continuity and minimization of spectrum related with the periodic camassa holm equation abstract an important point in looking for period solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y m t y the first aim of this paper is to study the dependence of eigenvalues for the periodic camassa holm equation on potentials as an infinitely dimensional parameter to be precise we prove that as nonlinear functionals of potentials eigenvalues for the periodic camassa holm equation are continuous in potentials with respect to the weak topologies in the l p lebesgue spaces the second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic camassa holm equation when the l 1 norm of potentials are given in order to make our results more applicable we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations

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

darboux transformation and multi soliton solutions of the camassa holm equation and modified camassa holm equation in this paper we propose a new approach to calculate multi soliton solutions of camassa holm ch equation and modified camassa holm mch equation with the aid of darboux transformation dt the new approach simplifies the approach presented by li and zhang proc r soc london ser a 460 2617 2627 2004 we first map the ch and mch equation to a negative order kdv nkdv equation by a reciprocal transformation then we proceed to apply the dt to solve the nkdv equation in the usual way finally we invert the reciprocal transformation to recover the solutions of the ch equation and mch equation

an isospectral problem for global conservative multi peakon solutions of the camassa holm equation we introduce a generalized isospectral problem for global conservative multi peakon solutions of the camassa holm equation utilizing the solution of the indefinite moment problem given by m g krein and h langer we show that the conservative camassa holm equation is integrable by the inverse spectral transform in the multi peakon case

extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls given a 1 periodic real potential q l1 r z we use 0 q to denote the smallest 1 periodic eigenvalue of the hill s equation x q t x 0 let b1 r be the ball centered at 0 of radius r in the l1 space l1 r z it is trivial that sup 0 q q b1 r r for all r 0 based on continuity of 0 q in q with the weak topology and continuous differentiability of 0 q in q with the l1 norm 1 we will apply scaling technique variational approach to extremal values in lp balls singular integrals and the limiting approach as p 1 to obtain i 0 q is bounded for q in any bounded set of l1 r z 1 and ii the minimal value l1 r inf 0 q q l1 r z q 1 r inf 0 q q l1 r z q 1 r is simply z0 1 r where z0 x 2 xtanh x 2 x 0 the extremal values of the smallest neumann eigenvalues for potentials in l1 balls are also found explicitly

heteroclinic bifurcation and singularly perturbed boundary value problems abstract we study a singularly perturbed boundary value problem in r m n x f x y e e y g x y e b 1 x 0 y 0 e 0 b 2 x 0 y 0 e 0 given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions similar results concerning the existence of periodic solutions relaxation oscillations are also presented several ideas from dynamical systems theory are employed e g exponential dichotomies fredholm alternatives and heteroclinic bifurcations

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

transversality of double pulse homoclinic orbits in the inviscid lorenz krishnamurthy system abstract we study a 5 dimensional system originally introduced by lorenz 1986 apart from its atmospheric interest this system also exhibits interesting dynamical behavior we introduce a general technique based on the fenichel coordinates introduced by jones and kopell 1994 which systematically tracks the evolution of the tangent space of the unstable manifold along a trajectory that spends a long time o log i where is a small parameter inside a neighborhood of a two dimensional normally hyperbolic invariant manifold the estimates for the components of tangent vectors are nontrivial precisely because of this long time more specifically we establish the transversality of the double pulse homoclinic orbits constructed by camassa 1993 transversality gives rise to chaotic behavior near the double pulse homoclinic orbits in particular there is a countable infinity of homoclinic orbits in the system with even number of pulses

the inverse spectral transform for the conservative camassa holm flow with decaying initial data we establish the inverse spectral transform for the conservative camassa holm flow with decaying initial data in particular it is employed to prove the existence of weak solutions for the corresponding cauchy problem

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

extremal values of eigenvalues of sturm liouville operators with potentials in l 1 balls abstract this paper is a continuation of zhang m zhang continuity in weak topology higher order linear systems of ode sci china ser a 51 2008 1036 1058 m zhang extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls j differential equations 246 2009 4188 4220 given a potential q l p 0 1 r p 1 we use m q to denote the m th dirichlet eigenvalue of the sturm liouville operator with potential q t where m n the minimal value l m p r and the maximal value m m p r of m q with potentials q in the l p ball of radius r are well defined in this paper we will exploit the continuity of m q in q with weak topologies and the variational method to give characterizations of l m p r and m m p r when p 1 by using the limiting approach as p 1 we find that the most important extremal values l m 1 r and m m 1 r can be evaluated explicitly using some elementary functions of r the corresponding extremal problems for neumann eigenvalues and some periodic eigenvalues will be reduced to l m p r and m m p r

on the geometry of an atmospheric slow manifold abstract we examine the hyperbolic structure and the invariant manifolds of a model proposed by lorenz to introduce the concept of an atmospheric slow manifold within the framework of dynamical system theory we address the question of the long time asymptotic behaviour of the system using the global geometric point of view it is shown that the model can be reduced to the classical example of a pendulum coupled to a harmonic oscillator the dynamical regimes of interest for the slow manifold hypothesis correspond to regions of phase space near the saddle center fixed point of this model which were not previously explored these phase space regions are analysed using a combination of melnikov type methods and ideas from singular perturbation theory by using the reversible symmetries of the model an extension of the melnikov theory is derived this extension allows us to find homoclinic orbits and determine their approximation by simply computing the zeros of a certain function which is constructed in terms of the usual melnikov function countable infinities of global homoclinic bifurcations and existence of chaotic dynamics can be shown to exist by using the new tool

dependence of solutions and eigenvalues of measure differential equations on measures abstract it is well known that solutions of ordinary differential equations are continuously dependent on finite dimensional parameters in equations in this paper we study the dependence of solutions and eigenvalues of second order linear measure differential equations on measures as an infinitely dimensional parameter we will provide two fundamental results which are the continuity and continuous frechet differentiability in measures when the weak topology and the norm topology of total variations for measures are considered respectively in some sense the continuity result obtained in this paper is the strongest one as an application we will give a natural simple explanation to extremal problems of eigenvalues of sturm liouville operators with integrable potentials

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

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

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

orbits homoclinic to resonances the hamiltonian case abstract in this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits hyperbolic fixed points or combinations of hyperbolic fixed points and or periodic orbits in a class of two degree of freedom integrable hamiltonian systems subject to arbitrary hamiltonian perturbations our methods differ from previous methods in that the invariant sets periodic orbits fixed points are created and become hyperbolic as a result of the interaction of the perturbation with a resonance in the unperturbed system this results in a very degenerate situation that requires a combination of geometric singular perturbation theory higher dimensional melnikov type methods and transversality theory we establish a simple energy phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance we apply our methods to a two mode truncation of the driven nonlinear schrodinger equation first studied by bishop et al in this example we show that as the energy is increased at resonance orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle node bifurcation of homoclinic orbits

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

on the spectral problem for the periodic camassa holm equation w x was recently derived 2 to describe the motion of solitary waves on shallow water it presents interesting features that do not arise in the well known korteweg de vries kdv model for example the solitary wave for the camassa holm equation has a corner at its peak the interaction of two w x solitary waves also presents striking differences with the case of kdv 2 let us consider the associated spectral problem in which the time t of the camassa holm equation appears only as a parameter

optimal lower bound for the first eigenvalue of the fourth order equation abstract in this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the l 1 norm of potentials is known we establish the minimization characterization for the first eigenvalue of the measure differential equation which plays an important role in the extremal problem of ordinary differential equation the conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

dynamical systems analysis of the maasch saltzman model for glacial cycles abstract this article is concerned with the internal dynamics of a conceptual model proposed by maasch and saltzman 1990 to explain central features of the glacial cycles observed in the climate record of the pleistocene epoch it is shown that in most parameter regimes the long term system dynamics occur on certain intrinsic two dimensional invariant manifolds in the three dimensional state space these invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the volume of the north atlantic deep water are well separated and they are center manifolds when these characteristic time scales are comparable in both cases the reduced dynamics on these manifolds are governed by bogdanov takens singularities and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles in addition knowledge of the reduced systems and their bifurcations is useful for understanding the effects of slowly varying parameters which cause passage through hopf bifurcations and of orbital milankovitch forcing both are central to the mechanism proposed by maasch and saltzman for the mid pleistocene transition in their model

multi bump orbits homoclinic to resonance bands we establish the existence of several classes of multi bump orbits homoclinic to resonance bands for completely integrable hamiltonian systems subject to small amplitude hamiltonian or dissipative perturbations each bump is a fast excursion away from the resonance band and the bumps are interspersed with slow segments near the resonance band the homoclinic orbits which include multi bump silnikov orbits connect equilibria and periodic orbits in the resonance band the main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

breakdown of the camassa holm equation the camassa holm equation is written in its eulerian form as displaystyle begin array rcl text ch quad frac partial v partial t v frac partial v partial x frac partial p partial x 0 end array with pressure p g v 2 frac 1 2 v 2 here g 1 d 2 1 frac 1 2 e vert x y vert it is easy to see that if at time t 0 v is odd with v x 0 for x 0 and v 0 0 then the slope s t v t 0 satisfies s bullet frac 1 2 s 2 and so is driven down to infty at some time t leq 2 s 0 in short the flow may break down this is the steepening lemma of camassa and holm 2 nothing worse happens v t x itself cannot jump like the usual kind of shock actually it is not the obvious v but rather the auxiliary function m v v that controls this m is a caricature of the vorticity that controls the behavior of honest euler in dimensions 2 and 3 see bertozzi and majda 1 for a full account

using the extended melnikov method to study the multi pulse global bifurcations and chaos of a cantilever beam the aim of this paper is to investigate the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended melnikov method in the resonant case first the extended melnikov method for studying the shilnikov type multi pulse homoclinic orbits and chaos in high dimensional nonlinear systems is briefly introduced in the theoretical frame then this method is utilized to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the cantilever beam how to employ this method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications is demonstrated through this example finally the results of numerical simulation are given and also show that the shilnikov type multi pulse chaotic motions can occur for the nonlinear non planar oscillations of the cantilever beam which verifies the analytical prediction

smooth linearization near a fixed point in this paper we extend a theorem of sternberg and bi leckii we study a vector field or a diffeomorphism in the vicinity of a hyperbolic fixed point we assume that the eigenvalues of the linear part a at the fixed point satisfy qth order algebraic inequalities where q 2 2 then there is ck linearization in the vicinity of the fixed point where k is the q smoothness of a we give an explicit and simple algorithm for com puting k an illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories we also show that if the vector field or diffeomorphism depends smoothly on a parameter then the linearizing conjugation de pends smoothly on the parameter

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

multipeakons of a two component modified camassa holm equation and the relation with the finite kac van moerbeke lattice abstract a spectral and the inverse spectral problem are studied for the two component modified camassa holm type for measures associated to interlacing peaks it is shown that the spectral problem is equivalent to an inhomogeneous string problem with dirichlet neumann boundary conditions the inverse problem is solved by stieltjes s continued fraction expansion leading to an explicit construction of peakon solutions sufficient conditions for global existence of solutions are given the large time asymptotics reveals that asymptotically peakons pair up to form bound states moving with constant speeds the peakon flow is shown to project to one of the isospectral flows of the finite kac van moerbeke lattice

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

lax integrability and the peakon problem for the modified camassa holm equation peakons are special weak solutions of a class of nonlinear partial differential equations modelling non linear phenomena such as the breakdown of regularity and the onset of shocks we show that the natural concept of weak solutions in the case of the modified camassa holm equation studied in this paper is dictated by the distributional compatibility of its lax pair and as a result it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the burgers type subsequently we give a complete construction of peakon solutions satisfying the modified camassa holm equation in the sense of distributions our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving stieltjes continued fractions and multi point pade approximations we propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds as well as toda like sorting property

filtered noise can mimic low dimensional chaotic attractors this contribution presents four results first calculations indicate that when examined by the grassberger procaccia algorithm alone filtered noise can mimic low dimensional chaotic attractors given the ubiquity of signal filtering in experimental investigations this is potentially important second a criterion is derived which provides an estimate of the minimum data accuracy needed to resolve the dimension of an attractor third it is shown that a criterion derived by eckmann and ruelle physica d 56 185 1992 to estimate the minimum number of data points required in a grassberger procaccia calculation can be used to provide a further check on these dimension estimates fourth it is shown that surrogate data techniques recently published by theiler and his colleagues in nonlinear modeling and forecasting edited by m casdagli and s eubanks addison wesley reading ma 1992 can successfully distinguish between linearly correlated noise and nonlinear structure these results and most particularly the first indicate that grassberger procaccia results must be interpreted with far greater circumspection than has previously been the case and that the algorithm should be used in combination with additional procedures such as calculations with surrogate data when filtered signals are examined by this algorithm alone a finite noninteger value of mathit d 2 is consistent with low dimensional chaotic behavior but it is certainly not a definitive diagnostic of chaos

on the isospectral problem of the dispersionless camassa holm equation abstract we discuss direct and inverse spectral theory for the isospectral problem of the dispersionless camassa holm equation where the weight is allowed to be a finite signed measure in particular we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite the results are applied to deduce several facts for the dispersionless camassa holm equation in particular we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by mckean

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

higher order melnikov theory for adiabatic systems in this paper we study adiabatic hamiltonian systems including those subject to small amplitude forcing and damping it is known that simple zeroes of the adiabatic poincare arnold melnikov function imply the existence of primary intersection points of the stable and unstable manifolds of hyperbolic orbits here we present an nth order melnikov function whose simple zeroes correspond to nth order transverse intersection points and hence to n pulse homoclinic orbits using this function it can be shown that n pulse homoclinic orbits arise in a plethora of adiabatic models including systems with slowly varying potentials the theory is illustrated on a damped hamiltonian system with a slowly varying cubic potential in addition the nth order adiabatic melnikov function is useful for showing the existence of multi pulse resonant periodic orbits in the special class of slow time periodic systems

multi pulse chaotic dynamics in non planar motion of parametrically excited viscoelastic moving belt abstract this paper investigates the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the parametrically excited viscoelastic moving belt using an extended melnikov method in the resonant case using the kelvin type viscoelastic constitutive law and hamilton s principle the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation applying the method of multiple scales and galerkin s approach to the partial differential governing equation the four dimensional averaged equation is obtained for the case of 1 1 internal resonance and primary parametric resonance from the averaged equations obtained the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues based on the explicit expressions of normal form the extended melnikov method is used for the first time to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics the paper demonstrates how to employ the extended melnikov method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications numerical simulations show that for the nonlinear non planar oscillations of the viscoelastic moving belt the shilnikov type multi pulse chaotic motions can occur overall both theoretical and numerical studies suggest that the chaos for the smale horseshoe sense in motion exists

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

quantitative analysis by renormalized entropy of invasive electroencephalograph recordings in focal epilepsy invasive electroencephalograph eeg recordings of ten patients suffering from focal epilepsy were analyzed using the method of renormalized entropy introduced as a complexity measure for the different regimes of a dynamical system the feature was tested here for its spatio temporal behavior in epilep tic seizures in all patients a decrease of renormalized entropy within the ictal phase of seizure was found furthermore the strength of this decrease is monotonically related to the distance of the recording location to the fo cus the results suggest that the method of renormalized entropy is a useful procedure for clinical applications like seizure detection and localization of epileptic foci 87 90 y

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

n pulse homoclinic orbits in perturbations of resonant hamiltonian systems in this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable two degree of freedom resonant hamiltonian systems our energy phase method applies to both hamiltonian and dissipative perturbations and reveals families of multi pulse solutions which are not amenable to melnikov type methods as an example we study a two mode approximation of the nonlinear nonplanar oscillations of a parametrically forced inextensional beam in this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

trace formulas and continuous dependence of spectra for the periodic conservative camassa holm flow abstract this article is concerned with the isospectral problem f 1 4 f z f z 2 f for the periodic conservative camassa holm flow where is a periodic real distribution in h loc 1 r and is a periodic non negative borel measure on r we develop basic floquet theory for this problem derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak topology

stability of peakons for the generalized modified camassa holm equation abstract in this paper we study orbital stability of peakons for the generalized modified camassa holm gmch equation which is a natural higher order generalization of the modified camassa holm mch equation and admits hamiltonian form and single peakons we first show that the single peakon is the usual weak solution of the pdes some sign invariant properties and conserved densities are presented next by constructing the corresponding auxiliary function h t x and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions the orbital stability of single peakon of the gmch equation is verified we introduce a new approach to prove the key inequality which is different from that used for the mch equation this extends the result on the stability of peakons for the mch equation qu et al 2013 36 successfully to the higher order case and is helpful to understand how higher order nonlinearities affect the dispersion dynamics

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

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere