continuous right inverses for the asymptotic borel map in ultraholomorphic classes via a laplace type transform abstract a new construction of linear continuous right inverses for the asymptotic borel map is provided in the framework of general carleman ultraholomorphic classes in narrow sectors such operators were already obtained by v thilliez by means of whitney extension results for non quasianalytic ultradifferentiable classes due to j chaumat and a m chollet but our approach is completely different resting on the introduction of a suitable truncated laplace type transform this technique is better suited for a generalization of these results to the several variables setting moreover it closely resembles the classical procedure in the case of gevrey classes so indicating the way for the introduction of a concept of summability which generalizes k summability theory as developed by j p ramis

dirichlet inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following complex ginzburg landau equation with cubic nonlinearity on r n for t 0 u t a i u b i u 2 u a b 0 under initial and dirichlet boundary conditions u x 0 h x for x u x t q x t on where h q are given smooth functions under suitable conditions we prove the existence of a global solution in h1 further this solution approaches to the solution of the nls limit under identical initial and boundary data as a b 0

local conditional lyapunov exponent characterization of consistency of dynamical response of the driven lorenz system local conditional lyapunov exponents are introduced and plotted in dynamical phase space to visualize the local susceptibility of the lorenz system driven by chaos and noise signals when the system produces consistent response outputs the dependence of the distribution of the local conditional lyapunov exponent on changes in the drive signal is investigated we consider the existence of multiple basins of consistent response for the same drive signal and the dependence of the basins on the drive signal

neumann inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following neumann inhomogeneous boundary value problem for the complex ginzburg landau equation on r n n 3 u t a i u b i u 2 u a b t 0 under initial condition u x 0 h x for x and neumann boundary condition u n k x t on where h k are given functions under suitable conditions we prove the existence of a global solution in h1

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

a dirichlet inhomogeneous boundary value problem for a generalized ginzburg landau equation abstract the ginzburg landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics physics and chemistry in this paper we study a generalized complex ginzburg landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives we prove a global existence and uniqueness theorem

on gevrey solutions of threefold singular nonlinear partial differential equations abstract we study gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain we deal with both fuchsian and irregular singularities and allow the presence of a singular perturbation parameter by means of the borel laplace summation method we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their gevrey asymptotic expansion in the perturbation parameter this result rests on the malgrange sibuya theorem and it requires to prove that the difference between two neighboring solutions is exponentially small what in this case involves an asymptotic estimate for a particular dirichlet like series

flat functions in carleman ultraholomorphic classes via proximate orders abstract whenever the defining sequence of a carleman ultraholomorphic class in the sense of h komatsu is strongly regular and associated with a proximate order flat functions are constructed in the class on sectors of optimal opening as consequences we obtain analogues of both borel ritt gevrey theorem and watson s lemma in this general situation

a neumann boundary value problem for a generalized ginzburg landau equation we study the following generalized 1d ginzburg landau equation on w 0 x 0 u t 1 i m u x x a 1 ib 1 u 2u x a 2 ib 2 u 2u x 1 i n u 4u with initial and neumann boundary conditions u x 0 h x u x 0 t p t under suitable conditions we prove that there is a unique weak solution that exists for all time

synchronization multistability and basin crisis in coupled pendula abstract the synchronization dynamics of two linearly coupled pendula is studied in this paper based on the lyapunov stability theory and linear matrix inequality lmi some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k th for the on set of full synchronization is obtained the numerical value of k th determined from the average energies of the systems is in good agreement with theoretical analysis prior to the on set of synchronization the boundary crisis of the chaotic attractor is identified in the bistable states where two asymmetric periodic attractors co exist it is shown that the coupled pendula can attain multistable states via a new dynamical transition the basin crisis that occur prior to the on set of stable synchronization the essential feature of basin crisis is that the two co existing attractors are destroyed while new three or more co existing attractors of the same or different periodicity are created in addition the linear perturbation technique and the routh hurwitz criteria are employed to investigate the stability of steady states and clearly identify the different types of bifurcations likely to be encountered finally two parameter phase plots show various regions of chaos hyperchaos and periodicity

finite dimensional behaviour for the derivative ginzburg landau equation in two spatial dimensions abstract in this paper we study the long time behaviour of the solutions for the generalized derivative ginzburg landau equation in two spatial dimensions we show the existence of the global attractor for this equation and establish the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor

inviscid limits of the complex ginzburg landau equation in the inviscid limit the generalized complex ginzburg landau equation reduces to the nonlinear schrodinger equation this limit is proved rigorously with h1 data in the whole space for the cauchy problem and in the torus with periodic boundary conditions the results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of l2 in the focusing case in any spatial dimension furthermore optimal convergence rates are proved the proofs are based on estimates of the schrodinger energy functional and on gagliardo nirenberg inequalities

summability in general carleman ultraholomorphic classes a definition of summability is put forward in the framework of general carleman ultraholomorphic classes in sectors so generalizing k summability theory as developed by j p ramis departing from a strongly regular sequence of positive numbers we construct an associated analytic proximate order and corresponding kernels which allow us to consider suitable laplace and borel type transforms both formal and analytic whose behavior closely resembles that of the classical ones in the gevrey case an application to the study of the summability properties of the formal solutions to some moment partial differential equations is included

transition from anticipatory to lag synchronization via complete synchronization in time delay systems the existence of anticipatory complete and lag synchronization in a single system having two different time delays that is feedback delay tau1 and coupling delay tau2 is identified the transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay tau2 with a suitable stability condition is discussed in particular it is shown that the stability condition is independent of the delay times tau1 and tau2 consequently for a fixed set of parameters all the three types of synchronizations can be realized further the emergence of exact anticipatory complete or lag synchronization from the desynchronized state via approximate synchronization when one of the system parameters b2 is varied is characterized by a minimum of the similarity function and the transition from on off intermittency via periodic structure in the laminar phase distribution

strongly regular sequences and proximate orders abstract summability methods for ultraholomorphic classes in sectors defined in terms of a strongly regular sequence m m p p n 0 have been put forward by a lastra s malek and the second author 28 we study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders in particular we give a simple condition that allows us to associate a proximate order with a strongly regular sequence under this assumption and through the characterization of strongly regular sequences in terms of so called regular variation we show that the growth index m defined by v thilliez 54 and the order of quasianalyticity m introduced by the second author 50 are the same

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

letter to the editor adaptive control for modified projective synchronization of a four dimensional chaotic system with uncertain parameters this article is concerned with the modified projective synchronization problem for a class of four dimensional chaotic system with uncertain parameters by utilizing lyapunov method an adaptive control scheme for the synchronization has been presented the control performances are verified by a numerical simulation

on the initial value problem for the generalized two dimensional ginzburg landau equation abstract the ginzburg landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics physics and chemistry most work so far concentrates on ginzburg landau type equations with one spatial dimension 1d in this paper the authors study a complex generalized ginzburg landau equation with two spatial dimensions 2d sufficient conditions for the existence and uniqueness of global solutions for the initial value problem of the generalized 2d ginzburg landau equation are obtained this rigorously establishes the foundation for further investigation of this type of model

an approach for characterizing coupling in dynamical systems abstract the study of coupling in dynamical systems dates back to christian hyugens who in 1665 discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other in that case the observed synchronous motion was out of phase in this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems the measure is based on quantifying the active degrees of freedom e g correlation dimension of the coupled system and the constituent subsystems the time delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system we use the coupled lorenz rossler and henon model systems with a coupling strength variable for evaluation of the proposed approach results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied furthermore using this approach the direction of coupling can be determined

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

zero delay synchronization of chaos in coupled map lattices we show that two coupled map lattices that are mutually coupled to one another with a delay can display zero delay synchronization if they are driven by a third coupled map lattice we analytically estimate the parametric regimes that lead to synchronization and show that the presence of mutual delays enhances synchronization to some extent the zero delay or isochronal synchronization is reasonably robust against mismatches in the internal parameters of the coupled map lattices and we analytically estimate the synchronization error bounds

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

injectivity and surjectivity of the asymptotic borel map in carleman ultraholomorphic classes abstract we consider roumieu carleman ultraholomorphic classes and classes of functions admitting asymptotic expansion in unbounded sectors defined in terms of a log convex sequence m departing from previous results by s mandelbrojt and b rodriguez salinas we completely characterize the injectivity of the borel map by means of the theory of proximate orders a growth index m turns out to put apart the values of the opening of the sector for which injectivity holds or not in the case of surjectivity we considerably extend partial results by j schmets and m valdivia and by v thilliez and prove a similar dividing character for the index m introduced by thilliez and generally different from m in some standard situations for example as far as m is strongly regular

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

multisummability in carleman ultraholomorphic classes by means of nonzero proximate orders abstract we introduce a general multisummability theory of formal power series in carleman ultraholomorphic classes the finitely many levels of summation are determined by pairwise comparable nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct thus we extend the powerful multisummability theory for finitely many gevrey levels developed by j p ramis j ecalle and w balser among others we provide both the analytical and cohomological approaches and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized laplace like operators

generalized synchronization of chaos in electronic circuit experiments abstract two examples of one way coupled electronic circuits displaying generalized synchronization of chaos are considered in one of them the dynamics of both the response and the driving systems represent a double scroll chaos oscillator in another example the double scroll oscillator is driven by an electronic analog of the mackey glass system to detect the generalized synchronization an auxiliary response system that is a replica of the original one is used in these systems we have discovered two types of generalized synchronization namely a strong and a weak synchronization which correspond to the existence of a smooth and a nonsmooth map from the trajectories of the driving attractor to the trajectories of the response system respectively

partial synchronization of chaotic systems with uncertainty we suggest an approach to partial synchronization of chaotic systems with uncertainty this method contains two steps i transforming the synchronization system into the canonical form by the well known feedback linearization theory and ii finding a control signal to ensure the asymptotic stability of the canonical system this partial synchronization approach requires very little system information by applying a finite time convergence technique to estimate uncertainties caused by unknown states parameters or structure we also argue in detail that this partial synchronization method can be extended to parameter identification sub structure estimation and even phase detection several examples are presented to illustrate the partial synchronization approach suggested

multisummability of formal solutions of some linear partial differential equations abstract let u t x n 1 un x tn be a formal power series in one variable t satisfying formally p t x t x u t x f t x t x c c d the purpose of the present paper is to introduce a class of linear partial differential operators and show that u t x is multisummable provided p t x t x belongs to this class

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

global existence of solutions to the derivative 2d ginzburg landau equation abstract in this paper we study a complex derivative ginzburg landau equation with two spatial variables 2d we obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2d ginzburg landau equation and improve the known results

multisummability of formal power series solutions of partial differential equations with constant coefficients we study gevrey properties and summability of power series in two variables that are formal solutions of a cauchy problem for general linear partial differential equations with constant coefficients in doing so we extend earlier results in two articles of balser and lutz miyake and schafke for the complex heat equation as well as in a paper of balser and miyake who have investigated the same questions for a certain class of linear pde with constant coefficients subject to some restrictive assumptions moreover we also present an example of a pde where the formal solution of the cauchy problem is not k summable for whatever value of k but instead is multisummable with two levels under corresponding conditions upon the cauchy data that this can occur has not been observed up to now

monomial summability and doubly singular differential equations abstract in this work we consider systems of differential equations that are doubly singular i e that are both singularly perturbed and exhibit an irregular singular point if the irregular singular point is at the origin they have the form e x r 1 d y d x f x e y f 0 0 0 0 with f analytic in some neighborhood of 0 0 0 if the jacobian d f d y 0 0 0 is invertible we show that the unique bivariate formal solution is monomially summable i e summable with respect to the monomial t e x r in a new sense that will be defined as a preparation poincare asymptotics and gevrey asymptotics in a monomial are studied

log convex sequences and nonzero proximate orders abstract summability methods for ultraholomorphic classes in sectors defined in terms of a strongly regular sequence m m p p n 0 have been put forward by a lastra s malek and the second author 10 and their validity depends on the possibility of associating to m a nonzero proximate order we provide several characterizations of this and other related properties in which the concept of regular variation for functions and sequences plays a prominent role in particular we show how to construct well behaved strongly regular sequences from nonzero proximate orders

route to hyperchaos in a system of coupled oscillators with multistability this work presents the results of a detailed experimental study into the transition between synchronized low dimensional and unsynchronized high dimensional dynamics using a system of coupled electronic chaotic oscillators novel data analysis techniques have been employed to reveal that a hyperchaotic attractor can arise from the amalgamation of two nonattracting sets these originate from initially multistable low dimensional attractors which experience a smooth transition from low to high dimensional chaotic behavior losing stability through a bubbling bifurcation numerical techniques were also employed to verify and expand on the experimental results giving evidence on the locally unstable invariant sets contained within the globally stable hyperchaotic attractor this particular route to hyperchaos also results in the possibility of phenomena such as unstable dimension variability that can be a major obstruction to shadowing and predictability in chaotic systems

paucity of attractors in nonlinear systems driven with complex signals we study the probability of multistability in a quadratic map driven repeatedly by a random signal of length n where n is taken as a measure of the signal complexity we first establish analytically that the number of coexisting attractors is bounded above by n we then numerically estimate the probability p of a randomly chosen signal resulting in a multistable response as a function of n interestingly with increasing drive signal complexity the system exhibits a paucity of attractors that is almost any drive signal beyond a certain complexity level will result in a single attractor response p 0 this mechanism may play a role in allowing sensitive multistable systems to respond consistently to external influences

detecting generalized synchronization between chaotic signals a kernel based approach a unified framework for analysing generalized synchronization in coupled chaotic systems from data is proposed the key of the proposed approach is the use of the kernel methods recently developed in the field of machine learning several successful applications are presented which show the capability of the kernel based approach for detecting generalized synchronization and dynamical change of the coupling strength between two chaotic systems can be captured by the proposed approach it is also discussed how the kernel parameter is suitably chosen from data

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

the ginzburg landau equation posed in a quarter plane abstract we study the following ginzburg landau equation gl ut i uxx i u 2u u 0 0 0 for a full line problem with u x 0 u0 x h2 global existence uniqueness is established for a half line problem with u x 0 u0 x h2 0 u 0 t q t c2 0 u0 0 q 0 the following results are available 1 local existence uniqueness 2 criteria for the existence of a small amplitude solution on any finite interval by means of small initial and boundary data 3 global existence in the case 3 or 0

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

multisummability of formal solutions of singular perturbation problems abstract we investigate the summability of the unique formal power series solution of a singular perturbation problem for certain linear systems of ordinary meromorphic differential equations using a new type of summability of power series in several here two variables