imposing boundary and interface conditions in multi resolution wavelet galerkin method for numerical solution of helmholtz problems abstract in this paper we propose a new formulation for numerical solution of boundary value problems using mesh free multi resolution wavelet galerkin wg method a difficulty with the wg method is imposing boundary condition constraints single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature ramp and cubic polynomial functions have been also used in the jump function approach described in element free galerkin context and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump but the cubic polynomial functions do not in this paper a multi resolution wg method is employed and both ramp and cubic polynomial jump functions are applied obtained results in one dimension are compared with analytical solutions results obtained with ramp jump functions in two dimensions are also reported the results obtained with the wg method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied numerical results are in good agreement with accurate solutions

on critical orbits and sectional hyperbolicity of the nonwandering set for flows in this note we introduce the notion of nonuniformly sectional hyperbolic set and use it to prove that any c1 open set which contains a residual subset of vector fields with nonuniformly sectional hyperbolic critical set also contains a residual subset of vector fields with sectional hyperbolic nonwandering set this not only extends theorem a of castro 11 but using suspensions we recover it

singular robustly chain transitive sets are singular volume partial hyperbolic for diffeomorphisms or for non singular flows there are many results relating properties persistent under mathcal c 1 perturbations and a global structures for the dynamics such as hyperbolicity partial hyperbolicity dominated splitting however a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point in bonatti star flows and multisingular hyperbolicity arxiv 1705 05799 2017 with christan bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities using this tool we recover the results in bonatti et al ann math 158 2 355 418 2003 for flows showing that robustly chain transitive sets have a weak form of hyperbolicity allowing us to conclude as well the kind of hyperbolicity carried by the examples in bonatti et al j inst math jussieu 12 3 449 501 2013 a robust chain transitive singular attractor with periodic orbits of different indexes along with the results in 8 this shows that the way we propose to interpret the effect of singularities has the potential to adapt to other settings in which there is coexistence of singularities and regular orbits with the goal of re obtaining the results that we already know for diffeomorphisms

quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions this paper deals with typical problems that arise when using wavelets in numerical analysis applications the first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions several types of quadratures are discussed and compared for different classes of wavelets since their construction using monomials is ill conditioned also a modified well conditioned construction using chebyshev polynomials is presented the second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions they are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

a multilevel wavelet collocation method for solving partial differential equations in a finite domain abstract a multilevel wavelet collection method for the solution of partial differential equations is developed two different approaches of treating general boundary conditions are suggested both are based on the wavelet interpolation technique developed in the present research the first approach uses wavelets as a basis and results in a differential algebraic system of equations where the algebraic part arises from boundary conditions the second approach utilizes extended wavelets which satisfy boundary conditions exactly this approach results in a system of coupled ordinary differential equations the method is tested on the one dimensional burgers equation with small viscosity the solutions are compared with those resulting from the use of other numerical algorithms the present results indicate that the method is competitive with well established numerical algorithms

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

equivalences between nonuniform exponential dichotomy and admissibility abstract relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems it was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy in this paper we find an appropriate admissible pair of classes of lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half lines r separately on both half lines r simultaneously and on the whole line r additionally the maximal admissibility is proved in the case on both half lines r simultaneously

persistently expansive geodesic flows we prove thatc1 persistently expansive geodesic flows of compact boundaryless riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set in the case of compact surfaces we deduce that the geodesic flow isc1 persistently expansive if and only if it is an anosov flow

hyperbolicity through stable shadowing for generic geodesic flows abstract we prove that the closure of the closed orbits of a generic geodesic flow on a closed riemannian n 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds c 2 robustly on the metric we obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization

a characterization of singular hyperbolicity via the linear poincare flow abstract singular hyperbolicity is a fundamental notion in dynamical systems that describes in particular lorenz like dynamics it is defined using the tangent flow and concerns both the regular and singular parts of the dynamics in this paper we give a characterization of singular hyperbolicity using the linear poincare flow which by definition is defined on the regular part usually non compact of the dynamics only

c 1 stably shadowable chain components are hyperbolic abstract let f be a diffeomorphism on a closed manifold and p be a hyperbolic periodic point of f denote c f p the chain component of f that contains p we say c f p is c 1 stably shadowable if there is a c 1 neighborhood u of f such that for every g u c g p g has the shadowing property where p g is the continuation of p we prove in this paper that if c f p is c 1 stably shadowable then c f p is hyperbolic

a gaussian sinc collocation approach for a whipping cantilever with a follower shear force at its tip a spatial discretization scheme based on a set of gaussian sinc functions is proposed for the temporal projection of a set of partial differential equations pdes describing the non linear dynamics of an elastic plastic hardening softening ephs cantilever subjected to a follower shear force at its tip the dynamics so described correspond to planar whipping of a pipe conveying fluid ruptured near a right angled bend the constitutive ephs moment curvature relationship used here follows the earlier work of reid et al proceedings of the royal society of london series a 1998 454 997 1029 compared to the more classical lagrangian polynomial based collocation functions the gaussian sine functions have better localization properties moreover for a relatively large number of collocation points use of such functions does not lead to numerical overflow or underflow problems often associated with the use of higher order polynomial collocation functions the spatial discretization leads to a set of non linear ordinary differential equations odes in time which are in turn integrated via a fourth order adaptive runge kutta scheme some numerical results for a cantilever whipping in a plane are presented to further illustrate the present approach the method is a step forward towards the development of a mesh free non linear beam element suitable for dynamic analyses of pipe networks and pipe on pipe impact problems

the star systems x and a proof of the c1ω stability conjecture for flows abstract in this paper we give a new proof for the c 1 stability conjecture for flows our approach is considerably different from that of hayashi or wen and a good deal of liao s obstruction sets theory is used

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

on the finiteness of uniform sinks abstract we study the finiteness of uniform sinks for flows precisely we prove that for 0 and t 0 if a vector field x has only hyperbolic singularities or sectionally dissipative singularities then x can have only finitely many t uniform sinks this is a generalized version of a theorem of liao 4

infinitesimal lyapunov functions for singular flows we present an extension of the notion of infinitesimal lyapunov function to singular flows and from this technique we deduce a characterization of partial sectional hyperbolic sets in absence of singularities we can also characterize uniform hyperbolicity these conditions can be expressed using the space derivative dx of the vector field x together with a field of infinitesimal lyapunov functions only and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region

on the existence of attractors on every compact 3 manifold we build a non empty open set u of diff m such that for every r 1 every cr generic diffeomorphism f u diffr m has no topological attractors on higher dimensional manifolds one may require that f has neither topological attractors nor topological repellers our examples have finitely many quasi attractors for flows we may require that these quasi attractors contain singular points finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics

an efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the daubechies wavelet basis an efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with daubechies wavelet and scaling function basis sets our quadrature is also applicable in the case of adaptive spatial resolution our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wavefunction of the harmonic oscillator in one dimension with and without adaptive resolution as a byproduct we derive a filter which upon application on the scaling function coefficients of a smooth function renders the approximate grid values of this function this also allows for a fast calculation of the charge density from the wavefunction

on the use of biorthogonal interpolating wavelets for large eddy simulation of turbulence recent work in the field of turbulence modelling has demonstrated the benefits of the wavelet based multiresolution analysis technique as a tool for the formulation of the large eddy simulation les equations in this formalism the les equations are obtained by projecting the navier stokes equations onto a hierarchy of wavelet spaces this paper investigates the use of biorthogonal interpolating wavelets as a basis for this projection placing special emphasis on the wavelet based differential operators that define this mapping a detailed analysis of their convergence properties is presented and compared to those of their orthogonal counterpart the daubechies wavelets based on this study we highlight the weaknesses of the unlifted interpolating wavelet representation for les sub grid modelling finally we establish a link between the unlifted framework and the sampling based les approach recently proposed in the literature

on c1 class local diffeomorphisms whose periodic points are nonuniformly expanding using a sifting shadowing combination we prove in this paper that an arbitrary c1 class local diffeomorphism f of a closed manifold mn is uniformly expanding on the closure mathrm cl m n mathrm per f of its periodic point set per f if it is nonuniformly expanding on per f

robust criterion for the existence of nonhyperbolic ergodic measures we give explicit c 1 open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy actually our criterion provides the existence of a partially hyperbolic compact set with one dimensional center and positive topological entropy on which the center lyapunov exponent vanishes uniformly

on herman s positive entropy conjecture abstract we show that any area preserving c r diffeomorphism of a two dimensional surface displaying an elliptic fixed point can be c r perturbed to one exhibiting a chaotic island whose metric entropy is positive for every 1 r this proves a conjecture of herman stating that the identity map of the disk can be c perturbed to a conservative diffeomorphism with positive metric entropy this implies also that the chirikov standard map for large and small parameter values can be c approximated by a conservative diffeomorphisms displaying a positive metric entropy a weak version of sinai s positive metric entropy conjecture finally this sheds light onto a herman s question on the density of c r conservative diffeomorphisms displaying a positive metric entropy we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable so conjecturally uniformly hyperbolic or display a chaotic island of positive metric entropy

stably average shadowable homoclinic classes abstract let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold m and let h f p be the homoclinic class of f containing p in this paper we show that if h f p is locally maximal and every hyperbolic periodic point in h f p is uniformly far away from being nonhyperbolic and h f p has the average shadowing property then h f p is hyperbolic

modulated amplitude waves and defect formation in the one dimensional complex ginzburg landau equation the transition from phase chaos to defect chaos in the complex ginzburg landau equation cgle is related to saddle node bifurcations of modulated amplitude waves maws first the spatial period p of maws is shown to be limited by a maximum psn which depends on the cgle coefficients maw like structures with period larger than psn evolve to defects second slowly evolving near maws with average phase gradients 0 and various periods occur naturally in phase chaotic states of the cgle as a measure for these periods we study the distributions of spacings p between neighbouring peaks of the phase gradient a systematic comparison of p and psn as a function of coefficients of the cgle shows that defects are generated at locations where p becomes larger than psn in other words maws with period psn represent critical nuclei for the formation of defects in phase chaos and may trigger the transition to defect chaos since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient the coefficients where the transition to defect chaos seems to occur depend on system size and integration time we conjecture that in the regime where the maximum period psn has diverged phase chaos persists in the thermodynamic limit 2001 published by elsevier science b v

a proof of stability conjecture for three dimensional flows we give a proof of the c1 stability conjecture for three dimensional flows i e prove that there exists a hyperbolic structure over the q set for the structurally stable three dimensional flows mafn s proof for the discrete case motivates our proof and we find his perturbation techniques crucial in proving this conjecture we have overcome several new difficulties e g the change of period after perturbation the ergodic closing lemma for flows the existence of dominated splitting over q y where 3 is the set of singularities for the flow the discontinuity of the contracting rate function on singularities etc based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems i e we create unstable saddle connections if there are accumulations of periodic orbits on the singularities

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

wamr an adaptive wavelet method for the simulation of compressible reacting flow part i accuracy and efficiency of algorithm abstract the wavelet adaptive multiresolution representation wamr method provides a robust method for controlling spatial grid adaptation fine grid spacing in regions where a solution varies greatly i e near steep gradients or near singularities and a much coarser grid where the solution varies slowly subsequently a wide range of spatial scales often demanded in challenging continuum physics problems can be efficiently captured furthermore the wavelet transform provides a direct measure of local error at each collocation point effectively producing automatically verified solutions the method is applied to the solution of unsteady compressible reactive flow equations and includes detailed diffusive transport and chemical kinetics models accuracy and performance of the method are examined on several test problems the sparse grids produced by the wamr method exhibit an impressive compression of the solution reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution

on the generalized wavelet galerkin method abstract in the frame of the traditional wavelet galerkin method based on the compactly supported wavelets it is important to calculate the so called connection coefficients that are some integrals whose integrands involve products of wavelets their derivatives as well as some known coefficients in considered differential equations however even for linear differential equations with non constant coefficient the computation of connect coefficients becomes rather time consuming and often even impossible in this paper we propose a generalized wavelet galerkin method based on the compactly supported wavelets which is computationally very efficient even for differential equations with non constant coefficients no matter linear or nonlinear problems some related mathematical theorems are proved based on which the basic ideas of the generalized wavelet galerkin method are described in details in addition some examples are used to illustrate its validity and high efficiency a nonlinear example shows that the generalized wavelet galerkin method is not only valid to solve nonlinear problems but also possesses the ability to find new solutions of multi solution problems this method can be widely applied to various types of both linear and nonlinear differential equations in science and engineering

a fast wavelet multigrid method to solve elliptic partial differential equations abstract in this paper we present a wavelet based multigrid approach to solve elliptic boundary value problems encountered in mathematical physics the system of equations arising from finite difference discretization is represented in wavelet basis these equations are solved using multiresolution properties of wavelets characterized by sparse matrices having condition number o 1 together with a multigrid strategy for accelerating convergence the filter coefficients of d 2 k k 2 3 4 from daubechies family of wavelets are used to demonstrate the effectiveness and efficiency of the method the distinguishing feature of the method is it works as both solver and preconditioner as a consequence it avoids instability minimizes error and speeds up convergence compared to the classical multigrid method this approach requires substantially shorter computation time at the same time meeting accuracy requirements it is found that just one cycle is enough for the convergence of wavelet multigrid scheme whereas normally 7 8 cycles are required in classical multigrid schemes to meet the same accuracy numerical examples show that the scheme offers a fast and robust technique for elliptic pde s

exact periodic solutions of the complex ginzburg landau equation three new exact periodic solutions of the complex ginzburg landau equation are obtained in terms of the weierstrass elliptic function furthermore the new periodic solutions and other shock solutions appear as their bounded limits along the real axis for particular relationships between the coefficients in the equation in the general case bounded limits are nothing but the already known pulse hole and shock solutions it is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution in particular the spatial structure of the new bounded periodic solutions varies in time while the pulse solution may exhibit breather like behavior

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

ensembles invariants non enlaces des diffeomorphismes du tore et de l anneau we introduce the notion of unlinked invariant set for a diffeomorphism of the 2 torus or the closed annulus and we look at the properties of these sets which generalize the aubry mather sets we prove that for any irrational number in the rotation set of an area preserving diffeomorphism of the annulus there exists an unliked invariant set whose rotation set is reduced to in the same way we prove that any minimal diffeomorphism of the 2 torus homotopic to the identity may be approximated in the c 0 topology by a periodic diffeomorphism

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

robust vanishing of all lyapunov exponents for iterated function systems given any compact connected manifold m we describe c 2 open sets of iterated functions systems ifs s admitting fully supported ergodic measures whose lyapunov exponents along m are all zero moreover these measures are approximated by measures supported on periodic orbits we also describe c 1 open sets of ifs s admitting ergodic measures of positive entropy whose lyapunov exponents along m are all zero the proofs involve the construction of non hyperbolic measures for the induced ifs s on the flag manifold

filter bank methods for hyperbolic pdes we use biorthogonal filter banks to solve hyperbolic pdes adaptively with a sparse multilevel representation of the solution the methods described are of finite difference type and the filter banks are used to give a sparse representation of signals and to transform between grids on different scales we derive bounds for the error and number of coefficients in the sparse representation these bounds also apply for filter banks that are not associated with any wavelets we develop algorithms for fast differentiation and multiplication in detail the strength of the method is shown in various test problems

symbolic extensions and dominated splittings for generic c 1 diffeomorphisms let mathrm diff 1 m be the set of all c 1 diffeomorphisms f m rightarrow m where m is a compact boundaryless d dimensional manifold d ge 2 we prove that there is a residual subset mathfrak r of mathrm diff 1 m such that if f in mathfrak r and if h p is the homoclinic class associated with a hyperbolic periodic point p then either h p admits a dominated splitting of the form e oplus f 1 oplus dots oplus f k oplus g where f i is not hyperbolic and one dimensional or f h p has no symbolic extensions

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

nonsingular star flows satisfy axiom a and the no cycle condition we give an affirmative answer to a problem of liao and mane which asks whether for a nonsingular flow to loose the stability it must go through a critical element bifurcation more precisely a vector field s on a compact boundaryless manifold is called a star system if s has a c 1 neighborhood mathcal u in the set of c 1 vector fields such that every singularity and every periodic orbit of every x in mathcal u is hyperbolic we prove that any nonsingular star flow satisfies axiom a and the no cycle condition

vector fields in the interior of kupka smale systems satisfy axiom a abstract in 1994 s hu conjectured in his paper 1994 trans amer math soc 342 753 772 that axiom a systems are dense in the interior of kupka smale systems we will show that if x is in the interior of kupka smale systems then x satisfies axiom a

geodesic flows with positive topological entropy twist maps and hyperbolicity we prove a perturbation lemma for the derivative of geodesic flows in high dimension this implies that a c 2 generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow

a note on the adaptive conservative dissipative discretization for evolutionary partial differential equations an adaptive conservative or dissipative numerical method for nonlinear partial differential equations is established the method not only inherits the conservation or dissipation property of the equation but also uses suitable non uniform grids at each time step our numerical experiments indicate that the method is useful especially for localized solutions such as solitary wave solutions

genericity of geodesic flows with positive topological entropy on s2 we show that the set of c riemannian metrics on s 2 or rp 2 whose geodesic flow has positive topological entropy is open and dense in the c 2 topology the proof is partially based on an analogue of franks lemma for geodesic flows on surfaces

an adaptive multilevel wavelet collocation method for elliptic problems an adaptive multilevel wavelet collocation method for solving multi dimensional elliptic problems with localized structures is described the method is based on multi dimensional second generation wavelets and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems int j comp fluid dyn 17 2003 151 wavelet decomposition is used for grid adaptation and interpolation while a hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid in order to accelerate the convergence of the solver an iterative procedure analogous to the multigrid algorithm is developed the overall computational complexity of the solver is o n where n is the number of adapted grid points the accuracy and computational efficiency of the method are demonstrated for the solution of two and three dimensional elliptic test problems

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

stationary modulated amplitude waves in the 1d complex ginzburg landau equation we reformulate the one dimensional complex ginzburg landau equation as a fourth order ordinary dierential equation in order to nd stationary spatiallyperiodic solutions using this formalism we prove the existence and stability of stationary modulated amplitude wave solutions approximate analytic expressions and a comparison with numerics are given

rotation sets for homeomorphisms and homology in this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds this approach is based on taking the suspended flow and using ideas of schwartzmann on homology and winding cycles for flows our main application is to give a generalisation of a theorem of libre and mackay for tori to the context of surfaces of higher genus

a characteristic based volume penalization method for general evolution problems applied to compressible viscous flows in order to introduce solid obstacles into flows several different methods are used including volume penalization methods which prescribe appropriate boundary conditions by applying local forcing to the constitutive equations one well known method is brinkman penalization which models solid obstacles as porous media while it has been adapted for compressible incompressible viscous and inviscid flows it is limited in the types of boundary conditions that it imposes as are most volume penalization methods typically approaches are limited to dirichlet boundary conditions in this paper brinkman penalization is extended for generalized neumann and robin boundary conditions by introducing hyperbolic penalization terms with characteristics pointing inward on solid obstacles this characteristic based volume penalization cbvp method is a comprehensive approach to conditions on immersed boundaries providing for homogeneous and inhomogeneous dirichlet neumann and robin boundary conditions on hyperbolic and parabolic equations this cbvp method can be used to impose boundary conditions for both integrated and non integrated variables in a systematic manner that parallels the prescription of exact boundary conditions furthermore the method does not depend upon a physical model as with porous media approach for brinkman penalization and is therefore flexible for various physical regimes and general evolutionary equations here the method is applied to scalar diffusion and to direct numerical simulation of compressible viscous flows with the navier stokes equations both homogeneous and inhomogeneous neumann boundary conditions are demonstrated through external flow around an adiabatic and heated cylinder theoretical and numerical examination shows that the error from penalized neumann and robin boundary conditions can be rigorously controlled through an a priori penalization parameter h the error on a transient boundary is found to converge as o h which is more favorable than the error convergence of the already established dirichlet boundary condition

partial hyperbolicity far from homoclinic bifurcations abstract we prove that any diffeomorphism of a compact manifold can be c 1 approximated by a diffeomorphism which exhibits a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by a diffeomorphism which is partially hyperbolic its chain recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two we also study in a more systematic way the central models introduced in crovisier in press 10

nonuniform hyperbolicity global dominated splittings and generic properties of volume preserving diffeomorphisms we study generic volume preserving diffeomorphisms on compact manifolds we show that the following property holds generically in the c1 topology either there is at least one zero lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component moreover if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold that is it has a positive measure intersection with any nonempty open set and there is a global dominated splitting for the proof we establish some new properties of independent interest that hold cr generically for any r 1 namely the continuity of the ergodic decomposition the persistence of invariant sets and the l1 continuity of lyapunov exponents

non separable splines and numerical computation of evolution equations by the galerkin methods we construct new non separable splines and we apply the spline sampling approximation to the computation of numerical solutions of evolution equations the non separable splines are basis functions which give a fine sampling approximation which enables us to compute numerical solutions by means of the method of lines combined with the galerkin method to demonstrate our approach we compute numerical solutions of the burgers equation and the kadomtsev petviashvili equation

the saga of a fish from a survival guide to closing lemmas abstract in the paper by d burago s ivanov and a novikov a survival guide for feeble fish it has been shown that a fish with limited velocity can reach any point in the possibly unbounded ocean provided that the fluid velocity field is incompressible bounded and has vanishing mean drift this result extends some known global controllability theorems though being substantially nonconstructive we give a fish a different recipe of how to survive in a turbulent ocean and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative in the sense of not having wandering sets of positive measure dynamics in particular this leads to the extension of c pugh s closing lemma to incompressible vector fields over unbounded domains the results are based on an extension of the poincare recurrence theorem to some finite measures and on specially constructed newtonian potentials

solving hyperbolic pdes using interpolating wavelets a method is presented for adaptively solving hyperbolic pdes the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids the adaptability is performed automatically by thresholding the wavelet coefficients operations such as differentiation and multiplication are fast and simple due to the one to one correspondence between point values and wavelet coefficients in the interpolating basis treatment of boundary conditions is simplified in this sparse point representation spr numerical examples are presented for one and two dimensional problems it is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops

an adaptive wavelet collocation method for fluid structure interaction at high reynolds numbers two mathematical approaches are combined to calculate high rey nolds number incompressible fluid structure interaction a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion and brinkman penalization to enforce solid boundaries of arbitrary complexity we also implement a wavelet based multilevel solver for the poisson problem for the pressure at each time step the method is applied to two dimensional flow around fixed and moving cylinders for reynolds numbers in the range 3 times 10 1 le re le 10 5 the compression ratios of up to 1000 are achieved for the first time it is demonstrated in actual dynamic simulations that the compression scales like re 1 2 over five orders of magnitude while computational complexity scales like re this represents a significant improvement over the classical complexity estimate of re 9 4 for two dimensional turbulence

non existence of elliptic travelling wave solutions of the complex ginzburg landau equation abstract we give a simple proof that for generic parameter values the cubic complex one dimensional ginzburg landau equation has no elliptic travelling wave solutions this is contrary to the expectations of musette and conte in physica d 181 2003 70 79 that elliptic solutions of zero codimension should exist the method of proof based on the residue theorem is very general and can be applied to determine necessary conditions for the existence of elliptic travelling waves for any autonomous partial differential equation as another application we prove that kudryashov s codimension one elliptic solution of the generalized kuramoto sivashinsky equation is the only one possible

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

perturbation of the lyapunov spectra of periodic orbits we describe all lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism the description is expressed in terms of the finest dominated splitting and lyapunov exponents that appear in the limit of a sequence of periodic orbits and involves the majorization partial order among the applications we give a simple criterion for the occurrence of universal dynamics

standing waves of the complex ginzburg landau equation abstract we prove the existence of nontrivial standing wave solutions of the complex ginzburg landau equation t e i e i k on a bounded domain with dirichlet boundary conditions our result requires that 0 be sufficiently small

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

existence of standing waves for the complex ginzburg landau equation abstract we study the existence of standing wave solutions of the complex ginzburg landau equation gl t e i i e i 0 in r n where 0 n 2 4 0 and r we show that for any 2 2 there exists e 0 such that gl has a non trivial standing wave solution if e analogous result is obtained in a ball r n for 1 where 1 is the first eigenvalue of the laplace operator with dirichlet boundary conditions

spatiotemporal chaos in the one dimensional complex ginzburg landau equation the dynamical behavior of a large one dimensional system obeying the cubic complex ginzburg landau equation is studied numerically as a function of parameters near a supercritical bifurcation two types of chaotic behavior can be distinguished beyond the benjamin feir instability a phase turbulence regime with a conserved phase winding number and no phase dislocations space time defects and a defect regime with a nonzero density of defects the transition between the two can either be continuous or discontinuous hysteretic depending on parameters the spatial decay of the phase correlation function is inferred to be exponential in both regimes with a sharp decrease of the correlation length upon entering the defect phase the temporal decay of correlations is exponential in the defect regime

lyapunov stability and sectional hyperbolicity for higher dimensional flows we study c 1 generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices we prove that these flows exhibit finitely many sinks and sectional hyperbolic transitive lyapunov stable sets whose basins form a residual subset of the ambient manifold this represents a partial positive answer to conjectures in arbieto and morales proc am math soc 141 2817 2827 2013 the palis conjecture palis nonlinearity 21 t37 t43 2008 and gives a flow version of crovisier and pujals essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms 2010

periodic orbits and expansiveness we prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map using this we reobtain a result due to sakai generic positively expansive maps are expanding our methods also show a global version of a result by gan and yang generic expansive diffeomorphisms are axiom a without cycles

a dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos capturing all scales of random modes on independent grids in stochastic computations or uncertainty quantification methods the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion the size of this system increases drastically when the number of independent random variables and or order of polynomial chaos expansions increases this is invariably the case for large scale simulations and or problems involving steep gradients and other multiscale features such features are variously reflected on each solution component or random uncertainty mode requiring the development of adaptive methods for their accurate resolution in this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space refinement on physical space that allows all scales of each solution component to be refined independently of the rest we exemplify this using the convection diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method thus we establish a new elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions

a c 2 generic trichotomy for diffeomorphisms hyperbolicity or zero lyapunov exponents or the c 1 creation of homoclinic bifurcations palis conjectured that densely in diff m r 1 diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations we prove a generic trichotomy for c2 diffeomorphisms an axiom a diffeomorphism with no cycles or kupka smale ones admitting zero lyapunov exponents or the c1 creation of homoclinic bifurcations i e the creation of homoclinic tangencies or heterodimensional cycles by some c1 small perturbations

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

hyperbolicity and stability for hamiltonian flows abstract we prove that a hamiltonian star system defined on a 2d dimensional symplectic manifold m d 2 is anosov as a consequence we obtain the proof of the stability conjecture for hamiltonians this generalizes the 4 dimensional results in bessa et al 2010 5

a c 1 generic dichotomy for diffeomorphisms weak forms of hyperbolicity or infinitely many sinks or sources we show that for every compact n dimensional manifold n 1 there is a residual subset of diff m of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities either it is contained in the closure of an infinite set of sinks or sources newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of mafine ma3 in particular we show that any cl robustly transitive diffeomorphism admits a dominated splitting

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse

connection coefficients on an interval and wavelet solutions of burgers equation a definition of connection coefficients is introduced and techniques of computation are presented we use semi implicit time difference scheme to solve burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form comparisons of accuracy and robustness of numerical solutions are mentioned in the examples