an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

homoclinic orbits in the near integrable double discrete sine gordon equation we establish the existence of homoclinic orbits for the near integrable double discrete sine gordon ddsg equation under periodic boundary conditions the hyperbolic structure and homoclinic orbits are constructed through the backlund transformation and lax pair a geometric perturbation method based on mel nikov analysis is used to establish necessary criteria for the persistent of temporally homoclinic orbits for the class of ddsg equations with dissipative perturbations

complex ginzburg landau equations as perturbations of nonlinear schrodinger equations a melnikov approach abstract we study the persistence of quasiperiodic and homoclinic solutions of generalized nonlinear schrodinger equations under ginzburg landau perturbations in this paper the first of a series melnikov criteria for the persistence of quasiperiodic and homoclinic solutions are derived directly from the governing partial differential equations via an averaging technique for families of tori of quasiperiodic solutions such as rotating waves and traveling waves that arise within critical sets of linear combinations of conserved functionals we find that usually only isolated tori will satisfy these selection criteria moreover in some simple cases these criteria are sufficient to conclude that a torus persists we also demonstrate the nonpersistence of solutions that are homoclinic to rotating waves under a broad class of ginzburg landau perturbations which satisfy a convexity condition

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

global stability analysis in delayed cellular neural networks in this paper the author analyzes further problems of global stability for a class of cellular neural networks with delays by means of the lyapunov functional method inequalities a 2 b 2 g 2ab and a 3 b 3 c 3 g 3abc a b cg 0 analysis technique some stability criteria are obtained under more general conditions these criteria can be applied to design globally stable networks and thus have important significance in both theory and applications

homoclinic jumping in the perturbed nonlinear schrodinger equation we study the damped periodically forced focusing nls equation with even periodic boundary conditions we prove the existence of complicated solutions that repeatedly leave and come back to the vicinity of a quasi periodic plane wave with two time scales for pure forcing we prove the existence of a complicated self similar family of homoclinic bifurcations for mode independent damping we construct jumping transients for mode dependent damping we find generalized silnikov type solutions that connect a periodic plane wave to itself through repeated jumps we also study the breakdown of the unstable manifold of plane waves through repeated jumping our results give a direct explanation for the numerical observations of bishop et al 1999 john wiley sons inc

a melnikov method for homoclinic orbits with many pulses we present an extension of the melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near integrable systems the melnikov function in this situation is the sum of the usual melnikov functions evaluated with some appropriate phase delays we show that a nonfolding condition which involves the logarithmic derivative of the melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist

robust output feedback model predictive control of constrained linear systems this paper provides a solution to the problem of robust output feedback model predictive control of constrained linear discrete time systems in the presence of bounded state and output disturbances the proposed output feedback controller consists of a simple stable luenberger state estimator and a recently developed robustly stabilizing tube based model predictive controller the state estimation error is bounded by an invariant set the tube based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the center of which is the solution of a nominal disturbance free system and the cross section of which is also invariant satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system the complexity of the resultant controller is similar to that required for nominal model predictive control

homoclinic and heteroclinic orbits for near integrable coupled nonlinear schrodinger equations in this paper a geometrical perturbation method is employed to prove the existence of heteroclinic orbits for the kinetic system of near integrable coupled nonlinear schrodinger cnls equations furthermore we obtain the persistence of homoclinic orbits for the perturbed cnls equations with even and periodic boundary conditions

a lax pair for the two dimensional euler equation a lax pair for the 2d euler equation is found a lax pair for the 2d euler equation is found

fourier spectral approximation to long time behavior of dissipative generalized kdv burgers equations in this paper we consider a generalized kdv burgers equation with periodic initial value condition firstly a fully discrete galerkin fourier spectral approximation scheme which is a linear one is constructed next the dynamical properties of the discrete system are analyzed for the autonomous case the existence and the convergence of the global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution the stability of the discrete scheme is also proved finally the long time stability and the convergence of the discrete scheme are proved for the nonautonomous case all results in this paper are obtained without any restriction on the time step size

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

a continuation gmres method for fast computation of nonlinear receding horizon control in this paper a fast numerical algorithm for nonlinear receding horizon control is proposed the control input is updated by a differential equation to trace the solution of an associated state dependent two point boundary value problem a linear equation involved in the differential equation is solved by the generalized minimum residual method one of the krylov subspace methods with jacobians approximated by forward differences the error in the entire algorithm is analyzed and is shown to be bounded under some conditions the proposed algorithm is applied to a two link arm whose dynamics is highly nonlinear simulation results show that the proposed algorithm is faster than the conventional algorithms

multi dimensional homoclinic jumping and the discretized nls equation we consider a class of dynamical systems that arise frequently in multi mode truncations and discretizations of partial differential equations including the perturbed nls we develop a general method to detect the existence of multi pulse solutions that are doubly asymptotic to an invariant manifold with two different time scales we use our method together with some recent results of li and mclaughlin to show the existence of several families of multi pulse orbits for the ablowitz ladik discretization of the perturbed nls these orbits include n pulse heteroclinic orbits and n pulse silnikov type orbits for arbitrarily large n

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

persistence of overflowing manifolds for semiflow this paper which is a sequel to a previous one 4 by the same authors is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a banach space we consider a c1 semiflow defined on a banach space x that is it is continuous on 0 x and for each t 0 t t x x is c1 and t t t x t t s x for all t s 0 and x x a typical example is the solution operator for a differential equation in 4 we proved that a compact normally hyperbolic invariant manifold m persists under small c1 perturbations in the semiflow we also showed that in a neighborhood of m there exist a center stable manifold and a center unstable manifold that intersect in the manifold m in 4 the compactness and invariance of the manifold m were important assumptions in the present paper we study the more general case where the manifold m is overflowing negatively invariant and the semiflow crosses the boundary transversally or inflowing positively invariant and the semiflow crosses the boundary transversally we do not assume that m is compact or finite dimensional also m is not necessarily an imbedded manifold but an immersed manifold as an example a local unstable manifold of an equilibrium point is an overflowing manifold in brief our main results on the overflowing manifolds may be summarized as follows the precise statements are given in section 2 we assume that the immersed manifoldm does not twist very much locally m is covered by the image under t t of a subset a positive distance away from boundary dt t contracts along the normal direction and does so more strongly than it does along the tangential direction and dt t has a certain uniform continuity in a neighborhood of m if the c1 perturbation t t of t t is sufficiently close to t t then t t has a unique c1 immersed overflowing manifold m nearm furthermore if t t isck and a spectral gap condition holds then m isck similar results for inflowing manifolds are also obtained and given in section 7

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

normally elliptic singular perturbations and persistence of homoclinic orbits abstract we consider a dynamical system possibly infinite dimensional or non autonomous with fast and slow time scales which is oscillatory with high frequencies in the fast directions we first derive and justify the limit system of the slow variables assuming a steady state persists we construct the stable unstable center stable center unstable and center manifolds of the steady state of a size of order o 1 and give their leading order approximations finally using these tools we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems

exponential stability of delayed bi directional associative memory networks some sufficient conditions are derived for the global exponential stability in delayed bi directional associative memory bam networks by constructing suitable lyapunov functional and the inequality 2ab

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

the center manifold and bifurcations of damped and driven sine gordon breathers abstract the generic bifurcations of breathers in the damped and driven sine gordon equation are investigated both numerically and analytically the linear stability analysis and information from periodic spectral theory suggest that three modes are relevant for the system they correspond to frequency and temporal phase changes and to the flat pendulum using these modes nonautonomous amplitude equations are derived and compared with numerical simulations of the perturbed sine gordon equation

efficient active set optimization in triple mode mpc an active set algorithm tailored to quadratically constrained quadratic programming in model predictive control mpc is presented it enables efficient use of augmented ellipsoidal invariant sets in conjunction with polytopic constraints in triple mode mpg the algorithm gives improved optimality and larger stabilizable initial condition sets than conventional quadratic programming mpc algorithms of comparable online computational burden

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

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

homoclinic orbits for a perturbed nonlinear schrodinger equation in recent years there have been extensive studies on the existence of homoclinic orbits for nearly integrable hamiltonian pdes which are closely related to chaos in this work we consider a perturbed nonlinear schrodinger equation for u even and periodic in x the diffusion ieuxx is an unbounded perturbation term when the diffusion is replaced by its bounded fourier truncation li mclaughlin shatah and wiggins 26 proved the existence of homoclinic orbits for the perturbed equation the method was based on invariant manifolds foliations and melnikov analysis the unboundedness of the diffusion prevents the equation from being solved for t 0 for general initial values and destroys some geometric structures however we overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed nls 2000 john wiley sons inc

efficient non linear model based predictive control for a large number of degrees of freedom and or large dimension systems non linear model based predictive control algorithms based on dual mode control can become intractable this paper proposes an alternative which deploys the closed loop paradigm that has proved to be very effective for the case of linear time varying or uncertain systems the various attributes and computational advantages of the approach are shown to carry over to the non linear case

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

a perturbation method for spinorial yamabe type equations on s m and its application for m ge 2 we prove the existence of non trivial solutions for a certain kind of nonlinear dirac equations with critical sobolev nonlinearities on s m via a perturbative variational method for the special case m 2 this establishes the existence of a conformal immersion s 2 rightarrow mathbb r 3 with prescribed mean curvature h which is close to a positive constant under an index counting condition on h

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

global asymptotic stability of bi directional associative memory networks with distributed delays in this paper several sufficient conditions are derived for the global asymptotic stability in bi directional associative memory bam networks with distributed delays by constructing suitable lyapunov functional and the young inequality technique these conditions have an important leading significance in the design and applications of globally asymptotically stable neural circuits for bam with distributed delays

zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear schrodinger equations zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear schrodinger equations nls are studied we start with spatially uniform and temporally periodic solutions the so called stokes waves we find that the spectra of the linear nls at the stokes waves often have surprising limits as dispersion or viscosity tends to zero when dispersion or viscosity is set to zero the size of invariant manifolds and or fenichel fibers approaches zero as viscosity or dispersion tends to zero when dispersion or viscosity is nonzero the size of invariant manifolds and or fenichel fibers approaches a nonzero limit as viscosity or dispersion tends to zero when dispersion is nonzero the center stable manifold as a function of viscosity is not smooth at zero viscosity a subset of the center stable manifold is smooth at zero viscosity the unstable fenichel fiber is smooth at zero viscosity when viscosity is nonzero the stable fenichel fiber is smooth at zero dispersion

on the stability of a domain wall brane model abstract we establish stability and nonstability results for a domain wall brane model arising in classical field theory in particular we show the nonexistence of nontrivial bounded solutions on the real line for a coupled pair of parameter dependent linear second order ordinary differential equations for an open set of those parameters moreover we establish the existence of nontrivial solutions for a hypersurface of the parameters we use fredholm theory for compact linear operators combined with the lyapunov schmidt method to prove our results the model is stable respectively unstable for those parameters for which the coupled system does not respectively does have nontrivial solutions

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

on the exponential stability and periodic solutions of delayed cellular neural networks a set of criteria is presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks dcnns by constructing suitable lyapunov functionals introducing many parameters and combining with the elementary inequality technique these criteria have important leading significance in the design and applications of globally stable dcnns and periodic oscillatory dcnns in addition earlier results are extended and improved other results are contained two examples are given to illustrate the theory