a kam theorem for one dimensional schrodinger equation with periodic boundary conditions abstract in this paper one dimensional 1 d nonlinear schrodinger equation iu t u xx mu g u u u 0 with periodic boundary conditions is considered m 1 12 z is a real parameter and the nonlinearity g u u j l j l 4 a jl u j u l a jl a lj r a 22 0 is a real analytic function in a neighborhood of the origin the kam machinery is adapted to fit the above equation so as to construct small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

completely degenerate lower dimensional invariant tori for hamiltonian system abstract we study the persistence of lower dimensional invariant tori for a nearly integrable completely degenerate hamiltonian system it is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness

degenerate lower dimensional tori in reversible systems abstract in this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems the normal variable is two dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

construction of response solutions for two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point under small perturbations abstract by developing two kam theorems in this paper we show that two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point a reversible system and a non conservative system admit a response solution under small perturbations for the degenerate reversible system applying special structure of unperturbed nonlinear term and herman method we successfully control the shift of equilibrium point which is difficult in view of the degenerate linear term for the degenerate non conservative system kam method is brought into force even in completely degenerate case because of the restrictions on the smallness and average of perturbation moreover arithmetic condition on the frequency is assumed to satisfy the brjuno russmann s non resonant condition by the poschel russmann kam method we prove that these two kinds of perturbed systems can be reduced to a suitable normal form with zero as equilibrium point by a quasi periodic transformation

existence of quasi periodic solutions of fast excited van der pol mathieu duffing equation the van der pol mathieu duffing equation x 02 h1cos 1t h2cos 2t x x2 x h3x3 h4 32cosxcos 3t is considered in this paper where h1 h2 h3 h4 1 2 are small parameters 0 the frequency 3 is large compared to 1 and 2 the above parameters are real for 0 we use kam kolmogorov arnold moser theory to prove that the van der pol mathieu duffing equation possesses quasi periodic solutions for most of the parameters 0 1 2 3 it verifies some phenomenon of fahsi and belhaq commun nonlinear sci 14 244 253 2009 and can be regarded as a extension of abouhazim et al nonlinear dyn 39 395 409 2005

umbilical torus bifurcations in hamiltonian systems we consider perturbations of integrable hamiltonian systems in the neighbourhood of normally umbilic invariant tori these lower dimensional tori do not satisfy the usual non degeneracy conditions that would yield persistence by an adaption of kam theory and there are indeed regions in parameter space with no surviving torus we assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a generalised umbilical catastrophe combining techniques of kam theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large cantor sets applications to gyrostat dynamics are pointed out

renormalization group and the melnikov problem for pde s we give a new proof of persistence of quasi periodic low dimensional elliptic tori in infinite dimensional systems the proof is based on a renormalization group iteration that was developed recently in bgk to address the standard kam problem namely persistence of invariant tori of maximal dimension in finite dimensional near integrable systems our result covers situations in which the so called normal frequencies are multiple in particular it provides a new proof of the existence of small amplitude quasi periodic solutions of nonlinear wave equations with periodic boundary conditions

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

stability of the solution of backward heat equation on a weak compactum the paper investigates the stability of the cauchy problem for the backward heat equation under the a priori assumption that the solution is bounded a special metrization of the weak topology in the space l 2 and the standard fourier series technique are applied to obtain stability bounds for the solution of the cauchy problem on the class of absolutely bounded functions

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

on the persistence of elliptic lower dimensional tori in hamiltonian systems under the first melnikov condition and russmann s non degeneracy condition abstract in this paper we formulate a theorem on the persistence of elliptic lower dimensional invariant tori for nearly integrable analytic hamiltonian systems under the first melnikov condition and russmann s non degeneracy condition and give the measure estimates of parameters for the non resonance conditions under russmann s non degeneracy condition which is essential for the proof of our result

elliptic type degenerate invariant tori for quasi periodically forced four dimensional non conservative systems abstract in this paper we will focus on the persistence of elliptic type degenerate invariant tori with the same frequency vector r d as the forcing in a class of quasi periodically forced four dimensional non conservative systems it is shown that under suitable hypothesis of smoothness with respect to and the brjuno russmann s non resonant condition instead of ordinary diophantine condition with respect to the frequency vector r d the invariant tori persist under small perturbations for most of the sufficiently small parameters in the sense of the lebesgue measure hence the system has a quasi periodic solution with the same frequency vector as the forcing the proof is based on the poschel russmann kam method which is a kam type method in which one uses the polynomial structure of function to truncate introduce a parameter q and make the steps of kam iteration infinitely small in the speed of function q n 0 q 1 rather than super exponential function like traditional kam method

lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms abstract for any analytic quasiperiodically forced circle diffeomorphisms p q e f where f is fixed and e is small we show that if is diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of e 0 i e there is a unilateral phase locking at e 0 then the diffeomorphism has at least one analytic q invariant torus provided e is small enough

invariant tori of full dimension for a nonlinear schrodinger equation abstract in this paper we consider the one dimensional nonlinear schrodinger equation i u t u x x m u f u 2 u 0 with periodic boundary conditions or dirichlet boundary conditions where f is a real analytic function in some neighborhood of the origin satisfying f 0 0 f 0 0 we prove that for each given constant potential m when the frequencies as a function of the amplitudes can be regarded as the independent parameters the equation admits a whitney smooth family of small amplitude time almost periodic solutions with all frequencies the proof is based on a birkhoff normal form reduction and an improved version of the kam theorem

degeneracy of the action frequency map a mechanism for homoclinic bifurcation of invariant tori a family of smooth invariant tori of a hamiltonian system can be parameterized by the values of the actions or the frequencies these parameterizations are related by the action frequency map the purpose of this paper is to show that when the action frequency map is degenerate it signals a homoclinic bifurcation remarkably the nonlinear properties of this homoclinic bifurcation to invariant tori are determined by the curvature of the action frequency map a homoclinic angle is also generated which is analogous to a hannay berry phase shift the theory is constructive and so can usefully be combined with computation some implications for quantization and the generation of solitary waves are also discussed

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

on small perturbation of two dimensional quasi periodic systems with hyperbolic type degenerate equilibrium point abstract in this paper we consider small quasi periodic perturbation of two dimensional nonlinear quasi periodic system with hyperbolic type degenerate equilibrium point by kam method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi periodic transformation hence the perturbed system has a quasi periodic solution near the equilibrium point

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

quasi periodic solutions in a nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 4 u 0 with the periodic boundary condition is considered it is proved that for each given constant potential m and each prescribed integer n 1 the equation admits a whitney smooth family of small amplitude time quasi periodic solutions with n diophantine frequencies the proof is based on a partial birkhoff normal form reduction and an improved kam method

a renormalization approach to lower dimensional tori with brjuno frequency vectors abstract we extend the renormalization scheme for vector fields on t d r m in order to construct lower dimensional invariant tori with brjuno frequency vectors for near integrable hamiltonian flows for every brjuno frequency vector r d and every vector r d satisfying a diophantine condition with respect to there exists an analytic manifold w of infinitely renormalizable hamiltonian vector fields each vector field on w is shown to have an analytic invariant torus with frequency vector

the hausdorff dimension of small divisors for lower dimensional kam tori given a bounded domain note see the image of page 359 for this formatted text omega subset r m and a lipschitz map note see the image of page 359 for this formatted text phi omega mapsto r n we determine the hausdorff dimension of sets of points omega in omega for which the inequality k cdot omega l cdot phi omega psi k l has infinitely many distinct integer solutions note see the image of page 359 for this formatted text k l in z m times z n satisfying l leq h where h is a fixed integer these sets interpolate between the cases h 0 and h infty which occur in the metric theory of diophantine approximation of independent and dependent quantities respectively they arise for example in the perturbation theories of lower dimensional tori in nearly integrable hamiltonian systems kam theory among others it turns out that their hausdorff dimension is independent of h and n it only depends on m and the lower order of psi at infinity part of this result even extends to the case n infty of infinite co dimension which is relevant in the kam theory of certain nonlinear partial differential equations

persistence of lagrange invariant tori at tangent degeneracy abstract in the present paper without russmann s nondegenderate condition we show the persistence of kam invariant tori using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three body problems especially for lagrange equilibrium position l2 with certain degeneracy

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

a modified quasi boundary value method for ill posed problems in this paper we study a final value problem for first order abstract differential equation with positive self adjoint unbounded operator coefficient this problem is ill posed perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter we show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution we also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions finally we give explicit convergence rates

persistence of lower dimensional invariant tori on sub manifolds in hamiltonian systems chow et al j non sci 12 2002 585 proved that the majority of the unperturbed tori on sub manifolds will persist for standard hamiltonian systems motivated by their work in this paper we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub manifolds for real analytic nearly integrable hamiltonian systems the surviving tori might be elliptic hyperbolic or of mixed type

persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable hamiltonian systems n n m m where the hamiltonian h s n q p x y u g t r r r n 1 f m n q where t is the usual n torus n s v y q 1f jf n j j m 2 2 1r2 v v u q is a normal form and p s p x y u e is a js1 j j j small perturbation term where e is a small perturbation parameter the n frequency vectors v s v v v g o r are also regarded as 1 2 n parameters where o is a bounded closed set of r with nonempty interior in general the normal frequency vector v v s v v v v 1 2 v v depends on v m if p s 0 system 1 1 is reduced to

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

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

a quasi periodic poincare s theorem we study the persistence of invariant tori on resonant surfaces of a nearly integrable hamiltonian system under the usual kolmogorov non degenerate condition by introducing a quasi linear iterative scheme to deal with small divisors we generalize the poincare theorem on the maximal resonance case i e the periodic case to the general resonance case i e the quasi periodic case by showing the persistence of majority of invariant tori associated to non degenerate relative equilibria on any resonant surface

observability with random observations recovery of the initial state of a high dimensional system can require a large number of measurements in this paper we explain how this burden can be significantly reduced when randomized measurement operators are employed our work builds upon recent results from compressive sensing cs in particular we make the connection to cs analysis for random block diagonal matrices by deriving concentration of measure com inequalities we show that the observability matrix satisfies the restricted isometry property rip a sufficient condition for stable recovery of sparse vectors under certain conditions on the state transition matrix for example we show that if the state transition matrix is unitary and if independent randomly populated measurement matrices are employed then it is possible to uniquely recover a sparse high dimensional initial state when the total number of measurements scales linearly in the sparsity level the number of non zero entries of the initial state and logarithmically in the state dimension we further extend our rip analysis for scaled unitary and symmetric state transition matrices we support our analysis with a case study of a diffusion process

persistence of invariant torus in hamiltonian systems with two degree of freedom in this paper with the kam iteration we prove a kam theorem for nearly integrable hamiltonian systems with two degrees of freedom without any non degeneracy condition

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

fractional lindstedt series the parametric equations of the surfaces on which highly resonant quasiperiodic motions develop lower dimensional tori cannot be analytically continued in general in the perturbation parameter e i e they are not analytic functions of e however rather generally quasiperiodic motions whose frequencies satisfy only one rational relation resonances of order 1 admit formal perturbation expansions in terms of a fractional power of e depending on the degeneration of the resonance we find conditions for this to happen and in such a case we prove that the formal expansion is convergent after suitable resummation

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

quasiperiodic solutions for the cubic complex ginzburg landau equation the existence of two dimensional kolmogorov arnold moser invariant tori is proved for cubic complex ginzburg landau equation of higher spatial dimension as a consequence the equation possesses a cantorian branch of nontraveling wave quasiperiodic solutions of two dimensional frequency vector

quasi periodic breathers in hamiltonian networks of long range coupling this work is concerned with hamiltonian networks of weakly and long range coupled oscillators with either variable or constant on site frequencies we derive an infinite dimensional kam like theorem by which we establish that given any n sites of the lattice there is a positive measure set of small amplitude quasi periodic breathers solutions of the hamiltonian network that are quasi periodic in time and exponentially localized in space having n frequencies which are only slightly deformed from the on site frequencies

identification of the initial condition in backward problem with nonlinear diffusion and reaction abstract in this paper we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient the problem is ill posed in the sense of hadamard under an a priori assumption on the exact solution in gevrey space we propose a new regularization method for stabilising the ill posed problem our results extend some earlier works on backward problems for nonlinear parabolic equations

persistence of lower dimensional hyperbolic tori for reversible system abstract in this paper we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system and prove that if the frequency mapping has nonzero topological degree at some diophantine frequency then the hyperbolic lower invariant dimensional torus with this frequency persists under small perturbations

measurement bounds for observability of linear dynamical systems under sparsity constraints in this paper we address the problem of observability of a linear dynamical system from compressive measurements and the knowledge of its external inputs observability of a high dimensional system state in general requires a correspondingly large number of measurements we show that if the initial state vector admits a sparse representation the number of measurements can be significantly reduced by using random projections for obtaining the measurements our analysis gives sufficient conditions for the restricted isometry property of the observability matrix to hold which leads to guarantees for the observability of the system our results depend only on the properties of system transfer and observation matrices and are derived using tools from probability theory and compressed sensing unlike the prior work in this direction our results are applicable to systems with an arbitrary nonzero system transfer matrix moreover our results are stronger than the existing results in the regime where they are comparable

on quasi periodic solutions for a generalized boussinesq equation abstract in this paper a one dimensional generalized boussinesq equation u t t u x x u 3 u x x x x 0 with hinged boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional hamiltonian system the proof is based on an infinite dimensional kam theorem and the birkhoff normal form

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

observability of a linear system under sparsity constraints consider an n dimensional linear system where it is known that there are at most nonzero components in the initial state the observability problem that is the recovery of the initial state for such a system is considered we obtain sufficient conditions on the number of available observations to be able to recover the initial state exactly for such a system both deterministic and stochastic setups are considered for system dynamics in the former setting the system matrices are known deterministically whereas in the latter setting all of the matrices are picked from a randomized class of matrices the main message is that one does not need to obtain full n observations to be able to uniquely identify the initial state of the linear system even when the observations are picked randomly when the initial condition is known to be sparse

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

multiscale kam theorem for hamiltonian systems abstract in this paper we study the persistence of invariant tori in nearly integrable multiscale hamiltonian systems with highorder degeneracy in the integrable part such hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems we introduce multiscale nondegenerate condition and multiscale diophantine condition comparable to the usual diophantine condition using quasilinear kam method we prove a multiscale kam theorem

quasi periodic solutions for the quasi periodically forced cubic complex ginzburg landau equation on td in this paper we discuss the existence of time quasi periodic solutions for quasi periodically forced cubic complex ginzburg landau equation of higher spatial dimension with basic frequency vector 1 2 m by constructing a kam kolmogorov arnold moser theorem for a dissipative system which depends on time in a quasi periodic way we obtain a cantorian branch of m 2 dimensional invariant tori for the equation

regularization and error estimate for the nonlinear backward heat problem using a method of integral equation abstract we consider the inverse time problem for the nonlinear heat equation in the form u t u x x f x t u x t x t 0 0 t u 0 t u t 0 t 0 t the nonlinear problem is severely ill posed we shall use the method of integral equation to regularize the problem and to get some error estimates we show that the approximate problems are well posed and that their solution u x t converges on 0 t if and only if the original problem has a unique solution we obtain several other results including some explicit convergence rates some numerical tests illustrate that the proposed method is feasible and effective

a kam theorem for hyperbolic type degenerate lower dimensional tori in hamiltonian systems a kam theorem for degenerate lower dimensional tori in nearly integrable hamiltonian systems is given in this paper for the non degenerate cases both hyperbolic and elliptic the kam theorem has been well established by many authors 8 9 11 13 14 17

moser s theorem for lower dimensional tori moser s c version of kolmogorov s theorem on the persistence of maximal quasi periodic solutions for nearly integrable hamiltonian system is extended to the persistence of non maximal quasi periodic solutions corresponding to lower dimensional elliptic tori of any dimension n between one and the number of degrees of freedom the theorem is proved for hamiltonian

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

on an inverse boundary value problem of a nonlinear elliptic equation in three dimensions abstract this work considers an inverse boundary value problem for a 3d nonlinear elliptic partial differential equation in a bounded domain in general the problem is severely ill posed the formal solution can be written as a hyperbolic cosine function in terms of the 2d elliptic operator via its eigenfunction expansion and it is shown that the solution is stabilized or regularized if the large eigenvalues are cut off in a theoretical framework a truncation approach is developed to approximate the solution of the ill posed problem in a regularization manner under some assumptions on regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type a local lipschitz case of source term for this nonlinear problem is obtained for numerical illustration two examples on the elliptic sine gordon and elliptic allen cahn equations are constructed to demonstrate the feasibility and efficiency of the proposed methods

quasi periodic solutions for 1d schrodinger equation with the nonlinearity u 2pu abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x u 2 p u 0 p n with periodic boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to 2 dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on infinite dimensional kam theory partial normal form and scaling skills

on a backward heat problem with time dependent coefficient regularization and error estimates in this paper we consider a homogeneous backward heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the final data a new regularization method is applied to formulate regularized solutions which are stably convergent to the exact ones with holder estimates a numerical example shows that the computational effect of the method is all satisfactory

on lower dimensional invariant tori in reversible systems abstract in this paper a result on the persistence of lower dimensional invariant tori in reversible systems is obtained under some weak non degenerate conditions such a result can be applied to the case where normal frequencies are not simple and some of them may be zero we also give an answer of a conjecture of m b sevryuk 1986 reversible systems lecture notes in math vol 1211 springer verlag new york berlin about the existence of lower dimensional invariant tori in resonant zones

real analytic quasi periodic solutions for the derivative nonlinear schrodinger equations in this paper we show that one dimension derivative nonlinear schrodinger equation admits a whitney smooth family of small amplitude real analytic quasi periodic solutions with two diophantine frequencies the proof is based on a partial birkhoff normal form reduction and an abstract infinite dimensional kolmogorov arnold moser kam theorem

a stability estimate on the regularized solution of the backward heat equation inverse problems where one looks for causes for observed or desired effects are usually ill posed i e their solution is not unique or unstable with respect to data perturbations we derive a stability estimate for the regularized solution of the backward heat equation which is needed for stable solution of this ill posed problem for this purpose we construct a regularizer for a certain minimization problem and in conclusion the stability of this minimization procedure

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

an approximate solution for nonlinear backward parabolic equations we consider the backward parabolic equation ut au f t u t 0 t t u t g where a is a positive unbounded operator and f is a nonlinear function satisfying a lipschitz condition with an approximate datum g the problem is severely ill posed using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces according to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type for all t 0 t an example on heat equations and numerical experiments are given

existence of kam tori in degenerate hamiltonian systems abstract this article is devoted to the proof that invariant tori still exist in perturbed integrable hamiltonian systems with n degrees of freedom h p q n p p p q if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping n p can be any positive integer smaller than n provided the image of any open set under the mapping includes a cn 2 curved one dimensional submanifold

persistence of lower dimensional hyperbolic invariant tori for generalized hamiltonian systems abstract in this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized hamiltonian systems here the generalized hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables in particular systems under consideration can be odd dimensional under russmann type non degenerate condition by introducing a modified linear kam iterative scheme we proved that the majority of the lower dimensional hyperbolic invariant tori persist under small perturbations for generalized hamiltonian systems

a numerical technique for backward inverse heat conduction problems in one dimensional space in this article we consider an inverse heat conduction problem in one dimensional space with two unknown terms in this problem the initial temperature and histories of temperature in the finite interval of time may be determined then by using a semi implicit finite difference method an estimation of these unknown terms will be found obviously this problem is ill posed but by applying a suitable parameter we can obtain the best of these estimations finally some numerical experiments are given in the final section of this paper

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

isochronous centers of lienard type equations and applications abstract in this work we study eq e x f x x 2 g x 0 with a center at 0 and investigate conditions of its isochronicity when f and g are analytic not necessary odd a necessary and sufficient condition for the isochronicity of 0 is given this approach allows us to present an algorithm for obtained conditions for a point of e to be an isochronous center in particular we find again by another way the isochrones of the quadratic loud systems l d f we also classify a 5 parameters family of reversible cubic systems with isochronous centers

lower dimensional invariant tori with prescribed frequency for the nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 2 u f u 2 u 0 subject to dirichlet boundary conditions is considered where the nonlinearity f is a real analytic function near u 0 with f 0 f 0 0 it is proved that for each given constant potential m and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude time quasi periodic solutions whose b dimensional frequencies are just small dilation of a prescribed diophantine vector accordingly we obtain the existence of lower dimensional invariant kam tori with tangential frequencies constrained to a given diophantine direction in an infinite dimensional phase space setting

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

lower dimensional invariant tori with prescribed frequency for nonlinear wave equation abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 3 0 subject to dirichlet boundary conditions is considered we show that for each given m 0 and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude quasi periodic solutions with b dimensional diophantine frequencies which correspond to b dimensional invariant tori of an associated infinite dimensional dynamical system in particular these diophantine frequencies are the small dilation of a prescribed diophantine vector the proof is based on a partial birkhoff normal form reduction and an improved kam method

on the persistence of invariant curves for fibered holomorphic transformations we consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations we define a fibered rotation number associated to an invariant curve we show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a brjuno type arithmetical condition nevertheless an extra complex parameter is added to the problem and the persistence becomes a one complex codimension property

the hyperbolic invariant tori of symplectic mappings abstract in this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings under a russmann type non degenerate condition by introducing a modified kam iteration scheme we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator