of depth threshold for the iet series while the ied series are characterized by a level of uncertainty approximately constant with the threshold magnitude the complexity measure calculated as the product of the fisher information measure and the shannon entropy power presents very similar pattern for both the types of seismic series indicating an increasing complexity with the decrease of the threshold magnitude and the increase of the threshold depth

spectral and informational analysis of pedestrian contact force in simulated overcrowding conditions abstract in this paper the time series of contact force among pedestrians in two different crowding conditions is analyzed by using the correlogram based periodogram and the fisher shannon method to investigate their spectral and informational properties respectively the overcrowding was simulated by controlled experiments carried out at university of science and technology of china hefei in three different walking styles our results show that 1 the contact force is characterized by a significant periodic behavior 2 a clear discrimination in the fisher shannon information plane is found between contact forces measured in the two conditions of lower and higher pedestrian density

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

kam for the nonlinear schrodinger equation we consider the d dimensional nonlinear schrodinger equation under periodic boundary conditions i dot u delta u v x u ep frac p f p bar u x u bar u quad u u t x x in t d where v x sum hat v a e i sc a x is an analytic function with hat v real and f is a real analytic function in re u im u and x this equation is a popular model for the real nls equation where instead of the convolution term v u we have the potential term vu for ep 0 the equation is linear and has time quasi periodic solutions u u t x sum a in a hat u a e i a 2 hat v a t e i sc a x quad hat u a 0 where a is any finite subset of z d we shall treat omega a a 2 hat v a a ina as free parameters in some domain u subset r a this is a hamiltonian system in infinite degrees of freedom degenerate but with external parameters and we shall describe a kam theory which under general conditions will have the following consequence if ep is sufficiently small then there is a large subset u of u such that for all omega in u the solution u persists as a time quasi periodic solution which has all lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients

lower dimensional invariant tori with prescribed frequency for the nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 2 u f u 2 u 0 subject to dirichlet boundary conditions is considered where the nonlinearity f is a real analytic function near u 0 with f 0 f 0 0 it is proved that for each given constant potential m and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude time quasi periodic solutions whose b dimensional frequencies are just small dilation of a prescribed diophantine vector accordingly we obtain the existence of lower dimensional invariant kam tori with tangential frequencies constrained to a given diophantine direction in an infinite dimensional phase space setting

lower dimensional invariant tori with prescribed frequency for nonlinear wave equation abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 3 0 subject to dirichlet boundary conditions is considered we show that for each given m 0 and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude quasi periodic solutions with b dimensional diophantine frequencies which correspond to b dimensional invariant tori of an associated infinite dimensional dynamical system in particular these diophantine frequencies are the small dilation of a prescribed diophantine vector the proof is based on a partial birkhoff normal form reduction and an improved kam method

a globally convergent parallel algorithm for zeros of polynomial systems certain classes of nonlinear systems of equations such as polynomial systems have properties that make them particularly amenable to solution on distributed computing systems some algorithms considered unfavorably on a single processor serial computer may be excellent on a distributed system this paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube some computational results are reported

on new higher order families of simultaneous methods for finding polynomial zeros abstract two one parameter families of iterative methods for the simultaneous determination of simple zeros of algebraic polynomials are presented the construction of these families are based on a one parameter family of the third order for finding a single root of nonlinear equation f x 0 some previously derived simultaneous methods can be obtained from the presented families as special cases we prove that the local convergence of the proposed families is of the order four numerical results are included to demonstrate the convergence properties of considered methods

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis