stability of discrete breathers abstract we approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling in the case of hamiltonian systems using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2 linearly stable for weak enough coupling we consider also dissipative networks of oscillators in this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p topology some examples are considered amongst which an instance of roto breather

intraband discrete breathers in disordered nonlinear systems ii localization abstract we find spatially localized time periodic solutions discrete breathers or dbs in disordered nonlinear systems with frequency inside the linear phonon spectrum under conditions that strictly prohibit their existence in their periodic counterparts for that purpose we develop a new in situ method for the accurate calculation of these solutions which does not make use of any continuation from an anticontinuous limit using this method we demonstrate that intraband localized modes intraband discrete breathers or idbs at a given site with frequencies inside the discrete linear spectrum do exist provided these frequencies do not belong to forbidden resonance gaps since there is a dense set of resonant frequencies we illustrate numerically in agreement with a theorem by albanese and frohlich that the localized dbs exist provided that their frequencies belong to fat cantor sets i e with finite measure such a set contains as accumulation points the linear frequency of the normal mode at the occupied site we check that many of these solutions are linearly stable and conjecture that their frequency belongs to another smaller fat cantor set our numerical methods provide a much wider set of exact solutions which are multisite breathers and suggest conjectures extending the existing theorems the physical implications of the existence of idbs and possible applications for glasses and the persistent spectral hole burning are discussed

the notion of persistence applied to breathers in thermal equilibrium abstract we study the thermal equilibrium of nonlinear klein gordon chains at the limit of small coupling anticontinuum limit we show that the persistence distribution associated to the local energy density is a useful tool to study the statistical distribution of the so called thermal breathers mainly when the equilibrium is characterized by long lived pinned excitations in that case the distribution of persistence intervals turns out to be a power law we demonstrate also that this generic behaviour has a counterpart in the power spectra where the high frequencies domains nicely collapse if properly rescaled these results are also compared to nonlinear klein gordon chains with a soft nonlinearity for which the thermal breathers are rather mobile entities finally we discuss the possibility of a breather induced anomalous diffusion law and show that despite a strong slowing down of the energy diffusion there are numerical evidences for a normal asymptotic diffusion mechanism but with exceptionally small diffusion coefficients

notions of stabilizability and controllability for a class of uncertain linear systems abstract this paper investigates the relationship between stabilizability and certain notions of controllability for a class of uncertain linear systems containing time varying unknown but bounded uncertain parameters the paper introduces a notion of controllability invariance roughly speaking an uncertain system is controllability invariant if it is controllable in the usual sense for each fixed value of the uncertain parameters the paper describes a class of uncertain systems for which controllability invariance implies stabilizability

when are the complex and the real stability radii equal characterization of those real stable matrices for which the unstructured complex and real stability radii are equal is presented special cases of this equality are discussed as application of this result it is shown that for 2 2 matrices with nonreal eigenvalues neither the case of equality nor the case of strict inequality is generic

on robust stability of 2 d discrete systems presents a study on robust stability of two dimensional 2 d discrete systems in the fornasini marchesini f m state space setting a measure of stability robustness of a stable f m model is defined the relation of this measure to its counterpart in the roessor state space and related computational issues are addressed three lower bounds of the stability robustness measure defined are derived using a one dimensional parameterization approach and a 2 d lyapunov approach a numerical example is included to illustrate the main results obtained

real stability radii of linear time invariant time delay systems in this paper the real stability radii for perturbed linear differential equations of the retarded and neutral type are investigated we present the corresponding readily computable formulae with respect to an arbitrary stability region in the complex plane the results are further extended to the case of linear discrete time delay systems

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

stability robustness of almost linear state equations sufficient conditions for stability robustness of finite dimensional autonomous systems are discussed the system comprises a stable linear nominal part and different types of unstructured norm bounded perturbations quantitative sufficient conditions for stability robustness are introduced for the case in which the perturbations are almost linear in the sense that both the size and the derivative of the perturbations are bounded these conditions describe a tradeoff between the size of the perturbations and their distance from linearity the arbitrary nonlinear case and a special linear case are shown to be limiting cases of the almost linear type as an illustration the same quantitative sufficient conditions for stability are applied to a system with slowly varying linear perturbations

a discrete analogue of lyapunov type inequalities for nonlinear difference systems in this paper we establish several new lyapunov type inequalities for the nonlinear difference system dx n email protected n x n 1 email protected n y n m 2y n dy n email protected n x n 1 n 2x n 1 email protected n y n when the end points are not necessarily usual zeros but rather generalized zeros our results improve almost all related existing ones

asymptotically efficient estimation of spectral moments the article studies parametric estimation of spectral moments of a zero mean complex gaussian stationary process immersed in independent gaussian noise with the merit of the maximum likelihood ml approach as motivation this work exploits a whittle s 1953 type objective function that is able to capture the relevant features of the log likelihood function while being much more manageable the resulting estimates are strongly consistent and asymptotically efficient as an example application to doppler weather radar data is considered

homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities abstract by using critical point theory and periodic approximations new sufficient conditions are obtained on the existence and nonexistence of homoclinic solutions for a class of discrete nonlinear periodic equations with asymptotically linear nonlinearities these results partially answer an open problem proposed by pankov 2006 2 under rather weaker conditions and greatly improve the related results before

robustness of stability of time varying linear systems abstract this paper introduces the concept of stability radius for time varying linear systems invariance properties of the stability radius are analysed for the group of bohl transformations we also explore the relationship between the stability radius the norm of a certain perturbation operator and the solvability of a nonstandard differential riccati equation as an application we construct robust lyapunov functions and show how they can be used to analyze robustness with respect to nonlinear perturbations

a pair of matrices sharing common lyapunov solutions a closer look abstract let a b be a pair of matrices with regular inertia if ha a h and hb b h are both positive definite for some hermitian matrix h then all matrices in conv a a 1 b b 1 have identical regular inertia this in turn implies that both conv a b and conv a b 1 consist of non singular matrices in general neither of the converse implications holds in this paper we seek situations where they do hold in particular when a and b are real 2 2 matrices several aspects of the above statements for n n matrices are discussed a connection to the characterization of the convex hull of matrices with regular inertia is introduced differences between the real and the complex case are indicated

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

nonlinear modes in coupled rotator models abstract we compare the properties of generalized nonlinear klein gordon equations with a sinusoidal coupling sine lattice equations to those of the usual nonlinear klein gordon equation with harmonic coupling for three equations leading to the same sine gordon equation in the continuum limit using hirota s bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations numerical simulations confirm this analogy even for very discrete cases however we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling the rotating modes they can be created thermally an approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable even in collisions with breathers possible implications for physics are discussed

demonstration of the stability or instability of multibreathers at low coupling abstract whereas there exists a mathematical proof for one site breathers stability and an unpublished one for two site breathers the methods for determining the stability properties of multibreathers rely on numerical computation of the floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear schrodinger equations here we present a set of multibreather stability theorems mst that provides a simple method to determine multibreathers stability in klein gordon systems these theorems are based in the application of degenerate perturbation theory to aubry s band theory we illustrate them with several examples

targeted energy transfer between a rotor and a morse oscillator a model for selective chemical dissociation abstract standard kramers theory of chemical reactions involves a coupling with a langevin thermal bath which intrinsically forbids the possible existence of discrete breathers dbs i e local modes however it is now known that in complex systems that energy may focus for long time as discrete breathers local mode in very special systems targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature the dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable chemical expressway a simple toy model illustrating this idea is reduced to a rotor weakly coupled to a morse oscillator supposed to represent two specific local modes in a complex system which are appropriately tuned for targeted energy transfer when the rotor is initially rotating with a frequency resonant with those of the morse oscillator at rest the energy of the rotor is almost completely transferred to the morse oscillator and induces chemical dissociation the periodic oscillations of the rotor and morse oscillator remain coherent and their frequencies simultaneously vary but always remain resonant this process is analytically described within an integrable approximation numerical investigations of this model confirm that in the appropriate conditions the particle in the morse oscillator is indeed promptly ejected at infinity with a finite velocity chemical dissociation despite some chaotic transient manifesting imperfect integrability

perturbation theory for periodic orbits in a class of infinite dimensional hamiltonian systems we consider a class of hamiltonian systems describing an infinite array of coupled anharmonic oscillators and we study the bifurcation of periodic orbits off the equilibrium point the family of orbits we construct can be parametrized by their periods which belong to cantor sets of large measure containing certain periods of the linearized problem as accumulation points the infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system we also have a result concerning the existence of solutions of arbitrarily large amplitude

discrete nonlinear schrodinger equations with superlinear nonlinearities abstract in this paper we study the discrete nonlinear equation lu n u n n g n u n n z where belongs to a finite spectral gap of the operator l or it is a lower edge of a finite spectral gap by using a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou we will study the existence of nontrivial solitons for this equation with gn is super linear

discrete breathers in vibroimpact chains analytic solutions we present exact analytic solutions for discrete breathers in essentially nonlinear oscillatory chains belonging to both of the most common universality classes klein gordon and fermi pasta ulam the exact solutions can be obtained due to use of vibroimpact potentials combining extreme nonlinearity with the possibility of description in terms of a forced linear model under conditions of self consistency a crossover between the cases of high and low energies can be studied directly the solutions obtained may be used as a high energy limit for models with other realistic potentials as well as benchmarks for the testing of approximate approaches in the theory of discrete breathers

effective hamiltonian for travelling discrete breathers hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers however solutions which look like moving discrete breathers for some time are not difficult to observe in numerics in this paper we propose an abstract framework for the description of approximate travelling discrete breathers in hamiltonian chains of oscillators the method is based on the construction of an effective hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers error estimate on the approximate dynamics is also studied the concept of the peierls nabarro barrier can be made clear in this framework we illustrate the method with two simple examples namely the salerno model which interpolates between the ablowitz ladik lattice and the discrete nonlinear schrodinger system and the fermi pasta ulam chain

supratransmission in a disordered nonlinear periodic structure abstract we study the interaction among dispersion nonlinearity and disorder effects in the context of wave transmission through a discrete periodic structure subjected to continuous harmonic excitation in its stop band we consider a damped nonlinear periodic structure of finite length with disorder disorder is introduced throughout the structure by small changes in the stiffness parameters drawn from a uniform statistical distribution dispersion effects forbid wave transmission within the stop band of the linear periodic structure however nonlinearity leads to supratransmission phenomenon by which enhanced wave transmission occurs within the stop band of the periodic structure when forced at an amplitude exceeding a certain threshold the frequency components of the transmitted waves lie within the pass band of the linear structure where disorder is known to cause anderson localization there is therefore a competition between dispersion nonlinearity and disorder in the context of supratransmission we show that supratransmission persists in the presence of disorder the influence of disorder decreases in general as the forcing frequency moves away from the pass band edge reminiscent of dispersion effects subsuming disorder effects in linear periodic structures we compute the dependence of the supratransmission force threshold on nonlinearity and strength of coupling between units we observe that nonlinear forces are confined to the driven unit for weakly coupled systems this observation together with the truncation of higher order nonlinear terms permits us to develop closed form expressions for the supratransmission force threshold in sum in the frequency range studied here disorder does not influence the supratransmission force threshold in the ensemble average sense but it does reduce the average transmitted wave energy

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

robustness of linear systems abstract in this paper we consider linear systems which are stable and examine the robustness of this property the perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability we begin with an abstract analysis of the problem but in later sections specialize to three types of systems those whose mild solutions are given in terms of 1 a strongly continuous semigroups 2 b resolvent operators integrodifferential systems 3 c evolution operators time varying systems for b a banach space l2 t0 t b denotes the space of all square integrable functions defined on t0 t with values in b for b1 b2 banach spaces t0 t denotes the space of strongly measurable functions f t0 t l b1 b2 with ess sup f b 1 t0 l b1 b2 denotes the space of strongly measurable functions f t0 l b1 b2 with f t 0 and k l1 t0

effects of using observers on stabilization of uncertain linear systems the effects of observers on robust linear feedback controllers are studied sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback the effects of the resulting high gain observers on the disturbance rejection bounds are also studied it is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection however if there is only additive plant disturbance and no measurement disturbance this degradation can be prevented

a note on some differences between real and complex stability radii abstract real and complex stability radii measure the robustness of stable linear systems under real and complex parameter perturbations respectively in this note we point out some basic differences between them in particular we investigate their different behaviour with respect to system interconnections and changes in system data

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

discrete breathers on symmetry determined invariant manifolds for scalar models on the plane square lattice a group theoretical approach for studying localized periodic and quasiperiodic vibrations in two and three dimensional lattice dynamical models is developed this approach is demonstrated for the scalar models on the plane square lattice the symmetry determined invariant manifolds admitting existence of localized vibrations are found and some types of discrete breathers are constructed on these manifolds a general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed this method allows one to decompose the corresponding system of linear differential equations with time dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

maximal stability robustness for state equations a measure for stability robustness of a linear time invariant finite dimensional system state equations is introduced an upper bound for the measure is derived using the characteristic values of the system it is shown that the set of optimal systems namely systems for which the stability robustness measure attains the bound contains the normal set which has been considered as the set of optimal robustness