synchronization time in a hyperbolic dynamical system with long range interactions abstract we show that the threshold of complete synchronization in a lattice of coupled non smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace we examine carefully the synchronization time and show that an inadequate observation of the system evolution leads to wrong results we present both careful numerical experiments and a rigorous mathematical explanation confirming this fact allowing for a generalization involving hyperbolic coupled map lattices

bubbling bifurcation loss of synchronization and shadowing breakdown in complex systems abstract complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high dimensional chaos this paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly non hyperbolic behavior the system undergoes on off intermittency with respect to the synchronization manifold this has been confirmed by numerical diagnostics of synchronization and non hyperbolic behavior the latter using the statistical properties of finite time lyapunov exponents

exponential stabilisation of a simple tree shaped network of timoshenko beams system the stabilisation problem of a forked triple system of timoshenko beams which is a simple tree shaped network is considered in this article suppose that the root of the fork is clamped at the interior node of the fork the displacement and rotational angle of the beams are continuous the shearing force and bending moments of the beams satisfy the transmission conditions the feedback controllers are designed at the exterior tips to stabilise this system then the closed loop system is proved to be well posed by the eigenfrequency analysis together with the asymptotic technique the system is shown to satisfy the spectrum determined growth condition finally this closed loop system is proved to be exponentially stable furthermore a simulation is given to support these results

a sufficient condition on riesz basis with parentheses of non self adjoint operator and application to a serially connected string system under joint feedbacks this paper gives an abstract sufficient condition on riesz basis with parentheses property for the generators of c0 groups in hilbert spaces whose eigenvalues are comprised of some finite unification of separable sets after taking the algebraic multiplicities into account the condition is then applied to the closed loop system of a serially connected string system under joint damping feedbacks to show that there is a family of generalized eigenfunctions that form a riesz basis with parentheses in the state space the spectrum determined growth condition is concluded as a consequence

the expansion of a semigroup and a riesz basis criterion problems on the expansion of a semigroup and a criterion for being a riesz basis are discussed in the present paper suppose that a is the generator of a c0 semigroup on a hilbert space and a 1 a 2 a with 2 a is consisted of isolated eigenvalues distributed in a vertical strip it is proved that if 2 a is separated and for each 2 a the dimension of its root subspace is uniformly bounded then the generalized eigenvectors associated with 2 a form an l basis under different conditions on the riesz projection the expansion of a semigroup is studied in particular a simple criterion for the generalized eigenvectors forming a riesz basis is given as an application a heat exchanger problem with boundary feedback is investigated it is proved that the heat exchanger system is a riesz system in a suitable state hilbert space

boundary control of the timoshenko beam it is shown that the timoshenko beam can be uniformly stabilized by means of a boundary control a numerical study on the spectrum is also presented

riddling and chaotic synchronization of coupled piecewise linear lorenz maps abstract we investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise linear lorenz maps riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale which has serious consequences on the predictability of the final state for such a coupled system we found that there are wide parameter intervals for which two piecewise linear lorenz maps exhibit riddled basins globally or locally which indicates that there are riddled basins in coupled lorenz equations as previously suggested by numerical experiments the use of piecewise linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins

predicting the synchronization time in coupled map networks an analytical expression for the synchronization time in coupled map networks is given by means of the expression the synchronization time for any given network can be predicted accurately furthermore for networks in which the distributions of nontrivial eigenvalues of coupling matrices have some unique characteristics analytical results for the minimal synchronization time are given

phase integral approximation for coupled ordinary differential equations of the schrodinger type four generalizations of the phase integral approximation pia to sets of ordinary differential equations of schrodinger type uj x k 1nrjk x uk x 0 j 1 2 n are described the recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix r x rjk x either hermitian or non hermitian for hermitian and negative definite r x matrices a wronskian conserving pia theory is formulated which generalizes fulling s current conserving theory pertinent to positive definite r x matrices the idea of a modification of the pia which is well known for one equation u x r x u x 0 is generalized to sets a simplification of wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections if the pia is generated by a nondegenerate eigenvalue of the r x matrix the eliminated integration is the only one present in that case the simplified theory becomes fully algorit

riesz basis property of serially connected timoshenko beams in this paper we study the riesz basis property of serially connected timoshenko beams with joint and boundary feedback controls suppose that the left end of the whole beam is clamped and the right end is free at intermediate nodes the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous the collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system we prove that the operator determined by the closed loop system has compact resolvent and generates a c 0 semigroup in an appropriate hilbert space we also show that there is a sequence of the generalized eigenvectors of the operator that forms a riesz basis with parentheses hence the spectrum determined growth condition holds therefore if the imaginary axis is not an asymptote of the spectrum then the closed loop system is exponentially stable finally we give a conclusion remark to explain that our result can be applied not

chaos synchronization in a lattice of piecewise linear maps with regular and random couplings abstract we investigate aspects of the spatio temporal dynamics exhibited by a one dimensional lattice of chaotic piecewise linear maps in a coupling prescription which includes both regular nearest and next to nearest neighbors and randomly chosen couplings we discuss the conditions for the existence of chaotic synchronized states and relate them to the coupling parameters the transition to synchronized behavior is described and we explore some statistical properties of the time it takes to achieve this regime

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

spatial recurrence plots we propose an extension of the recurrence plot concept to perform quantitative analyzes of roughness and disorder of spatial patterns at a fixed time we introduce spatial recurrence plots srps as a graphical representation of the pointwise correlation matrix in terms of a two dimensional spatial return plot this technique is applied to the study of complex patterns generated by coupled map lattices which are characterized by measures of complexity based on srps we show that the complexity measures we propose for srps provide a systematic way of investigating the distribution of spatially coherent structures such as synchronization domains in lattice profiles this approach has potential for many more applications e g in surface roughness analyzes

exponential stability analysis of xa long chain of coupled vibrating strings with dissipative linkage consider a long chain of coupled vibrating strings where a stabilizer is installed at each internal node and perhaps also at a boundary point the exponential stability of the stabilizers arrangement for this large dynamic structure will be determined through a careful transformation of the coupled wave equations for this structure into an equivalent hyperbolic system and analysis of the eigendeterminant it will be proven that the energy of the system decays uniformly exponentially if there is a stabilizer installed at a boundary point if the stabilizers are installed only at internal nodes it will be proven that the energy may decay either uniformly exponentially or nonuniformly or may not decay at all depending on the different wave speeds and the stabilizers arrangement all possible outcomes have been classified

transmission through cutoffs and resonances in the double phase integral approximation using the double phase integral approximation technique developed earlier for the wave equation d2 dz2 q2 z 0 we derive analytical formulas for the reflection r transmission t and absorption a coefficients they are valid to arbitrary order in the expansion parameter for functions q2 z having either two cutoffs or one cutoff and one resonance for two examples of this type the formulas for r t and a are checked against numerical results using approximations up to fifth order

tunneling in one dimensional ideal barriers general properties of the transmission coefficient of an ideal one dimensional potential barrier of arbitrary shape are studied it is proved that an arbitrary symmetric barrier is perfectly transparent for at least one energy in each energy band of the related band problem where the barrier potential is periodically continued on the whole real axis recursion relations are obtained for transmission coefficients of barriers consisting of 2k structural units they are used in a simple proof showing that transmission coefficients of finite barriers composed of m identical arbitrary structural units have chaotic behavior for almost all energies for m in each energy band there exists however becoming more dense with m a countable set of energies in each energy band where finite repeated barriers are perfectly transparent the results are illustrated by a numerical example

effects of microscopic disorder on the collective dynamics of globally coupled maps this paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps our analysis is complementary to the approach taken by teramae and kuramoto j teramae y kuramoto strong desynchronizing effects of weak noise in globally coupled systems phys rev e 63 2001 036210 who pointed out the anomalous scaling properties preceding the loss of coherence we focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one using properly defined order parameters an analytical approach is proposed for describing the collective dynamics in terms of an approximate low dimensional system the systematic derivation of the macroscopic equations provides a link between the microscopic features of the population single element dynamics and noise distribution and the properties of the emergent behaviour the macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population

validity of numerical trajectories in the synchronization transition of complex systems we investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly nonhyperbolic behavior the system undergoes on off intermittency with respect to the synchronization state there are potentially severe consequences of these facts on the validity of the computer generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties

statistics of finite time lyapunov exponents in the ulam map the statistical properties of finite time lyapunov exponents at the ulam point of the logistic map are investigated the exact analytical expression for the autocorrelation function of one step lyapunov exponents is obtained allowing the calculation of the variance of exponents computed over time intervals of length n the variance anomalously decays as 1 n 2 the probability density of finite time exponents noticeably deviates from the gaussian shape decaying with exponential tails and presenting 2 n 1 spikes that narrow and accumulate close to the mean value with increasing n the asymptotic expression for this probability distribution function is derived it provides an adequate smooth approximation to describe numerical histograms built for not too small n where the finiteness of bin size trims the sharp peaks

spectrum of a network of euler bernoulli beams abstract a network of n flexible beams connected by n vibrating point masses is considered the spectrum of the spatial operator involved in this evolution problem is studied if 2 is any real number outside a discrete set of values s and if is an eigenvalue then it satisfies a characteristic equation which is given the associated eigenvectors are also characterized if 2 lies in s and if the n beams are identical same mechanical properties another characteristic equation is available it is not the case for different beams no general result can be stated some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization at last the asymptotic behaviour of the eigenvalues is investigated by proving the so called weyl s formula

self organized memories in coupled map lattices abstract spatially extended dynamical systems subjected to a periodic external input are able to store short term memories as a result of self organization this phenomenon has been used to interpret sliding charge density wave experiments and has been found in lattices of coupled maps and oscillators in this paper we aim to describe the influence of a stochastic perturbation in the self organized memories in a lattice of coupled linear and weakly nonlinear maps we consider the case in which the coupling between maps depends on the lattice distance as a power law

output feedback stabilisation of a tree shaped network of vibrating strings with non collocated observation in this article the authors consider the simple tree shaped network of strings with non collocated observation suppose that the network consists of three strings and the displacements of these strings are continuous at the common vertex the sensors and actuators are required to be set at different vertices we propose an observer based output feedback controller to stabilise this network the exponential stability of the closed loop network is obtained based on the riesz basis approach together with estimates of the spectral distribution finally a numerical simulation is given to support these results

synchronization threshold in coupled logistic map lattices abstract we consider regular lattices of coupled chaotic maps whose local dynamics is ruled by the logistic map x 4 x 1 x a usual paradigm of chaotic systems through finite time exponents we scrutinize the lattice dynamics in the vicinity of the threshold of complete synchronization we connect dynamical features such as relaxation to the coherent state and intermittency with the statistics of finite time exponents focusing on the implications of the particular statistics related to the logistic map although numerical examples are given for lattice couplings decaying with distance as a power law our results are expected to be valid for a wider class of schemes coupling logistic maps

efficient integration of the one dimensional time independent wave equation for bound states and for wave propagation the wave equation is solved with the phase integral approximation in the asymptotic regions and integrated numerically elsewhere in this application the phase integral approximation is relatively simple in higher orders and its accuracy can be precisely controlled with the derived error estimates for the bound states the analytical formulas defining the termination error in the wave function are derived for the wave propagation an efficient formalism is given for calculating the reflection transmission and absorption coefficients

collective behavior in coupled chaotic map lattices with random perturbations abstract numerical simulations of coupled map lattices with non local interactions i e the coupling of a given map occurs with all lattice sites often involve a large computer time if the lattice size is too large in order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution this emulates a thermal bath where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude we demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied we considered one dimensional lattices of chaotic logistic maps at outer crisis x 4 x 1 x

stabilization of causally and non causally coupled map lattices abstract two dimensional coupled map lattices have global stability properties that depend on the coupling between individual maps and their neighborhood the action of the neighborhood on individual maps can be implemented in terms of causal coupling to spatially distant past states or non causal coupling to spatially distant simultaneous states in this contribution we show that globally stable behavior of coupled map lattices is facilitated by causal coupling thus indicating a surprising relationship between stability and causality the influence of causal versus non causal coupling for synchronous and asynchronous updating as a function of coupling strength and for different neighborhoods is analyzed in detail

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

periodic orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system chaotic dynamical systems with two or more attractors lying on invariant subspaces may provided certain mathematical conditions are fulfilled exhibit intermingled basins of attraction each basin is riddled with holes belonging to basins of the other attractors in order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest two species competition with extinction we have characterized quantitatively the intermingled basins using periodic orbit theory and scaling laws the latter results agree with a theoretical prediction from a stochastic model and also with an exact result for the scaling exponent we derived for the specific class of models investigated we discuss the consequences of the scaling laws in terms of the predictability of a final state extinction of either species in an ecological experiment

detailed behavior of the phase integral approximations at zeros and singularities of the square of the base function approximate solutions to the one dimensional time independent wave equation called the phase integral approximations are analyzed in the vicinity of characteristic points the approximations are of arbitrary order and are generated from an unspecified base function the general theory is illustrated by examples involving the power and or the exponential behavior of the square of the base function in these cases simple estimates are derived for the integrals which define the accuracy of the phase integral approximation and the optimum approximation order is determined

abstract second order hyperbolic system and applications to controlled network of strings in this paper we study an abstract second order hyperbolic system valued in mathbb c n with appropriate boundary conditions we prove that the system is well posed and associates with a c 0 semigroup in a hilbert state space under certain conditions we show that the spectra of the system operator are located in the vertical strip and that there is a sequence of eigenvectors and generalized eigenvectors that forms a riesz basis with parentheses for the hilbert state space and hence that the system satisfies the spectrum determined growth assumption as applications we investigate the exponential stability of a controlled tree shaped network of 7 strings and a network of n connected strings

analytical results for coupled map lattices with long range interactions we obtain exact analytical results for lattices of maps with couplings that decay with distance as r ensuremath ensuremath alpha we analyze the effect of the coupling range on the system dynamics through the lyapunov spectrum for lattices whose elements are piecewise linear maps we get an algebraic expression for the lyapunov spectrum when the local dynamics is given by a nonlinear map the lyapunov spectrum for a completely synchronized state is analytically obtained the critical line characterizing the synchronization transition is determined from the expression for the largest transversal lyapunov exponent in particular it is shown that in the thermodynamical limit such transition is only possible for sufficiently long range interactions namely for ensuremath alpha l ensuremath alpha c d where d is the lattice dimension

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

boundary feedback exponential stabilization of a timoshenko beam with both ends free in the present paper we consider the boundary feedback stabilization of a timoshenko beam with both ends free we propose boundary feedback control law that makes the closed loop system dissipative using asymptotic analysis techniques we give explicit asymptotic formula of eigenvalues of the closed loop system and prove the riesz basis property of eigenvectors and generalized eigenvectors by a detailed analysis of spectrum of the closed loop system we show that the closed system is exponentially stable

modeling stabilization and control of serially connected beams many flexible structures consist of a large number of components coupled end to end in the form of a chain in this paper we consider the simplest type of such structures which is formed by n seri

riddling bifurcations in coupled piecewise linear maps abstract a mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described we give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold it is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation