nonlinear inversion of geoelectric data acquired across 3d objects using a finite element approach we introduce a new finite element based scheme for the fast nonlinear inversion of large 3d geoelectric data sets acquired around isolated objects or across the earth s surface the principal novelty of this scheme is the combination of a versatile finite element approach with 1 a method involving minimization of an objective function using a conjugate gradient algorithm that includes an adjoint field technique for efficiently establishing the objective function gradient and 2 parabolic interpolation for estimating suitable inversion step lengths this scheme is capable of handling large volumes of data acquired using diverse electrode configurations located around or across 3d structures only three solutions to the forward problem are required for each iteration computation of the jacobian matrix which might require computers with a large amount of memory is not necessary to minimize artificial irregularities in the inverted models particularly near the electrodes we smooth the model parameter

bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

3d finite element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids abstractwe present a finite element solution to the 3d electromagnetic forward modeling problem in the frequency domain the method is based on decomposing the electric field into vector and scalar potentials in the helmholtz equation and in the equation of conservation of charge edge element and nodal element basis functions were used respectively for the vector and scalar potentials this decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries therefore finding an efficient solution to the problem the computational domain was subdivided into unstructured tetrahedral elements the system of equations was discretized using the galerkin variant of the weighted residuals method with the approximated vector and scalar potentials as the unknowns of a sparse linear system a generalized minimum residual solver with an incomplete lu preconditioner was use

stability and non linear responses of a rotor bearing system with pedestal looseness abstract vibration characteristics of a rotor bearing system with pedestal looseness are investigated a non linear mathematical model containing stiffness and damping forces with tri linear forms is considered the shooting method is used to obtain the periodic solutions of the system stability of these periodic solutions is analyzed by using the floquet theory period doubling bifurcation and naimark sacker bifurcation are found finally the governing equations are integrated using the fourth order runge kutta method different forms of periodic quasi periodic and chaotic vibrations are observed by taking the rotating speed and imbalance as the control parameter three kinds of routes to or out of chaos that is period to chaos quasi periodic route and intermittence are found

godel type space times in induced matter gravity theory five dimensional 5d generalized godel type manifolds are examined in the light of the equivalence problem techniques as formulated by cartan the necessary and sufficient conditions for local homogeneity of these 5d manifolds are derived the local equivalence of these homogeneous riemannian manifolds is studied it is found that they are characterized by three essential parameters k m2 and identical triads k m2 correspond to locally equivalent 5d manifolds an irreducible set of isometrically nonequivalent 5d locally homogeneous riemannian generalized godel type metrics are exhibited a classification of these manifolds based on the essential parameters is presented and the killing vector fields as well as the corresponding lie algebra of each class are determined it is shown that the generalized godel type 5d manifolds admit maximal group of isometry gr with r 7 r 9 or r 15 depending on the essential parameters k m2 and the breakdown of causality in all these classes of homogeneous

universal transition in a dynamical system forced at two incommensurate frequencies abstract the various routes to chaos are explored for a nonlinear mechanical system mode locked to one of two incommensurate external frequencies two types of transitions are seen in our model system a saddle node transition with its associated intermittency can occur when the mode locking is lost while the attracting two torus remains smooth a less trivial transition can occur in which the attracting torus roughens and the power spectrum of the time series develops singular low frequency components after the breakdown mode locking persists with noisy small scale motions about the former torus at the multicritical point where these two transition lines meet scaling and universal low frequency power spectra are observed a renormalization group treatment is proposed the analysis might also be applicable to transitions in rotationally invariant systems

persistent properties of bifurcations abstract this paper reviews results about bifurcations which occur in families of differential equations persistent properties are defined to be those which remain when the family of equations is perturbed we provide a list of such properties which is relevant for numerical studies of dynamical systems

an investigation of finite element modeling for electrical and electromagnetic data in three dimensions the finite element method can be used to solve the differential equations which describe electrical and electromagnetic em field behavior the equations are respectively poisson s equation and the vector damped wave equation the finite element equations are derived in both cases using the minimum theorem while both tetrahedral and hexahedral elements may be used for the modeling of the resistivity problem only hexahedral elements give satisfactory results for the em problem a disadvantage of the relatively simple mesh design used in the approach described here is the presence of long thin elements such elements have very poor interpolating properties and they adversely affect the rate of convergence of the overrelaxation technique used in solving the resulting system of linear equations for the modeling of resistivity data over an earth with one plane of symmetry the system of equations typically has about 9000 unknowns about 50 000 unknowns are needed to give a satisfactory solution to an

two periodical and quasi periodical wave solutions of the kuramoto sivashinsky equation and their stability and bifurcations abstract a nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media e g in problems of fluid film flow hydrodynamics wave solutions regular in space and both periodical and quasi periodical in time generating from steady state and steady state travelling waves have been found numerically stability of some solutions has been investigated and bifurcation analysis has been carried out the analysis has demonstrated that there is a sequence of bifurcations of solutions stable with respect to disturbances of the same spatial period and it has been shown that the bifurcations are related to the loss of some symmetries of the initial solution it is also shown that the regions of quiet time behaviour and those of intensive variations are separated in the temporal period of solutions with the bifurcation parameter increasing

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

riemannian space times of godel type in five dimensions the five dimensional 5d riemannian godel type manifolds are examined in light of the equivalence problem techniques as formulated by cartan the necessary and sufficient conditions for local homogeneity of these 5d manifolds are derived the local equivalence of these homogeneous riemannian manifolds is studied it is found that they are characterized by two essential parameters m2 and identical pairs m2 correspond to locally equivalent 5d manifolds an irreducible set of isometrically nonequivalent 5d locally homogeneous riemannian godel type metrics are exhibited a classification of these manifolds based on the essential parameters is presented and the killing vector fields as well as the corresponding lie algebra of each class are determined it is shown that apart from the m2 4 2 0 and m2 0 0 classes the homogeneous riemannian godel type manifolds admit a seven parameter maximal group of isometry g7 the special class m2 4 2 0 and the degenerated godel type class m2 0 0 a

the calculation of lyapunov spectra abstract linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems the naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations this information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration in this paper we extract information from this matrix through its singular values and the associated orthogonal matrices this given an algebraic foundation to the geometric methods that have been used previously the equations of the orthogonal principal stretching and contraction directions show that asymptotically after a long time they depend only on the location in phase space and not on the orbit as an example we show that these directions sketch out the attractor of the lorenz model

computation of invariant tori by the method of characteristics in this paper we present a technique for the numerical approximation of a branch of invariant tori of finite dimensional ordinary differential equations systems our approach is a discrete version of the graph transform technique used in analytical work by fenichel indiana univ math j 21 1971 pp 193 226 in contrast to our previous work l dieci j lorenz and r d russell siam j sci statist comput 12 1991 pp 607 647 the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori but determines and updates such a coordinate system during a continuation process we give general convergence results for the method and present its algorithmic description we also show how the method performs on two physically important nonlinear problems a system of two coupled oscillators and the forced van der pol oscillator in the latter case we discuss some modifications needed to approximate an invariant curve for the poincare map

computation continuation and bifurcation of torus solutions for dissipative maps and ordinary differential equations abstract a periodic solution can be considered as a fixed point of a poincare map we shall demonstrate how we can generalize this fixed point equivalence to quasi periodic solutions with two periods thus the quasi periodic solution or bi periodic solution will be a fixed point of a more general poincare map the stability of the bi periodic solution is then equivalent to the stability of the fixed point we demonstrate how we can follow the fixed point in dependence of any control parameter in the system furthermore we demonstrate both how the bi periodic solution may bifurcate to more complicated solutions and how it may arise as a bifurcation from a periodic solution our work is algorithmic in spirit and our statements are based on numerical evidence

the use of summary representation for electromagnetic modeling the method of summary representation developed by g n polozhii is a quasi analytical method for solving self adjoint finite difference boundary value problems expressed on regular meshes in principle the method should allow considerable savings in computing time as well as improved accuracy when compared to commonly used finite difference schemes we have used summary representation as the basis for a new hybrid scheme to solve the two dimensional helmholtz equation for electromagnetic modeling the theory behind this hybrid scheme is presented preliminary results for the two dimensional problem show that substantial computing time and storage savings can be made

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

computer aided study of a class of riemannian space times the riemannian godel type manifolds are examined in the light of the equivalence problem techniques by using algebraic computing the conditions for space time homogeneity of a riemannian manifold with a godel type metric are derived generalizing previous work on this subject matter a classification of the godel type riemannian manifolds based on the two relevant parameters m and is formulated it is shown that apart from the m2 4 2 case they are all petrov type d with a five parameter group of motions the special m2 4 2 manifold is shown to be conformally flat and to admit a seven parameter group of isometries an algebraic classification of the ricci spinor of all godel type riemannian manifolds is discussed possible sources for these space times are examined a generalization of the reboucas tiomno theorem on godel type manifolds is given

hybrid calculations of the three dimensional electromagnetic response of buried conductors the hybrid method for computing the electromagnetic response of a three dimensional conductor in a layered conducting half space consists of solving a finite element problem in a localized region containing the conductor and using integral equation methods to obtain the fields outside that region the original scheme obtains the boundary values by iterating between the integral equation solution and the finite element solution after making an initial guess based on primary values from the field a two dimensional interpolation scheme is then used to speed the evaluation of the 105 to 106 green s function convolution integrals required by most problems the two algorithms presented are modifications of the original scheme both contain a search routine to identify a set of unique points where the convolution integral evaluations are required by replacing the two dimensional interpolation with a one dimensional interpolation and reading the convolution integrals from a reference table computation time

quasi periodic solutions calculated with the simple shooting technique abstract a shooting type numerical method for calculating quasi periodic solutions in a multiexcited system by using derivatives and with an improved interpolation technique is presented three examples of ordinary differential equations and iterative maps show the efficiency of the new algorithm this algorithm can also be effectively applied to a periodic system

classification of second order symmetric tensors in five dimensional kaluza klein type theories an algebraic classification of second order symmetric tensors in five dimensional kaluza klein type lorentzian spaces is presented by using jordan matrices it is shown that the possible segre types are 1 1111 2111 311 zz111 and the degeneracies thereof a set of canonical forms for each segre type is found the possible continuous groups of symmetry for each canonical form are also studied

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

singularity removal a refinement of resistivity modeling techniques modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available a singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth the inability to adequately represent in discrete space a discontinuous function in this case the delta function describing the introduction of current at a point commonly results in numerical error near the source of a modeled singularity inspection of an integrated finite difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin a procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step thus preventing it from contributing to the numerical error using this procedure the average error in apparent resistivity values for a model of a polar dipole traverse over a nonconducting sphere is reduced by 40 percent for a dipole dipole traverse of a two layer model the error decreases by 75 percent and in the case of a wenner profile of a model of a vertically faulted earth the average error is diminished by 90 percent

high order synchronization transitions and competition among arnold tongues in a rotator under harmonic forcing we consider a rotator whose equation of motion for the angle theta consists of the zeroth and first fourier modes numerical analysis based on the trailing of saddle node bifurcations is used to locate the n 1 arnold tongues where synchronization occurs several of them are wide enough for high order synchronization to be seen in passive observations by sweeping the system parameters within a certain range we find that the stronger the dependence of theta over on theta the wider the regions of synchronization use of a synchronization index reveals a vast number of very narrow n m arnold tongues a competition phenomenon among the tongues is observed in that they push and squeeze one another as some tongues widen others narrow two mechanisms for transitions between different n m synchronization states are considered slow variation of the driving frequency and the influence of low frequency noise on the rotator

continuation of quasi periodic solutions with two frequency harmonic balance method the continuation of quasi periodic solutions for autonomous or forced nonlinear systems is presented in this paper the association of the asymptotic numerical method a robust continuation method and a two frequency harmonic balance method is performed thanks to a quadratic formalism there is no need for a priori knowledge of the solution the two pulsations can be unknown and can vary along the solution branch and the double fourier series are computed without needing a harmonic selection a norm criterion on fourier coefficients can confirm a posteriori the accuracy of the solution branch on a forced system frequency locking regions are approximated without blocking the continuation process the continuation of these periodic solutions can be done independently on an autonomous system an example of solution is shown where the number of fourier coefficients is increased to improve the accuracy of the solution

a note on godel type space times the godel type space times are examined under different conditions on the algebraic structure of the energy momentum tensor a generalization of the bampi zordan theorem on godel type models is given chakraborty s results are recast as a special case of the generalized bampi zordan theorem the reboucas tiomno solution is shown to be the unique godel type metric with an algebraic structure of the tachyon fluid type

quasiperiodically forced dynamical systems with strange nonchaotic attractors abstract we discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems we do this by examining a particular model differential equation cos e cos 2 t where is a two frequency quasiperiodic function of t when 0 the analysis of the equation is facilitated since then it can be related to the schrodinger equation with quasiperiodic potential we show that the equation does indeed exhibit strange nonchaotic attractors and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive lebesgue measure in parameter space the equation also exhibits two and three frequency quasiperiodic behavior we also show that the strange nonchaotic attractors have distinctive frequency spectrum this property might make them experimentally observable

fourier analysis of multi frequency dynamical systems abstract the fourier spectrum of a multiply periodic system appears complex due to resonances at integer combinations of the fundamental frequencies we show how to reorder the peaks in the fourier spectrum to yield a simpler structure which reveals important aspects of the time series which are not clearly seen in the ordinary fourier transform as examples we study time series from circle and torus maps and from experimental data by gollub and benson

fractional diffusion analysis of the electromagnetic field in fractured media part i 2d approach we address a 2d finite difference fd frequency domain modeling algorithm based on the theory of fractional diffusion of electromagnetic em fields which is generated by an infinite line source lying above a fractured geological medium the presence of fractures in the subsurface usually containing highly conductive pore fluids gives rise to spatially hierarchical flow paths of induced em eddy currents the diffusion of em eddy currents in such formations is anomalous generalizing the classical gaussian process described by the conventional maxwell equations based on the continuous time random walk ctrw theory the diffusion of em eddy currents in a rough medium is governed by the fractional maxwell equations here we model the em response of a 2d subsurface containing fractured zones based on the fractional maxwell equations the governing equation in the frequency domain is discretized using the fd approach the resulting equation system is solved by the multifrontal massively parallel solver mumps we find excellent agreement between the fd and analytic solutions for a rough half space model then fd solutions are calculated for a 2d fault zone model with variable conductivity and roughness we illustrate a case in which a rough fault zone would not be resolved by classical diffusion modeling even if its conductivity contrasts with the background

treatment of singularity in the method of boundary integral equations for 2 5d electromagnetic modeling abstractfast and accurate simulation of responses of electromagnetic logging while drilling tools in 2d and 3d formations is important for further development of the proactive geosteering service which is currently based on the 1d layered model inversion in search of an optimal method of forward simulation we have developed boundary integral equations in a 2d formation model with planar boundaries a peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy and hence the computation time increases as the transmitter approaches the boundary to retain efficiency in the case of transmitter boundary proximity we use the following modification of the method the main term of the calculated electromagnetic field namely the solution for the two layered model corresponding to the closest boundary is calculated by the well known explicit formulas whereas the anomalous field is found from the system of integral equations the modified method is implemented and tes

bifurcation analysis of a periodically forced relaxation oscillator differential model versus phase resetting map we compare the dynamics of the periodically forced fitzhugh nagumo oscillator in its relaxation regime to that of a one dimensional discrete map of the circle derived from the phase resetting response of this oscillator the phase resetting map the forcing is a periodic train of gaussian shaped pulses with the width of the pulses much shorter than the intrinsic period of the oscillator using numerical continuation techniques we compute bifurcation diagrams for the periodic solutions of the full differential equations with the stimulation period being the bifurcation parameter the period 1 solutions which belong either to isolated loops or to an everywhere unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes merge together to form a single branch at larger stimulation amplitudes as a consequence of the fast slow nature of the oscillator this merging occurs at virtually the same stimulation amplitude for all the period 1 loops again using continuation we show that this stimulation amplitude corresponds in the circle map to a change of topological degree from one to zero we explain the origin of this coincidence and also discuss the translational symmetry properties of the bifurcation diagram

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

the nonlinear dynamics of the damped and driven toda chain 1 energy bifurcation diagrams abstract the dynamics of a periodically driven damped toda chain with periodic boundary conditions is investigated with the help of energy cross sections it can be shown that besides periodic motions of different periodicity quasiperiodic and aperiodic chaotic motions exist the method of energy bifurcation diagrams is introduced which allows a survey of the resonance properties of the chain

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

numerical computation of invariant circles of maps abstract a method for the numerical computation of invariant circles of maps is presented along with appropriate techniques for its implementation the method involves solution of a functional equation by discretization and newton iteration the resulting algorithm is applied to a map of the cylinder and some examples of bifurcations of invariant circles are illustrated generalization of the method to the computation of invariant circles in more than two dimensions as well as sections of invariant tori is discussed

efficient 2 5d electromagnetic modeling using boundary integral equations abstractfast and accurate simulation of responses of logging while drilling lwd electromagnetic em tools in complex 2d and 3d formations is very important for reconstruction of the resistivity distribution in proactive geosteering currently real time interpretation is based on the 1d parametric inversion advances in fast simulation beyond the 1d model would open a way for real time 2d inversion we developed implemented numerically and tested an efficient method for simulation of lwd em tools in complex 2d formations with an arbitrary 3d position of the tool the method is based on the boundary integral equations for the tangential components of the field and the fourier transform which reduces the problem to a series of 1d integral equations computations are carried out simultaneously for the whole set of measurement points with the same matrix which provides a short computation time per point we verified the method by comparing our results with those obtained by the well known explicit 1d m

computation of quasi periodic solutions of forced dissipative systems ii abstract the problem of finding a t periodic solution of a forced dissipative system of ordinary differential equations is most conveniently reformulated as a fixed point problem of a poincare map mapping the phase space at t 0 into the phase space at t t and the stability of the periodic response is equivalent to the stability of the fixed point it is shown how the problem of determining quasi periodic solutions of forced systems with two forcing frequencies may be reformulated as a fixed point problem of a new type of poincare map that also opens the possibility of treating quasi periodic solutions with more than two frequencies as well as treating bifurcating quasi periodic solutions of autonomous equations three examples are considered one in which the exact solution is known and two others where the quasi periodic solutions have been determined by other means

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

notes on a class of homogeneous space times the breakdown of causality in homogeneous godel type space time manifolds is examined an extension of reboucas tiomno rt and accioly goncalves studies is made the existence of noncausal curves is also investigated under two different conditions on the energy momentum tensor an integral representation of the infinitesimal generators of isometries is obtained extending previous works on the rt geometry

on the global structure of period doubling flows abstract for a wide class of period doubling flows on r3 we analyze the global structure of the invariant manifolds and the topology of the bifurcating periodic orbits we emphasize aspects of the dynamics which are not visible in an analysis of the associated poincare return map the global manifold structure implies constraints for the subsequent bifurcational behavior of the flow the period doubled orbits are classified using the theory of iterated torus knots this classification reveals an infinite number of topologically distinct period doubling flows and raises the important question as to which sequences of torus knots appear in the asymptotic limit of a period doubling cascade if the asymptotic sequence of torus knots produced is not unique then the self similarity of the poincare map does not imply a self similar structure for the flow in particular at the accumulation point for the cascade one would not expect to observe a universal power spectrum for the flow this possibility is of some experimental interest

attractors on an n torus quasiperiodicity versus chaos the occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance however current understanding concerning the question of how prevalent such motions should be is incomplete with this in mind the types of attractors which can exist for flows on an n torus are studied numerically for n 3 and 4 specifically nonlinear perturbations are applied to maps representing n frequency quasiperiodic attractors these perturbations can cause the original n frequency quasiperiodic attractors to bifurcate to other types of attractors our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows n frequency quasiperiodic attractors are the most common followed by n 1 frequency quasiperiodic attractors followed by periodic attractors however as the nonlinearity is further increased n frequency quasiperiodicity becomes less common ceasing to occur when the map becomes noninvertible chaotic attractors are very rare for n 3 for small to moderate nonlinearity but are somewhat more common for n 4 examination of the types of chaotic attractors that occur for n 3 reveals a rich variety of structure and dynamics in particular we see that there are chaotic attractors which apparently fill the entire n torus i e limit sets of orbits on these attractors are the entire torus furthermore these are the most common types of chaotic attractors at moderate nonlinearities

mode locking in a periodically forced resonate and fire neuron model the resonate and fire rf model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system impact oscillator we analyze the response of the rf neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one dimensional time map based on such a firing map we describe mode locked solutions and their stability leading to the so called arnol d tongues the boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow despite the fact that the periodically driven rf system shows periodic firing its behavior may become chaotic when the forcing frequency is near the resonant frequency we compare these results to numerical simulations of the model undergoing sinusoidal forcing furthermore upon varying a system parameter the rf system can be reduced to the integrate and fire system and in this case we show the consistency of the results on mode locked solutions

the nonlinear dynamics of the damped and driven toda chain iii classification of the nonlinear resonances and local bifurcations abstract the dynamics of damped and driven toda chains is investigated it is shown that the resonance and bifurcation scenario of alternating saddle node and period doubling bifurcations presented in the first part of this series on the nonlinear dynamics of the damped and driven toda chain physica d 31 1988 103 116 can be explained by the specific structure of the eigenvalue spectrum of the associated linearized flow map dx t

a finite element solution for the transient electromagnetic response of an arbitrary two dimensional resistivity distribution we present a numerical method for solving maxwell s equations in the case of an arbitrary two dimensional resistivity distribution excited by an infinite current line the electric field is computed directly in the time domain the computations are carried out in the lower half space only because exact boundary conditions are used on the free surface the algorithm follows the finite element approach which leads after space discretization to an equation system with a sparse matrix time stepping is done with an implicit time scheme at each time step the solution of the equation system is provided by the fast system iccg 0 the resulting algorithm produces good results even when large resistivity contrasts are involved we present a test of the algorithm s performance in the case of a homogeneous earth with a reasonable grid the relative error with respect to the analytical solution does not exceed 1 percent even 2 s after the source is turned off

the classification of the ricci and plebanski tensors in general relativity using newman penrose formalism a list is given of a canonical set of the newman penrose quantities ab the tetrad components of the trace free ricci tensor for each plebanski class according to plebanski s classification of this tensor this comparative list can easily be extended to cover the classification in tetrad language of any second order trace free symmetric tensor in a space time a fourth order tensor which is the product of two such tensors was defined by plebanski and used in his classification this has the same symmetries as the weyl tensor the petrov classification of this tensor here called the plebanski tensor is discussed along with the classification of the ricci tensor the use of the plebanski tensor in a couple of areas of general relativity is also briefly discussed

bifurcations de points fixes elliptiques ii orbites periodiques et ensembles de cantor invariants formes normales orbites bien ordonnees estimation a priori sur c estimations lipschitziennes sur les orbites bien ordonnees elimination des problemes de bord existence d ensembles invariants d aubry mather ensembles translates d aubry mather

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

2 d resistivity modeling an approach to arrays parallel to the strike direction an algorithm for two dimensional electrical resistivity modeling using the finite element method with mixed boundary conditions can calculate the electrical potential along any arbitrary direction in the particular case of the direction parallel to the strike of the structure a numerical singularity occurs we resolved it by calculating the potential near the singularity and improving the involved interpolation of the transformed potential investigating the shape of potential in the fourier transformed space we conclude that this interpolation may combine logarithmic and exponential functions

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

on the continuability of periodic orbits of parametrized three dimensional differential equations on considere un systeme autonome a n dimensions d equations differentielles ordinaires a 1 parametre x f x on suppose que f est definie et continue sur un sous ensemble ouvert o de r r n et que f x 0 sur o on etudie les orbites periodiques du systeme contenues dans o

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