complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

automatic 3 d interpretation of potential field data using analytic signal derivatives a new method has been developed for the automatic and general interpretation of gravity and magnetic data this technique based on the analysis of 3 d analytic signal derivatives involves as few assumptions as possible on the magnetization or density properties and on the geometry of the structures it is therefore particularly well suited to preliminary interpretation and model initialization processing the derivatives of the analytic signal amplitude instead of the original analytic signal amplitude gives a more efficient separation of anomalies caused by close structures moreover gravity and magnetic data can be taken into account by the same procedure merely through using the gravity vertical gradient the main advantage of derivatives however is that any source geometry can be considered as the sum of only two types of model contact and thin dike models in a first step depths are estimated using a double interpretation of the analytic signal amplitude function for these two basic models second the most suitable solution is defined at each estimation location through analysis of the vertical and horizontal gradients practical implementation of the method involves accurate frequency domain algorithms for computing derivatives with an automatic control of noise effects by appropriate filtering and upward continuation operations tests on theoretical magnetic fields give good depth evaluations for derivative orders ranging from 0 to 3 for actual magnetic data with borehole controls the first and second derivatives seem to provide the most satisfactory depth estimations

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

evanescent coupling between discs a model for near integrable tunnelling an analysis is provided of the splitting due to tunnelling of the energy levels in two dimensional double disk potentials the formal setting of this problem is similar to that of tunnelling between tori in the near integrable regime but is free of the difficulties arising from the existence of natural boundaries that are generic to such problems enabling a systematic investigation of the relevant semiclassical theory the semiclassical predictions are found to be consistent with exponentially accurate results obtained following reduction to a boundary value problem a numerical quantization is also performed and found to agree with the approximate results

chaos in hamiltonian systems modern developments in hamiltonian dynamics are described showing the change of view that has occurred in the last few decades the properties of mixed systems which exhibit both regular and chaotic motion are contrasted with those of the integrable systems for which the motion is entirely regular and of anosov systems for which it is almost everywhere chaotic the k a m theorem and problems of convergence are discussed

fourth order difference methods for hyperbolic ibvps abstract in this paper we consider fourth order difference approximations of initial boundary value problems for hyperbolic partial differential equations we use the method of lines approach with both explicit and compact implicit difference operators in space the explicit operator satisfies an energy estimate leading to strict stability for the implicit operator we develop boundary conditions and give a complete proof of strong stability using the laplace transform technique we also present numerical experiments for the linear advection equation and burgers equation with discontinuities in the solution or in its derivative the first equation is used for modeling contact discontinuities in fluid dynamics the second one is used for modeling shocks and rarefaction waves the time discretization is done with a third order runge kutta tvd method for solutions with discontinuities in the solution itself we add a filter based on second order viscosity in case of the non linear burgers equation we use a flux splitting technique that results in an energy estimate for certain difference approximations in which case also an entropy condition is fulfilled in particular we shall demonstrate that the unsplit conservative form produces a non physical shock instead of the physically correct rarefaction wave in the numerical experiments we compare our fourth order methods with a standard second order one and with a third order tvd method the results show that the fourth order methods are the only ones that give good results for all the considered test problems

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

a new conservative unsplit method for the solution of the vlasov equation we have developed a new conservative method for solving the vlasov equation without using any splitting technique our goal is to maintain the positivity of the distribution function and to avoid un physical oscillations which might lead to numerical instabilities based on a finite volume conservative discretization of the conservative form of the vlasov equation we implemented a highly accurate second order upwind scheme in order to avoid un physical oscillations and their possible numerical instability we apply a flux limiter in the second order we validate our new vlasov solver by considering standard cases of one dimensional current driven ion acoustic instabilities solving a vlasov ampere set of equations

outflow boundary conditions for the fourier transformed two dimensional vlasov equation in order to facilitate numerical simulations of plasma phenomena where kinetic processes are important we have studied the technique of fourier transforming the vlasov equation analytically in velocity space and solving the resulting equation numerically particular attention has been paid to the boundary conditions of the fourier transformed system by using outgoing wave boundary conditions in the fourier transformed space small scale information in velocity space is carried outside the computational domain and is removed representing a dissipative loss mechainsm thereby the so called recurrence phenomenon is reduced in the present article a previously developed method in one spatial and one velocity dimension plus time is generalised to two spatial and two velocity dimensions plus time different high order methods are used for computing derivatives as well as for the time stepping

critical kam circles and the brjuno function the authors consider a complex hamiltonian map and show how to approximate the critical value of the perturbation parameter at which a given kam circle disappears by means of a function the brjuno function which only depends on the continued fraction expansion of the rotation number

singularity analysis by pade approximants of some holomorphic maps the analysis of the singularity structure of some series arising in the conjugation theory of holomorphic maps is performed using the pade approximants the validity of pa is checked for functions whose analytic structure is known one observes that the numerical results are extremely sensitive to errors due to the use of a finite precision normal or double precision used in fortran codes is in most cases not sufficient to perform a numerical analysis of the singularity structure in order to have significant results at high orders all the computations were carried out using codes which allow us to operate with a sufficiently large number of digits the case of linearizable diffeomorphisms and of mappings tangent to the identity of c o was considered pa of both the direct and the inverse conjugating functions of a quadratic map to its normal form were computed and a comparison was performed with analytical results when available when the singularity pattern was unknown the poles and the zeros of the pa provided a coherent picture which in some cases allowed rigorous results to be established

tunnelling in near integrable systems tunnelling rates are qualitatively different for integrable and near integrable systems here a uniform result is derived which interpolates between the two regimes and is applied successfully to two dimensional double well and pendulum potentials when the underlying symmetry is reflection in a single coordinate the splitting remains positive after perturbation but for potentials whose underlying symmetry is reflection through the origin the splitting is predicted to oscillate as a function of a perturbation parameter with regularly spaced zeros where tunnelling switches off

singularity spectrum of a fractal diagram for a hamiltonian system we determine numerically the relevant spectrum of scaling indices for the fractal diagram of the standard map infinite partitions related by the gauss transformation and an appropriate measure had to be used in order to obtain convergent results this choice of the measure and the partitions is motivated by the method of modular smoothing

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

greene s residue criterion for the breakup of invariant tori of volume preserving maps abstract invariant tori play a fundamental role in the dynamics of symplectic and volume preserving maps codimension one tori are particularly important as they form barriers to transport such tori foliate the phase space of integrable volume preserving maps with one action and d angles for the area preserving case greene s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits even though kam theory applies to the three dimensional case the robustness of tori in such systems is still poorly understood we study a three dimensional reversible volume preserving analogue of chirikov s standard map with one action and two angles we investigate the preservation and destruction of tori under perturbation by computing the residue of nearby periodic orbits we find tori with diophantine rotation vectors in the spiral mean cubic algebraic field the residue is used to generate the critical function of the map and find a candidate for the most robust torus

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

toward a three dimensional automatic interpretation of potential field data via generalized hilbert transforms fundamental relations the paper extends to three dimensions 3 d the two dimensional 2 d hilbert transform relations between potential field components for the 3 d case it is shown that the hilbert transform is composed of two parts with one part acting on the x component and one part on the y component as for the previously developed 2 d case it is shown that in 3 d the vertical and horizontal derivatives are the hilbert transforms of each other the 2 d cauchy riemann relations between a potential function and its hilbert transform are generalized for the 3 d case finally the previously developed concept of analytic signal in 2 d can be extended to 3 d as a first step toward the development of an automatic interpretation technique for potential field data

combining spatial components and hilbert transforms to interpret ground time domain electromagnetic data abstractwe have developed a method for displaying or imaging data from a ground time domain electromagnetic system and for extracting the geometric parameters of a small conductor the parameters are determined directly from the data using combinations of the spatial components of the secondary fields and their hilbert transforms the position of the target coincides with the peaks of the energy envelope ee or the t component of the response here the ee is the square root of the sum of the squares of the three spatial components and their hilbert transforms whereas the t component response is an analogous quantity that excludes the hilbert transform terms studies on synthetic models indicate that the t component response is sharper than the ee in most possible target orientations once the position of a body has been determined using the peak of the t component response the dip of the target can be quantified using the ratio of the full width at half magnitude fwhm of the t component response and

nonresonant limit for sequences of resonant orbits the case of holomorphic maps the approximation of a nonresonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane the problem is motivated by hamiltonian dynamics greene s conjecture and we consider a complexified hamiltonian map in the region far from the section of real dynamics where it can be reduced to a holomorphic map of a single complex variable for a sequence of maps in normal form with linear resonant frequencies the limit to a linear map with nonresonant diophantine frequency has a simple interpretation the flower like resonant orbits become circles due to the increase of the number of petals and the freezing of radial motion a similar non trivial result is proved for small perturbations of the normal forms by investigating the behaviour of the conjugation functions

an obstruction method for the destruction of invariant curves abstract an important point in a conservative dynamical system which is a parameter depending perturbation of an integrable system is the destruction of the invariant curves for small perturbations many invariant curves subsist according to kam theorem an increase in the size of perturbations produces eventually the destruction of the curves the stochastic zone is then very large in this paper we present for the standard map a geometrical approach to obtain values of the parameter for which no invariant curves exist this is due to the existence of heteroclinic points as it has been observed using other methods there are several scaling properties in the phenomena which are also discussed here

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

multi dimensional fully implicit spectral method for the vlasov maxwell equations with exact conservation laws in discrete form a spectral method for the numerical solution of the multi dimensional vlasov maxwell equations is presented the plasma distribution function is expanded in fourier for the spatial part and hermite for the velocity part basis functions leading to a truncated system of ordinary differential equations for the expansion coefficients moments that is discretized with an implicit second order accurate crank nicolson time discretization the discrete non linear system is solved with a preconditioned jacobian free newton krylov method it is shown analytically that the fourier hermite method features exact conservation laws for total mass momentum and energy in discrete form standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically the whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities some preconditioning strategies are presented showing that the number of linear iterations of the krylov solver can be drastically reduced and a significant gain in performance can be obtained

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

a splitting algorithm for vlasov simulation with filamentation filtration a fourier fourier transformed version of the splitting algorithm for simulating solutions of the vlasov poisson system of equations is introduced it is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm it is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle in cell counterpart it is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

the usefulness of multicomponent time domain airborne electromagnetic measurements time domain airborne electromagnetic aem systems historically measure the inline horizontal x component new versions of the electromagnetic systems are designed to collect two additional components the vertical z component and the lateral horizontal y component to provide greater diagnostic information in areas where the geology is near horizontal the z component response provides greater signal to noise particularly at late delay times this allows the conductivity to be determined to greater depth in a layered environment the symmetry implies that the y component will be zero hence a nonzero y component will indicate a lateral inhomogeneity the three components can be combined to give the energy envelope of the response over a vertical plate the response profile of this envelope has a single positive peak and no side lobes the shape of the energy envelope is dependent on the flight direction but less so than the shape of the x component response profile in the interpretation of

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

a critical comparison of eulerian grid based vlasov solvers a common problem with direct vlasov solvers is ensuring that the distribution function remains positive a related problem is to guarantee that the numerical scheme does not introduce false oscillations in velocity space in this paper we use a variety of schemes to assess the importance of these issues and to determine an optimal strategy for eulerian split approaches to vlasov solvers from these tests we conclude that maintaining positivity is less important than correctly dissipating the fine scale structure which arises naturally in the solution to many vlasov problems furthermore we show that there are distinct advantages to using high order schemes i e third order rather than second a natural choice which satisfies all of these requirements is the piecewise parabolic method ppm which is applied here to vlasov s equation for the first time

invariant curves for area preserving twist maps far from integrable one parameter families of area preserving twist maps of the formf e x y x y ef x y ef x are considered various invariant curves for the maps corresponding tof x sin andf x sinx 1 50 sin 5x are rigorously constructed forlarge values of the nonlinearity parameter for larger values of close to critical some numerical experiments are briefly discussed

breakdown of lindstedt expansion for chaotic maps in a previous paper of one of us europhys lett 59 330 336 2002 the validity of greene s method for determining the critical constant of the standard map sm was questioned on the basis of some numerical findings here we come back to that analysis and we provide an interpretation of the numerical results by showing that the conclusions of that paper were wrong as they relied on a plausible but untrue assumption hence no contradiction exists with respect to greene s method we show that the previous results based on the expansion in lindstedt series do correspond to the critical constant but for a different map the semi standard map ssm for such a map no greene s method analog is at disposal so that methods based on lindstedt series are essentially the only possible ones moreover we study the expansion for two simplified models obtained from the sm and ssm by suppressing the small divisors we call them the simplified sm and simplified ssm respectively the first case turns out to be re

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

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

outflow boundary conditions for the fourier transformed three dimensional vlasov maxwell system a problem with the solution of the vlasov equation is its tendency to become filamented oscillatory in velocity space which in numerical simulations can give rise to unphysical oscillations and recurrence effects we present a three dimensional vlasov maxwell solver three spatial and velocity dimensions plus time in which the vlasov equation is fourier transformed in velocity space and the resulting equations solved numerically by designing absorbing outflow boundary conditions in the fourier transformed velocity space the highest fourier modes in velocity space are removed from the numerical solution this introduces a dissipative effect in velocity space and the numerical recurrence effect is strongly reduced the well posedness of the boundary conditions is proved analytically while the stability of the numerical implementation is assessed by long time numerical simulations well known wave modes in magnetized plasmas are shown to be reproduced by the numerical scheme

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

the measure of chaos by the numerical analysis of the fundamental frequencies application to the standard mapping abstract the method of analysis of the chaotic behaviour of a dynamical system by the numerical analysis of the fundamental frequencies developed for the study of the stability of the solar system j laskar icarus 88 1990 is presented here with application to the standard mapping this method is well suited for weakly chaotic motion with any number of degrees of freedom and is based on the analysis of the variations with time of the fundamental frequencies of an hamiltonian system it allows to give an analytical representation of the solution when it is regular to detect if an orbit is chaotic over a smaller time span than with the lyapunov exponents and gives also an estimate of the size of the chaotic zones in the frequency domain the frequency analysis also provides a numerical criterion for the destruction of invariant curves its application to the standard mapping shows that the golden curve does not survive for a 0 9718 which is very close and compatible with greene s value ac 0 971635

rigorous estimates for a computer assisted kam theory nonautonomous hamiltonian systems of one degree of freedom close to integrable ones are considered let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given kam kolmogorov arnold moser torus breaks down a computer assisted method that allows one to give rigorous lower bounds for e c is presented this method has been applied in celletti falcolini porzio to be published in ann inst h poincare to the escande and doveil pendulum yielding a bound which is within a factor 40 2 of the value indicated by numerical experiments