a numerical method for computing initial conditions of lagrangian invariant tori using the frequency map we present a numerical method for computing initial conditions of lagrangian quasi periodic invariant tori of hamiltonian systems and symplectic maps such initial conditions are found by solving using the newton method a nonlinear system obtained by imposing suitable conditions on the frequency map the basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi periodic signal allowing to compute frequencies and their derivatives with respect to parameters roughly speaking this method consists in computing suitable weighted averages of the iterates of the signal and using the richardson extrapolation method the proposed approach performs with high accuracy at a moderate computational cost we illustrate the method by considering a discrete fpu model and the vicinity of the point l 4 in a rtbp c 2016 elsevier b v all rights reserved

non generic cusps we find the order of contact of the boundaries of the cusp for two parameter families of vector fields on the real line or diffeomorphisms of the real line for cusp bifurcations of codimensions 1 and 2 moreover we create a machinery that can be used for the same problem in higher codimensions and perhaps for other similar problems

an algorithm to generate canonical transformations application to normal forms we describe a new method to generate symplectic diffeomorphisms this method is based on primitive functions and it was introduced in nonlinearity 13 2000 1483 we apply the algorithm to the construction of normal forms of kam tori a concrete application developed in detail and implemented numerically is normal forms near invariant circles

functional equations for circle homeomorphisms with golden ratio rotation number the investigation of a scaling limit for mappings of the circle to itself with golden ratio rotation number leads to a pair of functional equations with at least a formal resemblance to the functional equation using the accumulation of period doubling bifurcations we discuss the general theory of these functional equations assuming that solutions exist

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

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

numerical investigations of the structure around an invariant kam torus using the frequency map analysis abstract the hierarchical structure around kam tori is studied numerically on the standard map as a model problem applying the laskar frequency map analysis the theoretical prediction recently given by morbidelli and giorgilli j stat phys 78 1994 about the existence of a neighborhood almost completely full of slave tori around a chief torus are recovered and extended to the near critical regime

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

a note on permanents and generalized complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices 3 the multiplication of the g i s is intrinsic connections between the 2 2 matrices c k and the resulting cb matrix g k have been explored in particular the properties which are inherited from the c k to the g k two situations were considered for the ordinary real cb matrices and for the corresponding sign pattern matrices in this paper we consider generalized complementary basic matrices where the c k matrices are replaced by square matrices of arbitrary sizes some interesting facts and results about the permanents and permanental polynomials of such matrices are obtained

a reversible bifurcation analysis of the inverted pendulum the inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting parameters are given by the size of the forcing and the frequency ratio normal form theory provides an integrable approximation of the poincare map generated by a planar vector field genericity of the model is studied by a perturbation analysis where the spatial symmetry is optional here equivariant singularity theory is used

a finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh a finite volume fv procedure is described for solving the elastic solid mechanics equations in three dimensions on an unstructured mesh for bodies undergoing thermal or mechanical loads the fv procedure is developed in parallel with the conventional fe galerkin procedure so that the differences in each approach may be clearly distinguished the matrix form of the fv procedure is described and is implemented in parallel with the fe procedure both for two dimensional quadrilateral and three dimensional brick meshes the fv and fe procedures are then compared against a range of benchmark problems that test the basic capability of the fv technique it is shown to be approximately as accurate as the fe procedure on similar meshes though its system matrix set up time is twice as long for a node by node set up procedure

renormalization of cocycles and linear ode with almost periodic coefficients in the current paper we address the problem of classification of cocycles over an irrational rotation we use the renormalization group approach

numerical computation of the asymptotic size of the rotation domain for the arnold family abstract we consider the arnold tongue of the arnold family of circle maps associated to a fixed diophantine rotation number the corresponding maps of the family are analytically conjugate to a rigid rotation this conjugation is defined on a maximal complex strip of the circle and after a suitable scaling the size of this strip is given by an analytic function of the perturbative parameter the main purpose of this paper is to perform a numerical accurate computation of this function and of its taylor expansion this allows us to verify previous theoretical results the rotation numbers we select are quadratic irrationals mainly the golden mean by introducing a nonstandard extrapolation process especially suited for the problem we compute all the quantities required rotation numbers arnold tongues fourier and taylor coefficients with high precision

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

on the numerical approximation of the rotation number abstract the starting point of this paper is a polygonal approximation of an invariant curve of a map using this polygonal approximation an approximation for the circle map the restriction of the map to the invariant curve is obtained the rotation number of the circle map is then approximated by the rotation number of the approximated circle map the error in the obtained approximate rotation number is discussed and related to the error in the polygonal approximation of the invariant curve simple algorithms for the approximation of the rotation number are described a numerical example illustrates the theory

stability of the phase motion in race track microtrons we model the phase oscillations of electrons in race track microtrons by means of an area preserving map with a fixed point at the origin which represents the synchronous trajectory of a reference particle in the beam we study the nonlinear stability of the origin in terms of the synchronous phase the phase of the synchronous particle at the injection we estimate the size and shape of the stability domain around the origin whose main connected component is enclosed by an invariant curve we describe the evolution of the stability domain as the synchronous phase varies we also clarify the role of the stable and unstable invariant curves of some hyperbolic fixed or periodic points

on the numerical computation of diophantine rotation numbers of analytic circle maps in this paper we present a numerical method to compute diophantine rotation numbers of circle maps with high accuracy we mainly focus on analytic circle diffeomorphisms but the method also works in the case of enough finite differentiability the keystone of the method is that under these conditions the map is conjugate to a rigid rotation of the circle moreover although it is not fully justified by our construction the method turns out to be quite efficient for computing rational rotation numbers we discuss the method through several numerical examples

determination of the threshold of the break up of invariant tori in a class of three frequency hamiltonian systems we consider a class of hamiltonians with three degrees of freedom that can be mapped into quasi periodically driven pendulums the purpose of this paper is to determine the threshold of the break up of invariant tori with a specific frequency vector we apply two techniques the frequency map analysis and renormalization group methods the renormalization transformation acting on a hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the hamiltonian and a rescaling of phase space around the considered torus we give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break up of invariant tori furthermore we obtain by these methods numerical values of the threshold of the break up of the last invariant torus

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

poisson lie structures on poincare and euclidean groups in three dimensions the complete list of poisson lie structures on poincare and euclidean groups in three dimensions is presented some new solutions for inhomogenous are given

a new proof of m herman s theorem a new proof of the m herman theorem on the smooth conjugacy of a circle map is presented here it is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables

a finite volume format for structural mechanics summary a general finite volume method fvm for the analysis of structural problems is presented it is shown that the fvm can be considered to be a particular case of finite elements with a non galerkin weighting for structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals both displacement and mixed fv formulations are presented for static and dynamic problems the finite volume method fvm evolved in the early seventies via finite difference approxima tions on non orthogonal grids the popularity of the fvm has been extensive in the field of computational fluid dynamics cfd and heat transfer on the contrary in the field of computational solid mechanics csm the use of the fvm has never achieved such acceptance an early attempt to use fv concepts in csm is due to wilkins6 as an alternative approximation to derivatives in a cell in this he defines the average gradient of a function u in a volume l2 as using the well known divergence theorem such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early hydrocodes of the lawrence livermore laboratory the reasons for the unpopularity of the fvm in structural mechanics is understandable as finite volumes are well known to be less accurate than galerkin based finite elements for self adjoint elliptic problems a comparison between fvm and fem has been recently pre sented by zienkiewicz and oiiate here the authors show that fvm and fem share concepts such as mesh discretization and interpolation giving precisely the same discretized systems of equations for some particular cases this coincidence also shows clearly for 2 d and 3 d structural problems as detailed in this work here however surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of sides in the mesh this leads to an overall solution cost very similar to that of fe computations note that for a refined mesh of three node triangles the number of sides is 1 5 times that of elements this fact suggests that computational speed is not a prior one of the keys to the possible success of fvm in structural problems however the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which in the authors opinion may be worth exploring in detail indeed it

the numerical approximation of the rotation number of planar maps abstract given a convex jordan curve c we present a method which computes the numerical approximation of the rotation number of a planar map whose invariant curve is c it does not require any numerical approximation of the invariant curve but just exploits the order by which the iterates of the map are generated numerical examples are reported and discussed

computation of derivatives of the rotation number for parametric families of circle diffeomorphisms in this paper we present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averages of iterates of the map and richardson extrapolation we focus on analytic circle diffeomorphisms but the method also works if the maps are differentiable enough in order to justify the method we also require the family of maps to be differentiable with respect to the parameters and the rotation number to be diophantine in particular the method turns out to be very efficient for computing taylor expansions of arnold tongues of families of circle maps finally we adapt these ideas to study invariant curves for parametric families of planar twist maps

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

a continuation method to study periodic orbits of the froeschle map abstract the dynamics of many hamiltonian systems with three degrees of freedom is represented by the froeschle map which is symplectic and four dimensional in this paper we study sequences of periodic orbits approaching the invariant tori in order to obtain information about its stability a homotopic method is used to continue the branches of periodic orbits we found that the existence of turning points is related to the linear stability of the periodic orbit and its rotation vector

finite volumes and finite elements two good friends in this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic convective diffusion and fluid flow problems the paper shows that both procedures share a number of features like mesh discretization and approximation moreover it is shown that in many cases both techniques are completely equivalent

equivariant singularity theory with distinguished parameters two case studies of resonant hamiltonian systems we consider hamiltonian systems near equilibrium that can be formally reduced to one degree of freedom spatiotemporal symmetries play a key role the planar reduction is studied by equivariant singularity theory with distinguished parameters the method is illustrated on the conservative spring pendulum system near resonance where it leads to integrable approximations of the iso energetic poincare map the novelty of our approach is that we obtain information on the whole dynamics regarding the quasi periodic solutions the global configuration of their invariant manifolds and bifurcations of these

numerical determination of the continued fraction expansion of the rotation number abstract the rotation number is an important invariant of homeomorphisms on the circle in this paper an algorithm is described which computes the continued fraction expansion of this number it therefore detects rational rotation numbers and in general determines a sequence of intervals of decreasing length containing the rotation number it will be shown that for almost all rotation numbers the convergence is of third order also error analyses of some other algorithms and numerical examples are given including a computation of resonance tongues in the arnold family

multifractal analysis of tori destruction in a molecular hamiltonian system in this paper an analysis of the phase space structure of the isomerizing molecular system linc licn using poincare surfaces of section and frequency analysis is presented the scaling structure of the frequency map in the chaotic region next to the regular part corresponding to the stable linear isomer linc is studied using multifractal analysis this approach is a way to characterize quantitatively the complexity in the mechanism of the tori destruction in a molecular hamiltonian system that exhibits soft chaos as the vibrational energy of the system increases

stability diagram for 4d linear periodic systems with applications to homographic solutions we consider a family of 4 dimensional hamiltonian time periodic linear systems depending on three parameters 1 2 and e such that for e 0 the system becomes autonomous using normal form techniques we study stability and bifurcations for e 0 small enough we pay special attention to the d alembert case the results are applied to the study of the linear stability of homographic solutions of the planar three body problem for some homogeneous potential of degree 0 2 including the newtonian case

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

numerical computation of rotation numbers of quasi periodic planar curves abstract recently a new numerical method has been proposed to compute rotation numbers of analytic circle diffeomorphisms as well as derivatives with respect to parameters that takes advantage of the existence of an analytic conjugation to a rigid rotation this method can be directly applied to the study of invariant curves of planar twist maps by simply projecting the iterates of the curve onto a circle in this work we extend the methodology to deal with general planar maps our approach consists in computing suitable averages of the iterates of the map that allow us to obtain a new curve for which the direct projection onto a circle is well posed furthermore since our construction does not use the invariance of the quasi periodic curve under the map it can be applied to more general contexts we illustrate the method with several examples

an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects this paper presents the development and validation of a new parallel unstructured multi grid preconditioned implicit method for the simulation of three dimensional 3d unsteady compressible flows using a so called immersed membrane method imm g h xia y zhao j h yeo an immersed membrane method for simulation of flui structure interaction in bio fluid flows in 1st international bioengineering conference ibec 2004 08 10 september 2004 the novel feature of the method presented is that it employs a unique combination of a parallel multi grid scheme with low mach number preconditioning and the imm so that 3d unsteady low mach number flows with arbitrarily moving objects can be efficiently simulated

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

birkhoff averages and rotational invariant circles for area preserving maps abstract rotational invariant circles of area preserving maps are an important and well studied example of kam tori john greene conjectured that the locally most robust rotational circles have rotation numbers that are noble i e have continued fractions with a tail of ones and that of these circles the most robust has golden mean rotation number the accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry and such high accuracy has not been obtained in more general maps in this paper we develop a method based on a weighted birkhoff average for identifying chaotic orbits island chains and rotational invariant circles that does not rely on these symmetries we use chirikov s standard map as our test case and also demonstrate that our methods apply to three other well studied cases

intrinsic products and factorizations of matrices we say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition we present several examples together with important applications these applications include companion matrices and sign nonsingular matrices

incremental approach for radial basis functions mesh deformation with greedy algorithm mesh deformation is an important element of any fluidstructure interaction simulation in this article a new methodology is presented for the deformation of volume meshes using incremental radial basis function rbf based interpolation a greedy algorithm is used to select a small subset of the surface nodes iteratively two incremental approaches are introduced to solve the rbf system of equations 1 block matrix inversion based approach and 2 modified lu decomposition approach the use of incremental approach decreased the computational complexity of solving the system of equations within each greedy algorithm s iteration from o n3 to o n2 results are presented from an accuracy study using specified deformations on a 2d surface mesh deformations for bending and twisting of a 3d rectangular supercritical wing have been demonstrated outcomes showed the incremental approaches reduce the cpu time up to 67 as compared to a traditional rbf matrix solver finally the proposed mesh deformation approach was integrated within a fluidstructure interaction solver for investigating a flow induced cantilever beam vibration

some inheritance properties for complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices m fiedler a note on companion matrices linear algebra appl 372 2003 325 331 9 3 the multiplication of the g i s is intrinsic we explore connections between the 2 2 matrices c k and the resulting cb matrix g k in particular which properties are inherited from the c k to the g k we consider two situations for the ordinary real cb matrices and for the corresponding sign pattern matrices

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

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

finite volume method for thermo elasto plastic stress analysis abstract in this paper a new numerical method for prediction of stresses and displacements in thermo elasto plastic material is presented the method is based on the direct solution of the governing equations for thermal energy and momentum balance expressed in terms of temperature and displacements as dependent variables via corresponding constitutive relations and discretized by the finite volume technique as a result the unsteady fields of temperature displacements and stresses as well as residual stresses in a solid body subjected to a non uniform temperature and or external loads are obtained the method has been applied to a number of test cases and the results are compared with the analytical solution calculations obtained by another numerical method or experimental data results show that the method is accurate stable and efficient and that it can be used as a useful tool for predicting thermo mechanical deformation processes in a solid body

some graph theoretic properties of generalized complementary basic matrices abstract this paper investigates graph theoretic properties of the generalized complementary basic matrices these include cyclic properties hamiltonian properties block connected properties and minimal strongly connected digraph properties

halo orbits around the collinear points of the restricted three body problem abstract we perform an analytical study of the bifurcation of the halo orbits around the collinear points l 1 l 2 l 3 for the circular spatial restricted three body problem following a standard procedure we reduce to the center manifold constructing a normal form adapted to the synchronous resonance introducing a detuning which measures the displacement from the resonance and expanding the energy in series of the detuning we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio in most cases the analytical results thus obtained are in very good agreement with the numerical expectations providing the bifurcation threshold with good accuracy care must be taken when dealing with l 3 for small values of the mass ratio between the primaries in that case the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold

the inverted pendulum a singularity theory approach the inverted pendulum with small parametric forcing is considered as an example of a wider class of parametrically forced hamiltonian systems the qualitative dynamics of the poincare map corresponding to the central periodic solution is studied via an approximating integrable normal form at bifurcation points we construct local universal models in the appropriate symmetry context using equivariant singularity theory in this context structural stability can be proved under generic conditions

asymptotic behaviour of the stability parameter for a family of singular limit hill s equation abstract under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular limit hill s equation which depends on three parameters we use the blow up techniques introduced in r martinez a sama c simo analysis of the stability of a family of singular limit linear periodic systems in r 4 applications j differential equations 226 2006 652 686 the main contribution of this paper concerns the study of the nondegeneracy conditions we give a geometrical interpretation of them in terms of heteroclinic orbits for some related systems in this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied as a motivation and application we consider the vertical stability of homographic solutions in the three body problem

on aubry maher sets abstract let f be a two dimensional area preserving twist map for each irrational rotation number in a certain nontrivial interval there is an f invariant minimal set which preserves order with respect to that rotation number for large nonlinearity these sets are typically cantor sets and they are referred to as aubry mather sets we prove that under some assumptions these sets are ordered vertically according to ascending rotation number monotonicity furthermore if f satisfies certain conditions the right hand points of the gaps in an irrational cantor set lie on a single orbit single gap and diffusion through these aubry mather sets can be understood as a limit of resonance overlaps convergence of turnstiles these conditions essentially establish the existence of a hyperbolic structure and limit the number of homoclinic minimizing orbits some other results along similar lines are given such as the continuity at irrational rotation numbers of the lyapunov exponent on aubry mather sets

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators

resonance tongues in hill s equations a geometric approach abstract the geometry of resonance tongues is considered in mainly reversible versions of hill s equation close to the classical mathieu case hill s map assigns to each value of the multiparameter the corresponding poincare matrix by an averaging method the geometry of hill s map locally can be understood in terms of cuspoid whitney singularities this adds robustness to the result the algorithmic nature of the averaging method enables a pull back to the resonance tongues of the original system