chaos in the hamiltonian mean field model rst show that the largest lyapunov exponent remains strictly positive in the innite size limit converging to its asymptotic value with 1 lnn corrections we then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered clustered phase and the disordered phase present at high energy densities we also show that the full spectrum of lyapunov exponents consists of a bulk component converging to the zero value taken by a test oscillator forced by the mean eld plus subextensive bands of o lnn exponents taking nite values we nally investigate the robustness of these results by studying a 2d extension of the hmf model where each particle is endowed with 4 degrees of freedom thus allowing the emergence of chaos at the level of single particle altogether these results illustrate the subtle eects of global or long range coupling and the importance of the order in which the innite time and innite size limits are taken for an innite size hmf system represented by the vlasov equation no chaos is present while chaos exists and subsists for any nite system size

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

an abstract result on asymptotically positively homogeneous differential equations abstract following from the proof of spectrum theory of completely continuous linear operators we give an abstract result concerning the estimates of norms of positively homogeneous operators in this paper using this result and some duality theorems for coincidence degree due to 1 the periodic boundary value problem of asymptotically positively homogeneous differential equations will be studied and the present results sharply improve those obtained earlier

contributions to coincidence degree theory of asymptotically homogeneous operators in this paper we give contributions to the coincidence degree theory of asymptotically homogeneous operators applications are given to the periodic problem for second order functional differential equations

nonresonance conditions for asymptotically positively homogeneous differential systems the fucik spectrum and its generalization in this article we propose how to define the fucik spectra for differential systems in rnwith jumping nonlinearities the propertyp a generalization of the non fucik spectra to positively homogeneous nonautonomous differential systems is also defined in the most general setting a quantitative estimate result concerning the parameterized completely continuous positively homogeneous perturbations of the identity in banach spaces is given in this article such a result is a generalization of one property in the spectrum theory for completely continuous linear operators using this estimate result we will give some significantly improved nonresonance results from the fucik spectra and propertypfor asymptotically positively homogeneous differential systems

intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators a theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit cycle oscillators with distributed native frequencies in particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above this peculiar feature is demonstrated by numerical simulations mainly through finite size scaling analyses in the course of exploring its origin we encounter a new concept termed a correlation frequency which provides a natural interpretation of the finite size scaling laws a comment is given on a recent theory by kuramoto and nishikawa to clarify why it contradicts our results

scaling of lyapunov exponents of coupled chaotic systems we develop a statistical theory of the coupling sensitivity of chaos the effect was first described by daido prog theor phys 72 853 1984 it appears as a logarithmic singularity in the lyapunov exponent in coupled chaotic systems at very small couplings using a continuous time stochastic model for the coupled systems we derive a scaling relation for the largest lyapunov exponent the singularity is shown to depend on the coupling and the systems mismatch generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered too the analytical results are confirmed by numerical simulations

uwb sparse diffuse channels part i channel models and bayesian estimators in this two part paper the problem of channel estimation in ultra wide band uwb systems is investigated due to the large transmission bandwidth the channel has been traditionally modeled as sparse however some propagation phenomena e g scattering from rough surfaces and frequency distortion are better modeled by a diffuse channel herein a novel hybrid sparse diffuse hsd channel model is proposed tailored to the hsd model channel estimators are designed for different scenarios that vary in the amount of side information available at the receiver an expectation maximization algorithm to estimate the power delay profile of the diffuse component is also designed the proposed methods are compared to unstructured and purely sparse estimators the numerical results show that the hsd estimation schemes considerably improve the estimation accuracy and the bit error rate performance over conventional channel estimators in part ii the new channel estimators are evaluated with more realistic geometry based channel emulators the numerical results show that even when the channel is generated in this manner the new estimation strategies achieve high performance moreover a mean squared error analysis of the proposed estimators is performed in the high and low signal to noise ratio regimes thus quantifying in closed form the achievable performance gains

clustered motion in symplectic coupled map systems clustered motion of particles is found in hamiltonian dynamics of symplectic coupled map systems particles assemble and move with strong correlation the motion is chaotic but is distinguishable from random chaotic motion lyapunov analysis distinguishes global instability from local fluctuations clustered motions have finite lifetime they have fractal geometric structure in the phase space as the orbits are trapped to ruins of kam tori and islands

uwb sparse diffuse channels part ii estimator analysis and practical channels in this two part paper the problem of channel estimation in ultra wide band uwb systems is investigated in part i a novel hybrid sparse diffuse hsd model is proposed for the uwb channel and new channel estimation strategies are designed for this model in this paper part ii a mean squared error mse analysis of the generalized mmse and generalized thresholding estimators developed in part i is performed for the asymptotic regimes of low and high snr the analysis quantifies the achievable mse performance of these schemes over unstructured estimators specifically we prove that it is beneficial to be conservative in the estimation of the sparse component i e to assume that the sparse component is sparser than it actually is moreover we analyze the scenario with a non orthogonal pilot sequence and establish a connection between the generalized thresholding estimator and conventional sparse approximation algorithms proposed in the literature in addition to the theoretical analysis these channel estimation schemes are evaluated in a more realistic geometry based channel emulator for which the hsd model developed in part i is an approximation the numerical results are shown to match the expected asymptotic mse behavior moreover the proposed estimation techniques are shown to outperform conventional unstructured and purely sparse estimators from both an mse and a bit error rate perspectives even for the realistic geometry based channel model

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

generalization of fredholm alternative for nonlinear differential operators on considere le probleme aux valeurs limites u t p 2 u t f t u t g t t 0 u 0 u 0 on demontre que si lim u inf f t u u p 2 u lim u sup f t u u p 2 u sont entre des valeurs propres du probleme aux valeurs propres u p 2 u u p 2 u sur 0 u 0 u 0 alors le probleme aux limites est resoluble pour tout g l 1

dependence of solutions and eigenvalues of measure differential equations on measures abstract it is well known that solutions of ordinary differential equations are continuously dependent on finite dimensional parameters in equations in this paper we study the dependence of solutions and eigenvalues of second order linear measure differential equations on measures as an infinitely dimensional parameter we will provide two fundamental results which are the continuity and continuous frechet differentiability in measures when the weak topology and the norm topology of total variations for measures are considered respectively in some sense the continuity result obtained in this paper is the strongest one as an application we will give a natural simple explanation to extremal problems of eigenvalues of sturm liouville operators with integrable potentials

on the peroidic bvp for the forced duffing equation with jumping nonlinearity on etudie le probleme suivant u cu f t u e t p p sur 0 t u t u 0 0 u t u 0 0 c est une fonction arbitraire dans l 1 0 t r c r et l application non lineaire f est a saut

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

nonresonance for one dimensional p laplacian with regular restoring abstract in this paper the existence of periodic solutions of the equation p x f t x 0 is proved where f 0 t r r has some nonlinear growth near x and the nonresonance condition sp which can be explained using the periodic eigenvalues concerning p laplacian is satisfied

existence and multiplicity results for periodic solutions of semilinear duffing equations we obtain existence an multiplicity of harmonic and subharmonic solutions to the periodically forced nonlinear duffing s equation x g x p t in the semi linear case 0 a g x x b for x large we introduce new conditions of nonresonance with respect to the spectrum of the associated linear differential operator x x and show by some nonlinear counterexamples that the results are sharp

periodic solutions for p laplacian neutral functional differential equation with deviating arguments by using the theory of coincidence degree we study a kind of periodic solutions to p laplacian neutral functional differential equation with multiple deviating arguments a new result on the existence of periodic solutions is obtained

a continuation approach to superlinear periodic boundary value problems this paper deals with the problem of the existence of t periodic solutions for the first order differential system x i t x 1 1 where f 0 t x r r is a caratheodory function in what follows we prove some results for the solvability of the periodic bvp in the case when the dimension of the space is even such a limitation is motivated by our interest in applications to the second order equation d g t u u o 1 2 which takes the form of 1 1 when it is written as the equivalent system u v 0 g t u v 1 3

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

statistical mechanical formulation of lyapunov exponents we show how the lyapunov exponents of a dynamic system can in general be expressed in terms of the free energy of a non hermitian quantum many body problem this puts their study as a problem of statistical mechanics whose intuitive concepts and techniques of approximation can hence be borrowed

entropy complexity analysis in some globally coupled systems globally coupled n body systems are well known to possess an intricate dynamics when n is large collective effects may drastically lower the effective dimension of the dynamics breaking the conditions on ergodicity necessary for the applicability of statistical mechanics these problems are here illustrated and discussed through an entropy complexity analysis of the repulsive hamiltonian mean field model using a poincare section of the mean field time series provides a natural sampling time in the entropy complexity treatment this approach is shown to single out the out of equilibrium dynamical features and to uncover a transition of the system dynamics from low energy non boltzmann quasi stationary states to high energy stochastic like behavior

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

eigenvalue estimates for the p laplace operator on manifolds abstract in this paper some new geometric estimates for the first eigenvalue and the fundamental tone of the p laplacian on manifolds in terms of admissible vector fields are obtained also a new spectral invariant and its relation with the geometry of the manifold is defined

n tree approximation for the largest lyapunov exponent of a coupled map lattice the n tree approximation scheme introduced in the context of random directed polymers is applied here to the computation of the maximum lyapunov exponent in a coupled map lattice we discuss both an exact implementation for small tree depth n and a numerical implementation for larger n we find that the phase transition predicted by the mean field approach shifts towards larger values of the coupling parameter when the depth n is increased we conjecture that the transition eventually disappears

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

hyperbolic decoupling of tangent space and effective dimension of dissipative systems we show using covariant lyapunov vectors that the tangent space of spatially extended dissipative systems is split into two hyperbolically decoupled subspaces one comprising a finite number of frequently entangled physical modes which carry the physically relevant information of the trajectory and a residual set of strongly decaying spurious modes the decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators we conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor

boundedness in forced isochronous oscillators abstract in this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator x v x g x f t where v is a so called t isochronous potential the perturbation g is assumed to be bounded and the 2 periodic function f t is smooth using the resonant small twist theorem and averaged small twist theorem established by ortega we will prove the boundedness of all solutions for the above forced isochronous oscillator in the resonant and non resonant cases under some reasonable assumptions respectively

periodic solutions for ordinary differential equations with sublinear impulsive effects abstract the continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects the applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations for instance the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with landesman lazer conditions

chaos and relaxation to equilibrium in systems with long range interactions in the thermodynamic limit systems with long range interactions do not relax to equilibrium but become trapped in nonequilibrium stationary states for a finite number of particles a nonequilibrium state has a finite lifetime so that eventually a system will relax to thermodynamic equilibrium the time that a system remains trapped in a quasistationary state qss scales with the number of particles as n with 0 and diverges in the thermodynamic limit in this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a qss we discover that chaos measured by the lyapunov exponents favors faster relaxation to equilibrium surprisingly weak chaos favors faster relaxation than strong chaos

the fredholm alternative for the one dimensional p laplacian abstract in this paper we consider the solvability of the boundary value problem p u 1 p u f u h t u 0 u t 0 where p 1 p u u p 2 u by using generalized polar coordinates transformation method we improve and generalize some results obtained recently in j differential equations 151 1999 386

scaling law for the maximal lyapunov exponent the authors study the scaling law for epsilon to 0 of the maximal lyapunov exponent for coupled chaotic map lattices and for products of random jacobi matrices to this purpose they develop approximate analytical treatments of the random matrix problem inspired by the theory of directed polymers in a random medium a type of mean field method and a tree approximation which introduces correlations the theoretical results suggest a leading mod log epsilon mod 1 increase in the maximal lyapunov exponent near epsilon 0 which is confirmed by numerical simulations also for coupled map lattices a dynamical mechanism responsible for this behaviour is investigated for a 2 2 random matrix model the theory also predicts a phase transition at a critical value of the coupling epsilon e which is not observed in numerical simulations and might be an artifact of the approximation

the rotation number approach to the periodic fuc caron ik spectrum abstract in this paper we study the fucik spectrum of the problem q t x q t x 0 with the 2 periodic boundary condition where q t are 2 periodic after introducing a rotation number function for we prove using the hamiltonian structure and the positive homogeneity of that for any positive integer n the two boundary curves of the domain 1 n 2 in the plane are fucik curves of the result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional fucik spectrum with the dirichlet condition in particular it remains open if the fucik spectrum of is composed of only these curves

an extension of mawhin s continuation theorem and its application to boundary value problems with a p laplacain abstract in order to study boundary value problems bvps with a p laplacian an extension of mawhin s continuation theorem is given then some existence theorems for bvps with a p laplacian at resonance are proved

nonuniform nonresonance of semilinear differential equations abstract consider the dirichlet problem of nonlinear differential equations with the principal part the p laplacian when the nonlinearity satisfies some semilinearity conditions the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues in this article we will introduce some weighted eigenvalues the nonuniform nonresonance conditions proved in this article using weighted eigenvalues will improve the usual ones significantly

a p laplacian problem with a multi point boundary condition in this paper we will use the degree theory to give an existence result to the boundary value problem of nonlinear perturbations of the scalar p laplacian with the multi point boundary condition involving the lebesgue stieltjes integral among the boundary conditions considered in this paper we will find also the optimal bounds on the growth of the nonlinear perturbations in our existence result 2004 elsevier ltd all rights reserved

scaling laws for the largest lyapunov exponent in long range systems a random matrix approach we investigate the laws that rule the behavior of the largest lyapunov exponent lle in many particle systems with long range interactions we consider as a representative system the so called hamiltonian ensuremath alpha ensuremath mathrm xy model where the adjustable parameter ensuremath alpha controls the range of the interactions of n ferromagnetic spins in a lattice of dimension d in previous work the dependence of the lle with the system size n for sufficiently high energies was established through numerical simulations in the thermodynamic limit the lle becomes constant for ensuremath alpha gd whereas it decays as an inverse power law of n for ensuremath alpha ld a recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results here we show that the scaling behavior can also be explained by a random matrix approach in which the tangent mappings that define the lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble

resonant and nonresonant oscillations for some third order nonlinear ordinary differential equations on etudie l existence de solutions 2 periodiques des equations differentielles d ordre 3 x ax bx g t x p t x x x x ax bx t x x x x p t x x x x ax f x x g t x p t x x x

existence of periodic solutions for quasilinear differential equations in this paper we find some sufficient conditions for the existence of periodic solutions to a class of p laplacian by using a fixed point theorem by ding chin ann math 2 1981 281

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

existence of periodic solutions of duffing equations on considere le probleme aux valeurs limites periodiques de type duffing u t ku t g t u t f t t 0 2 u 0 u 2 u 0 u 2 on demontre un resultat d existence general qui montre que si la croissance de g t u a l infini appartient a un ensemble avec la propriete p alors pour tout f l 1 0 2 le probleme a au moins une solution

analytic estimation of the lyapunov exponent in a mean field model undergoing a phase transition the parametric instability contribution to the largest lyapunov exponent lle is derived for a mean field hamiltonian model with attractive long range interactions this uses a recent riemannian approach to describe hamiltonian chaos with a large number n of degrees of freedom through microcanonical estimates of suitable geometrical observables the mean field behavior of the lle is analytically computed and related to the second order phase transition undergone by the system it predicts that chaoticity drops to zero at the critical temperature and remains vanishing above it with the lle scaling as n 1 3 to the leading order in n