the geometry of mixing in time periodic chaotic flows i asymptotic directionality in physically realizable flows and global invariant properties abstract this paper demonstrates that the geometry and topology of material lines in 2d time periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality this property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the jacobian matrix of the n th iterative of the poincare map associated with the flow asymptotic directionality also determines the geometry of the invariant unstable manifolds which are everywhere tangent to the field of asymptotic eigendirections this fact is used to derive simple non perturbative methods for reconstructing the global invariant manifolds to any desired level of detail the geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a poincare map of the flow defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation the connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the poincare map in a sense that will be made clear in the article the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures two physically realizable systems are analyzed closed cavity flow and flow between eccentric cylinders asymptotic directionality provides evidence of a global self organizing structure characterizing chaotic flow which is analogous to that of anosov diffeomorphisms which turns out to be the basic prototype of mixing systems in this framework we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure different from the ergodic measure associated with the dynamical system generated by the field of asymptotic unstable eigenvectors

a numerical study of the small dispersion limit of the korteweg de vries equation and asymptotic solutions abstract we study numerically the small dispersion limit for the korteweg de vries kdv equation u t 6 u u x 2 u x x x 0 for 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small in the whole x t plane the matching of the asymptotic solutions is studied numerically

a numerical study of the 3 periodic wave solutions to kdv type equations abstract in this paper by using the direct method of calculating periodic wave solutions proposed by akira nakamura we present a numerical process to calculate the 3 periodic wave solutions to several kdv type equations the korteweg de vries equation the sawada koterra equation the boussinesq equation the ito equation the hietarinta equation and the 2 1 dimensional kadomtsev petviashvili equation some detailed numerical examples are given to show the existence of the three periodic wave solutions numerically

periodic and chaotic behaviour in a reduction of the perturbed korteweg de vries equation the dynamical behaviour of a reduction of the forced and damped korteweg de vries equation is studied numerically chaos arising from subharmonic instability and homoclinic crossings are observed both period doubling bifurcations and the melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour here characterised by a positive lyapunov exponent supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits and their behaviour under perturbation using melnikov functions

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

finite time transport in aperiodic flows abstract we study the transport of particles in a general two dimensional incompressible flow in the presence of a transient eddy i e a bounded set of closed streamlines with a finite time of existence using quantities obtained from eulerian observations we provide explicit conditions for the existence of a hyperbolic structure in the flow which induces mixing between the eddy and its environment our results can be used directly to study finite time transport in numerically or experimentally generated vector fields with general time dependence

homoclinic crossings and pattern selection the correlations between a homoclinic orbit hmo and coherent patterns in the nonlinear schr odinger model are discussed in a hamiltonian situation two independent patterns are revealed to exist one corresponds to a motion within an hmo and the other outside of an hmo the study further illustrates when the hamiltonian constraint is released the significance of an hmo on the patterns dynamics by presenting the irregular hmo crossings and the resulting chaotic selections between the patterns

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

galois module structure of the square root of the inverse different in even degree tame extensions of number fields abstract let g be a finite group and let n e be a tamely ramified g galois extension of number fields whose inverse different c n e is a square let a n e denote the square root of c n e then a n e is a locally free z g module which is in fact free provided n e has odd order as shown by erez using m taylor s theorem we can rephrase this result by saying that when n e has odd degree the classes of a n e and o n the ring of integers of n in cl z g are equal and in fact both trivial we show that the above equality of classes still holds when n e has even order assuming that n e is locally abelian this result is obtained through the study of the frohlich representatives of the classes of some torsion modules which are independently introduced in the setting of cyclotomic number fields jacobi sums together with the hasse davenport formula are involved in this study finally when g is the binary tetrahedral group we use our result in conjunction with taylor s theorem to exhibit a tame g galois extension whose square root of the inverse different has nontrivial class in cl z g

intermittencies and power law low frequency divergencies in a nonlineat oscillator abstract we have studied the different intermittent regimes in a nonlinear quadratic oscillator by means of analog electronic simulations the intermittency region has been located in parameter space the power spectrum exhibits a 1 behavior with 1 2 the intermittent region has been numerically studied and two different bifurcation schemes leading to pomeau manneville intermittencies of type i and type iii have been identified furthermore a continuous change in the slope of the power law low frequency power spectrum has been found and interpreted in terms of competition between two unstable fixed points finally a one dimensional map which displays the whole phenomenology of the physical system is provided

on twisted forms and relative algebraic k theory this paper introduces a new approach to the study of certain aspects of galois module theory by combining ideas arising from the study of the galois structure of torsors of finite group schemes with techniques coming from relative algebraic k theory

computing riemann theta functions the riemann theta function is a complex valued function of g complex variables it appears in the construction of many quasi periodic solutions of various equations of mathematical physics in this paper algorithms for its computation are given first a formula is derived allowing the pointwise approximation of riemann theta functions with arbitrary user specified precision this formula is used to construct a uniform approximation formula again with arbitrary precision

optimal perturbation for enhanced chaotic transport the issue of determining the best perturbation which results in optimal chaotic flux across a separatrix is addressed using the melnikov function and lobe dynamics this theoretical analysis is motivated mainly through micro fluidic devices for which this problem has become important recently both two and three dimensional flows are analysed utilising a fourier transform representation the nature of the perturbation which maximises this flux for each frequency value is obtained the resulting optimally attainable flux is computed a concise bound on this flux is presented in terms of the supremum norm of the normal component of the perturbing velocity and the size of the heteroclinic manifold in this instance where the spatial part of the perturbation is permitted to be chosen based on the frequency it is shown that greater flux is achievable for smaller frequencies the theory is illustrated through two examples

detecting coherent structures using braids abstract the detection of coherent structures is an important problem in fluid dynamics particularly in geophysical applications for instance knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills existing methods detect lagrangian coherent structures which are barriers to transport by examining the stretching field as given by finite time lyapunov exponents these methods are very effective when the velocity field is well determined but in many applications only a small number of flow trajectories are known for example when dealing with oceanic float data we introduce a topological method for detecting invariant regions based on a small set of trajectories in this method we regard the two dimensional trajectory data as a braid in three dimensions with time being the third coordinate invariant regions then correspond to trajectories that travel together and do not entangle other trajectories we detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories and searching for loops that show negligible growth

nonautonomous analysis of steady korteweg de vries waves under nonlocalised forcing abstract recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the korteweg de vries kdv equation arbitrary but small nonlocalised bottom topographies are amenable to this method two classes of free surface solutions hyperbolic and homoclinic solutions of the associated augmented dynamical system are characterised the first of these corresponds to near uniform free surface flows for which explicit formulae are developed for a range of topographies the second corresponds to solitary waves on the free surface and a method for determining their number is developed formulae for the shape of these solitary waves are also obtained theoretical free surface profiles are verified using numerical kdv solutions and excellent agreement is obtained

uniformly attracting solutions of nonautonomous differential equations abstract understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory by means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible we introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations in particular for asymptotically autonomous polynomial and periodic equations our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems we provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations

wave dynamics in the extended forced korteweg de vries equation the extended forced korteweg de vries equation efkdv is a mathematical model for simulating the interaction of a shallow layer of fluid with external forcing agents herein we consider the dynamics of the efkdv when the forcing is assumed to be small and spatially periodic with a time periodic variation of the phase we show that a good heuristic understanding of the dynamics for a certain class of initial data can be found by studying a one degree of freedom hamiltonian system one feature associated with this hamiltonian system is that if the phase varies slowly with time then to leading order the dynamics for the resonant solutions are governed by the forced nonlinear pendulum equation furthermore we show that resonant solutions can correspond to waves which are trapped i e waves which do not travel but instead oscillate the theory is illustrated by numerical simulations

solutions to the kaup broer system and its 2 1 dimensional integrable generalization via the dressing method abstract in this paper we formulate the nonlocal dbar problem dressing method of manakov and zakharov zakharov and manakov 1984 1985 zakharov 1989 for the 4 scaling classes of the 1 1 dimensional kaup broer system broer 1975 kaup 1975 the applications of the method for the 1 1 dimensional kaup broer systems are reductions of a method for a complex valued 2 1 dimensional completely integrable partial differential equation first introduced in rogers and pashaev 2011 this method allows computation of solutions to all scaling classes of the kaup broer system we then consider the case of non capillary waves with gravitational forcing and use the dressing method to compute n soliton solutions and more general solutions in the closure of the n soliton solutions in the topology of uniform convergence in compact sets called primitive solutions these more general solutions are analogous to the solutions derived in dyachenko and zakharov 2016 zakharov and dyachenko 2016 zakharov et al 2016 for the kdv equation we derive dressing functions for finite gap solutions and compute counter propagating dispersive shockwave type solutions numerically

structure of motion near saddle points and chaotic transport in hamiltonian systems generic symmetry and transport properties of near separatrix motion in 11 2 degree of freedom hamiltonian systems are studied first the rescaling invariance of motion near saddle points with respect to the transformation epsilon lambdaepsilon chi chi pi of the amplitude epsilon and phase chi of the time periodic perturbation is recalled the rescaling parameter lambda depends only on the frequency of the perturbation and the behavior of an unperturbed hamiltonian near a saddle point additional rescaling symmetry of the motion with respect to transformation epsilon lambda 1 2 epsilon chi chi pi 2 is found for some hamiltonian systems possessing symmetry in the phase space it is shown that these rescaling invariance properties of motion lead to strong periodic or quasiperiodic dependencies of all statistical characteristics of the chaotic motion near the separatrix on log 10 epsilon with the period log 10 lambda these properties are examined for different models of chaotic motion by direct numerical integrations of equations of motion as by well as using a computationally efficient method of the separatrix mapping

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

digitization of nonautonomous control systems abstract methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time varying coefficients is digitized e g approximated by piecewise autonomous systems on small time subintervals

nonautonomous control of stable and unstable manifolds in two dimensional flows abstract we outline a method for controlling the location of stable and unstable manifolds in the following sense from a known location of the stable and unstable manifolds in a steady two dimensional flow the primary segments of the manifolds are to be moved to a user specified time varying location which is near the steady location we determine the nonautonomous perturbation to the vector field required to achieve this control and give a theoretical bound for the error in the manifolds resulting from applying this control the efficacy of the control strategy is illustrated via a numerical example

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

galois module structure of elementary abelian extensions abstract let k be an algebraic number field o k its ring of integers and g an elementary abelian group of order l k in this article we determine which classes in the locally free class group cl g of the group ring g are realizable as galois module classes of rings of integers o l in tame galois extensions l k with gal l k g the set r g of such realizable classes is described in terms of the action on cl g of a stickelberger ideal j in the integral group ring z c where c f x lk is a cartan subgroup of the automorphism group aut g gl k f l

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

absolute flux optimising curves of flows on a surface abstract given a flow on a surface we consider the problem of connecting two distinct trajectories by a curve of extremal absolute instantaneous flux we develop a complete classification of flux optimal curves accounting for the possibility of the flux having spatially and temporally varying weight this weight enables modelling the flux of non equilibrium distributions of tracer particles pollution concentrations or active scalar fields such as vorticity our results are applicable to all smooth autonomous flows area preserving or not

weak finite time melnikov theory and 3d viscous perturbations of euler flows abstract the ordinary differential equations related to fluid particle trajectories are examined through a 3d melnikov approach this theory assesses the destruction of 2d heteroclinic manifolds such as that present in hill s spherical vortex under a perturbation which is neither differentiable in the perturbation parameter e nor defined for all times the rationale for this theory is to analyse viscous flows that are close to steady euler flows such closeness in e can only reasonably be expected in a weak sense for finite times an expression characterising the splitting of the two dimensional separating manifold is derived

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

hyperelliptic theta functions and spectral methods a code for the numerical evaluation of hyperelliptic theta functions is presented characteristic quantities of the underlying riemann surface such as its periods are determined with the help of spectral methods the code is optimized for solutions of the ernst equation where the branch points of the riemann surface are parameterized by the physical coordinates an exploration of the whole parameter space of the solution is thus only possible with an efficient code the use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision the case of almost degenerate riemann surfaces is addressed tests of the numerics using identities for periods on the riemann surface and integral identities for the ernst potential and its derivatives are performed it is shown that an accuracy of the order of machine precision can be achieved these accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary einstein equations the resulting solution agrees with the theta functional solution to very high precision

price of anarchy and an approximation algorithm for the binary preference capacitated selfish replication game abstract we consider the capacitated selfish replication csr game with binary preferences over general undirected networks we study the price of anarchy of such games and show that it is bounded above by 3 we develop a quasi polynomial algorithm o n 2 ln d where n is the number of players and d is the diameter of the network which can find in a distributed manner an allocation profile that is within a constant factor of the optimal allocation and hence of any pure strategy nash equilibrium ne of the game proof of this result uses a novel potential function we further show that when the underlying network has a tree structure every globally optimal allocation is an ne which can be reached in only linear time we formulate the optimal solutions and ne points of the csr game using integer linear programs finally we introduce the lcsr game as a localized version of the csr game wherein the actions of the players are restricted to only their close neighborhoods

selfish routing in capacitated networks according to wardrop s first principle agents in a congested network choose their routes selfishly a behavior that is captured by the nash equilibrium of the underlying noncooperative game a nash equilibrium does not optimize any global criterion per se and so there is no apparent reason why it should be close to a solution of minimal total travel time i e the system optimum in this paper we offer positive results on the efficiency of nash equilibria in traffic networks in contrast to prior work we present results for networks with capacities and for latency functions that are nonconvex nondifferentiable and even discontinuous the inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows in this more general model multiple nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient nonetheless our main result shows that the best equilibrium is as efficient as in the model without capacities moreover this holds true for broader classes of travel cost functions than considered hitherto

non autonomous perturbation of autonomous semilinear differential equations continuity of local stable and unstable manifolds abstract in this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a banach space the situation considered is such that the perturbed dynamical system is non autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium then we prove that the local unstable and stable manifolds associated to them are given as graphs roughness of dichotomy plays a fundamental role here moreover we prove the continuity of this local unstable and stable manifolds with respect to the perturbation with that result we conclude the lower semicontinuity of pullback attractors

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

computation of stable and unstable manifolds of hyperbolic trajectories in two dimensional aperiodically time dependent vector fields in this paper we develop two accurate and fast algorithms for the computation of the stable and unstable manifolds of hyperbolic trajectories of two dimensional aperiodically time dependent vector fields first we develop a benchmark method in which all the trajectories composing the manifold are integrated from the neighborhood of the hyperbolic trajectory this choice although very accurate is not fast and has limited usage a faster and more powerful algorithm requires the insertion of new points in the manifold as it evolves in time its numerical implementation requires a criterion for determining when to insert those points in the manifold and an interpolation method for determining where to insert them we compare four different point insertion criteria and four different interpolation methods we discuss the computational requirements of all of these methods we find two of the four point insertion criteria to be accurate and robust one is a variant of a criterion originally proposed by hobson the other is a slight variant of a method due to dritschel and ambaum arising from their studies of contour dynamics the preferred interpolation method is also due to dritschel these methods are then applied to the computation of the stable and unstable manifolds of the hyperbolic trajectories of several aperiodically time dependent variants of the duffing equation

control of mixing in fluid flow a maximum entropy approach in many technological processes a fundamental stage involves the mixing of two or more fluids as a result the design of optimal mixing protocols is a problem of both fundamental and practical importance in this paper the authors formulate a prototypical mixing problem in a control framework where the objective is to determine the sequence of fluid flows that will maximize entropy by developing the appropriate ergodic theoretic tools for the determination of entropy of periodic sequences they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other the authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols

exact solitary water waves with vorticity the solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field the main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity each solution has a supercritical parameter value and decays exponentially at infinity the proof is based on a generalized implicit function theorem of the nash moser type the first approximation to the surface profile is given by the kdv equation with a supercritical value of the surface tension coefficient a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity

chaotic behavior in differential equations driven by a brownian motion abstract in this paper we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a brownian motion we prove that for almost all sample paths of the brownian motion in the classical wiener space the forced equation admits a topological horseshoe of infinitely many branches this result is then applied to the randomly forced duffing equation and the pendulum equation

dynamical survey of the dual power zakharov kuznetsov burgers equation with external periodic perturbation abstract we investigate the dynamical behavior of the newly introduced dual power zakharov kuznetsov burgers equation using bifurcation theory of planar dynamical systems we study bifurcations of traveling wave solutions of the dual power zakharov kuznetsov burgers equation in presence and absence of viscosity effect in presence of an external periodic perturbation we discuss the periodic and chaotic motions of the perturbed dual power zakharov kuznetsov burgers equation by analyzing phase projection analysis time series analysis poincare section and sensitivity analysis the effect of viscosity plays an important role in the transition from chaotic motion to periodic motion of the perturbed dual power zakharov kuznetsov burgers equation

numerical inverse scattering for the korteweg de vries and modified korteweg de vries equations abstract recent advances in the numerical solution of riemann hilbert problems allow for the implementation of a cauchy initial value problem solver for the korteweg de vries equation kdv and the defocusing modified korteweg de vries equation mkdv without any boundary approximation borrowing ideas from the method of nonlinear steepest descent this method is demonstrated to be asymptotically accurate the method is straightforward for the case of defocusing mkdv due to the lack of poles in the riemann hilbert problem and the boundedness properties of the reflection coefficient solving kdv requires the introduction of poles in the riemann hilbert problem and more complicated deformations the introduction of a new deformation for kdv allows for the stable asymptotic computation of the solution in the entire spacial and temporal plane kdv and mkdv are dispersive equations and this method can fully capture the dispersion with spectral accuracy thus this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion this method can easily be adapted to other integrable equations with riemann hilbert formulations such as the nonlinear schrodinger equation

a riemann hilbert problem for the finite genus solutions of the kdv equation and its numerical solution abstract we derive a riemann hilbert problem satisfied by the baker akhiezer function for the finite gap solutions of the korteweg de vries kdv equation as usual for riemann hilbert problems associated with solutions of integrable equations this formulation has the benefit that the space and time dependence appears in an explicit linear and computable way we make use of recent advances in the numerical solution of riemann hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi periodic finite genus solutions of the kdv equation

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

chaotic perturbations of kdv i rational solutions abstract rational pole solutions of a perturbed kdv equation describing nonlinear ion acoustic plasma waves become chaotic in time when a small perturbation is periodically driven a mcgehee transformation blows up a degenerate stationary point at infinity and the smale birkhoff homoclinic theorem adapted to submanifolds in phase permits the use of the melnikov method with pole solutions being homoclinic orbits

chaotic phenomena triggering the escape from a potential well this paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold the aim is to show how the well known atoms of dissipative dynamics saddle node folds period doubling flips cascades to chaos boundary crises etc assemble to form molecules of overall response hierarchies of cusps incomplete feigenbaum trees etc particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle the latter being accurately predicted by a melnikov analysis after escape chaotic transients are shown to conform to recent scaling laws analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

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

acceleration of one dimensional mixing by discontinuous mappings abstract the paper considers a simple class of models for mixing of a passive tracer into a bulk material that is essentially one dimensional we examine the relative rates of mixing due to diffusion stretch and fold operations and permutation of sections of the sample in particular we show how a combination of diffusion with permutation of sections of the sample chopping and shuffling can achieve a faster rate of mixing than pure diffusion this is done by numerical approximation of eigenvalues of certain linear operators

controlling the unsteady analogue of saddle stagnation points it is well known that saddle stagnation points are crucial flow organizers in steady autonomous flows due to their accompanying stable and unstable manifolds these have been extensively investigated experimentally numerically and theoretically in situations related to macro and micromixers in order to either restrict or enhance mixing saddle points are also important players in the dynamics of mechanical oscillators in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior the entity analogous to a saddle point in an unsteady nonautonomous flow is a time varying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time within the context of nearly steady flows the unsteady velocity perturbation required to ensure that such a hyperbolic saddle trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained

nonlinear steepest descent and numerical solution of riemann hilbert problems the effective and efficient numerical solution of riemann hilbert problems has been demonstrated in recent work with the aid of ideas from the method of nonlinear steepest descent for riemann hilbert problems the resulting numerical methods have been shown in practice to retain accuracy as values of certain parameters become arbitrarily large remarkably this numerical approach does not require knowledge of local parametrices rather the deformed contour is scaled near stationary points at a specific rate the primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved to do so we first construct a general theoretical framework for the numerical solution of riemann hilbert problems second we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes in particular we prove sufficient conditions for numerical methods to retain accuracy finally we compute solutions to the homogeneous painleve ii equation and the modified korteweg de vries equation to explicitly demonstrate the practical validity of the theory 2014 wiley periodicals inc

nearly linear dynamics of nonlinear dispersive waves abstract dispersive averaging effects are used to show that the korteweg de vries kdv equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly numerical simulations are presented which indicate the high accuracy of this approximation furthermore this result is applied to shallow water wave dynamics in the limit of kdv approximation which is obtained by asymptotic analysis in combination with numerical simulations of kdv

self duality and torsion galois modules in number fields let k k be a galois extension of number fields with galois group r and let d o be the rings of integers of k k respectively in this paper we investigate the structure of 0 as an or module by comparing 0 with its dual horn 0 m j taylor 11 theorem p 1731 proved in 1978 that these two modules are stably isomorphic over zr assuming that all the primes of o dividing the order of r are unramified in k this result had been conjectured by frohlich 5 p 4231 in the case k q its generalization to all tame extensions is an easy consequence of taylor s subsequent proof of frohlich s conjecture describing the class of 0 in the class group of locally free ztmodules in terms of the artin root number 12 theorem 1 p 4 1 5 proposition 3 p 4391 to study the galois module structure of 0 in the wild case queyrut 7 introduced the s grothendieck groups gi ot and k ot where s is a set of primes of o namely g ot is the grothendieck group corresponding to the category of finitely generated o torsion free or modules with relations arising from short exact sequences splitting outside s ki ot is the grothendieck group of all finitely generated o torsion free or modules which are locally projective outside s with relations arising from short exact sequences with cassou nogues 3 corollaire 6 3 p 231 queyrut generalized taylor s duality result to the wild case by showing that if s is a set of rational primes containing those with a divisor in o wildly ramified in k then the class of 0 is equal to the class of its dual in k u as in the tame case this result could be obtained as a consequence of queyrut s conjecture concerning the class of 0 in k zi 3 p 81 which was proved in several cases 3 the proofs of the above results used the description of certain grothendieck groups as quotients of groups of character functions as well as results on modules introduced by swan in lo

algebro geometric finite gap solutions to the korteweg de vries equation as primitive solutions abstract in this paper we show that all algebro geometric finite gap solutions to the korteweg de vries equation can be realized as a limit of n soliton solutions as n diverges to infinity see remark 1 for the precise meaning of this statement this is done using the primitive solution framework initiated by dyachenko et al 2016 and zakharov et al 2016 25 26 one implication of this result is that the n soliton solutions can approximate any bounded periodic solution to the korteweg de vries equation arbitrarily well in the limit as n diverges to infinity we also study primitive solutions numerically that have the same spectral properties as the algebro geometric finite gap solutions but are not algebro geometric solutions

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

on the impact of combinatorial structure on congestion games we study the impact of combinatorial structure in congestion games on the complexity of computing pure nash equilibria and the convergence time of best response sequences in particular we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time we show that if the strategy space of each player consists of the bases of a matroid over the set of resources then the lengths of all best response sequences are polynomially bounded in the number of players and resources we also prove that this result is tight that is the matroid property is a necessary and sufficient condition on the players strategy spaces for guaranteeing polynomial time convergence to a nash equilibrium in addition we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games including first results about the hardness of market sharing games and congestion games for overlay network design our approach also yields a short proof for the pls completeness of network congestion games in particular we show that network congestion games are pls complete for directed and undirected networks even in case of linear latency functions

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

dynamics of the generalized kp mew burgers equation with external periodic perturbation the generalized kadomtsevpetviashvili modified equal width burgers kp mew burgers equation is introduced for the first time the qualitative change of the traveling wave solutions of the kp mew burgers equation is studied using numerical simulations considering an external periodic perturbation the periodic and chaotic motions of the perturbed kp mew burgers equation are investigated by using the phase projection analysis time series analysis poincare section and bifurcation diagram the parameter a nonlinear coefficient plays a crucial role in the periodic motions and chaotic motions through period doubling route to chaos of the perturbed kp mew burgers equation

evolutionary semigroups and dichotomy of linear skew product flows on locally compact spaces with banach fibers abstract we study evolutionary semigroups generated by a strongly continuous semi cocycle over a locally compact metric space acting on banach fibers this setting simultaneously covers evolutionary semigroups arising from non autonomous abstract cauchy problems and c 0 semigroups and linear skew product flows the spectral mapping theorem for these semigroups is proved the hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew product flow to this end a banach algebra of weighted composition operators is studied the results are applied in the study of roughness of the dichotomy dichotomy and solutions of nonhomogeneous equations green s function for a linear skew product flow pointwise dichotomy versus global dichotomy and evolutionary semigroups along trajectories of the flow

bifurcations phenomena in a nonlinear oscillator approximate analytical studies versus computer simulation results abstract the behavior of a driven single well oscillator with quadratic nonlinearity is studied and attention is focused on exploring a relationship between classic concepts of the approximate theory of nonlinear oscillations and the complex bifurcations and escape phenomena obtained by computer simulations an analytical analysis of the hill s type variational equation leads to approximate estimates for the blue sky catastrophe and period doubling instability and eventually results in the approximate predictive criteria for the main escape region the theoretical study and computer simulations are extended to the 1 2 subharmonic resonance and the analysis brings a new insight into the nonlinear phenomena which occur in the higher frequency region

an extension of the steepest descent method for riemann hilbert problems the small dispersion limit of the korteweg de vries kdv equation this paper extends the steepest descent method for riemann hilbert problems introduced by deift and zhou in a critical new way we present in particular an algorithm to obtain the support of the riemann hilbert problem for leading asymptotics applying this extended method to small dispersion kdv korteweg de vries equation we i recover the variational formulation of p d lax and c d levermore 1979 proc natl acad sci usa 76 3602 3606 for the weak limit of the solution ii derive without using an ansatz the hyperelliptic asymptotic solution of s venakides that describes the oscillations and iii are now able to compute the phase shifts integrating the modulation equations exactly the procedure of this paper is a version of fully nonlinear geometrical optics for integrable systems with some additional analysis the theory can provide rigorous error estimates between the solution and its computed asymptotic expression

on the harmonic response of an oscillator with unsymmetric restoring force abstract the approximate fundamental response of a driven mechanical oscillator with a quadratic stiffness characteristic is studied by using the method of harmonic balance the accuracy of the periodic solutions based on first approximation theory and their stability are compared with exact results obtained from direct time integration

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper

on chaotic wave patterns in periodically forced steady state kdvb and extended kdvb equations the periodically forced kdvb and extended kdvb equations are considered we investigate the structure of the totality of steady profiles the existence of profiles that are close to any shuffling o

periodic and chaotic behaviour in a reduced form of the perturbed generalized korteweg de vries and kadomtsev petviashvili equations abstract the dynamical behaviour of a reduced form of the perturbed generalized korteweg de vries and kadomtsev petviashvili equations extension of the korteweg de vries equation to two space variables are studied in this paper harmonic solutions of non resonance and primary resonance are obtained using the perturbation method chaotic motion under harmonic excitations is studied using the melnikov method a wide range of solutions for the reduced perturbed generalized korteweg de vries equations in which non linear phenomena appearing within transition from regular harmonic response periodic solutions to chaotic motion are obtained using the time integration runge kutta method when chaos is found it is detected by examining the phase plane the poincare map the sensitivity solution of the solution to initial conditions and by calculating the largest lyapunov exponent

chaos for a damped and forced kdv equation abstract the attractor for a damped and forced korteweg de vries equation is studied for large damping it is shown that the attractor is trivial with all solutions converging to a unique fixed point for small damping however the dynamics can be complicated it is investigated numerically how the attractor changes when the damping term decreases in particular a period doubling cascade of periodic solutions is found culminating with a chaotic regime the ratio between consecutive points in this cascade converging to feigenbaum s constant beyond this cascade the frequency spectrum of the solution is continuous and one lyapunov exponent is positive this seems to be the first work to obtain the onset of temporal chaos for this equation