the meromorphic non integrability of the three body problem we study the planar three body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the lagrangian solution we use the ziglin s method and study the monodromy group of the corresponding normal variational equations

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

melnikov method and transversal homoclinic points in the restricted three body problem abstract in this paper we show by melnikov method the existence of the transversal homoclinic orbits in the circular restricted three body problem for all but some finite number of values of the mass ratio of the two primaries this implies the existence of a family of oscillatory and capture motion this also shows the non existence of any real analytic integral in the circular restricted three body problem besides the well known jacobi integral for all but possibly finite number of values of the mass ratio of the two primaries this extends a classical theorem of poincare 10 because the resulting singularities in our equation are degenerate a stable manifold theorem of mcgehee 7 is used

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

h controller synthesis by j lossless coprime factorization this paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the mathcal h infty norm of the closed loop is strictly less than a given bound by the use of coprime factorizations it is shown that the mathcal h infty control problem is fundamentally related to the so called analytic systems considered by helton et al regional conference series in mathematics 68 1987 such problems admit a solution if and only if a certain j lossless factorization exists interpreted in the mathcal h infty control context this means that for a controller of the requisite type to exist the plant must admit a certain j lossless coprime factorization the full mathcal h infty synthesis problem requires that two nested j lossless factorizations exist the results are independent of whether discrete time or continuous time systems are being considered it is then shown that j lossless factorization is equivalent to the existen

exact results for a noise induced bistable system a stochastic system where bistability is caused by noise has been recently investigated by biancalani et al phys rev lett 112 038101 2014 they have computed the mean switching time for such a system using a continuous fokker planck equation derived from the taylor expansion of the master equation to estimate the parameter of such a system from experiment in this article we provide the exact solution for the full discrete system without resorting to continuous approximation and obtain the expression for the mean switching time we further extend this investigation by solving exactly the master equation and obtaining the expression of other quantities of interests such as the dynamics of the moments and the equilibrium time

on a general structure of the stabilizing controllers based on stable range in this paper we prove that some stabilizing controllers of a plant which admits a left right coprime factorization have a special form where their stable and unstable parts are separated the dimension of the unstable part depends on the algebraic concept of stable range of the ring a of siso stable plants moreover we prove that if the stable range of a is equal to 1 then every plant defined by a transfer matrix with entries in the quotient field of a and admitting a left right coprime factorization can be stabilized by a stable controller strong stabilization in particular using a result of treil proving that the stable range of h infty mathbb d is equal to 1 we show that every stabilizable plant defined by a transfer matrix with entries in the quotient field of h infty mathbb d or h infty mathbb c is strongly stabilizable and equivalently every couple of stabilizable plants can be simultaneously stabilized by a controller simultaneous stabilization finally using the fact that the topological stable range of h infty mathbb d is equal to 2 a result due to suarez we show that every unstabilizable siso plant defined by a transfer function with entries in the quotient field of h infty mathbb d is as close as we want to a stabilizable plant in the product topology

homoclinic orbits and oscillation for the planar three body problem sitnikov proved the existence of oscillation and capture for a special motion of the restricted three body problem m 0 141 alekseev made a systematic study of this situation 11 he also proved the same result for all nonzero masses see section 5 below for a discussion of this case a good exposition of this example following lectures of conley is contained in 9 using the stable manifold result for a degenerately hyperbolic closed orbit of mcgehee 81 easton and mcgehee proposed a planar three body example with negative energy which could possibly exhibit similar oscillation and capture 3 while sitnikov s example has two degrees of freedom this planar example after all the integrals and symmetries are removed has three degrees of freedom they studied a model example which completely decouples when the third body is near infinity and proved oscillation and capture exist for this model example their work left the following three steps undone 1 show the parabolic orbits form a submanifold for the real equations 2 show the a parabolic orbits and w parabolic orbits are transverse in a strong sense see the definition of a hyperbolic homoclinic orbit defined below and 3 show that symbolic dynamics can be used to show oscillation and capture exist using only the fact that the binary asymptotically decouples and not that it completely decouples when the third body is near infinity for negative energy h as the third particle goes to infinity parabolically the asymptotic motion of q r2 rl is that of a two body problem with energy h after regularization the set of all two body motions with energy h 0 is the hopf flow on the three sphere s3 let ivs s3 resp wu s be the set of w parabolic orbits resp w parabolic orbits easton has now shown that ws s3 and wl s3 are lipschitz manifolds 2 this paper proves they are real analytic manifolds and cm at infinity at s

chaos in the gylden problem we consider the gylden problem a perturbation of the kepler problem via an explicit function of time for certain general classes of planar periodic perturbations after proving a poincare melnikov type criterion we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi infinite sequences with infinitely many symbols we achieve the main result by computing the melnikov integral explicitly

infinite dimensional linear systems a distributional approach we introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems the class of distributional resolvent linear systems includes all linear time invariant systems that have been studied in the control literature we show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory

real solutions to control approximation and factorization problems during the past decades much of finite dimensional systems theory has been generalized to infinite dimensions however there is one important flaw in this theory it guarantees only complex solutions even when the data is real we show that the standard solutions of many classical problems with real data are also real we call a possibly matrix or operator valued holomorphic function g real real symmetric if g bar z overline g z for every z we show that if such a function can be presented as g nm 1 where n m in mathcal h infty then we have g n r m r 1 where n r m r in mathcal h infty are real and weakly right coprime consequently if a real function g has a stabilizing compensator i e a function k such that begin smallmatrix i k g i end smallmatrix 1 in mathcal h infty then g has a real doubly coprime factorization and a youla parameterization of all real stabilizing controllers if a system of the form dot x ax bu y cx du or of the form x

from brake to syzygy in the planar three body problem we study solutions with zero initial velocity brake orbits following such a solution until the three masses become collinear syzygy we obtain a continuous flow induced poincare map we study the image of the map in the set of collinear configurations and define a continuous extension to the lagrange triple collision orbit in addition we provide a variational characterization of some of the resulting brake to syzygy orbits and find simple examples of periodic brake orbits

mixed dynamics in a parabolic standard map abstract we use numerical and analytical tools to provide arguments in favor of the existence of a family of smooth symplectic diffeomorphisms of the two dimensional torus that have both a positive measure set with positive lyapunov exponent and a positive measure set with zero lyapunov exponent the family we study is the unfolding of an almost hyperbolic diffeomorphism on the boundary of the set of anosov diffeomorphisms proposed by lewowicz

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

covering relations cone conditions and the stable manifold theorem abstract we show how to effectively link covering relations with cone conditions we give a new geometric proof of the stable manifold theorem for hyperbolic fixed point of a map

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

oscillatory motions for the restricted planar circular three body problem the restricted three body problem models the motion of a massless body under the influence of the newtonian gravitational force caused by two other bodies called the primaries when they move along circular keplerian orbits and the third body moves in the same plane one has the restricted planar circular three body problem rpc3bp in suitable coordinates it is a hamiltonian system of two degrees of freedom the conserved energy is usually called the jacobi constant llibre and simo math ann 248 2 153 184 1980 proved the existence of oscillatory motions for this system that is orbits which leave every bounded region but which return infinitely often to some fixed bounded region to prove their existence they had to assume the ratio between the masses of the primaries to be small enough in this paper we prove the existence of such motions for any value of the mass ratio mu closing the problem of existence of oscillatory motions in the rpc3bp to obtain such motions we restrict ourselves to the level sets of the jacobi constant we show that for any value of the mass ratio and for large values of the jacobi constant there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets these transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits the main achievement is to prove the existence of these orbits without assuming the mass ratio mu small when mu is not small this transversality can not be checked by means of classical perturbation theory since our method is valid for all values of mu we are able to detect a curve in the parameter space formed by mu and the jacobi constant where cubic homoclinic tangencies between the invariant manifolds of infinity appear

optimal control on the doubly infinite continuous time axis and coprime factorizations we study the problem of existence of weak right or left or strong coprime factorizations in h infinity over the right half plane of an analytic function defined in some subset of the right half plane we give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous time axis in particular we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic riccati equation of an arbitrary realization that need not be well posed have a solution in general unbounded and not even densely defined and that a coupling condition involving these two solutions is satisfied the proofs that we give are partly based on corresponding discrete time results which we have recently obtained

the stable rank of the algebra h equals 1 abstract it is proved that if the functions f g h f 1 g 1 satisfy the carleson corona condition z inf f z g z def o then there exists a function g h such that the fucntion f gg is invertible in h and moreover g c 1 c where the constant c depends only on

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

parabolic orbits in the elliptic restricted three body problem abstract the main goal of this paper is to describe the parabolic orbits of the planar restricted elliptic three body problem the method used is based on a standard blow up of the periodic orbits at the infinity and the perturbation of the integrable case corresponding to mass parameter equal to zero we give an asymptotic formula for the distance between the stable and unstable manifolds of the infinity which allows us to describe how the heteroclinic orbits are created changing the eccentricity of the primaries some conclusions of the quantitative study are given at the end

invariant manifolds of parabolic fixed points ii approximations by sums of homogeneous functions abstract we study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields if the dimension of these manifolds is two or greater in general it is not possible to obtain polynomial approximations here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations we deal with both the differentiable and analytic cases we also study the dependence on parameters in the companion paper 4 these approximations are used to obtain the existence of true invariant manifolds close by examples are provided

non gaussian noise weakened stability in a foraging colony system with time delay abstract in this paper the dynamical properties in a foraging colony system with time delay and non gaussian noise were investigated using delay fokker planck approach the stationary probability distribution spd the associated relaxation time art and normalization correlation function ncf are obtained respectively the results show that i the time delay and non gaussian noise can induce transition from a single peak to double peaks in the spd i e a type of bistability occurring in a foraging colony system where time delay and non gaussian noise not only cause transitions between stable states but also construct the states themselves numerical simulations are presented and are in good agreement with the approximate theoretical results ii there exists a maximum in the art as a function of the noise intensity this maximum for art is identified as the characteristic of the non gaussian noise weakened stability of the foraging colonies in the steady state iii the art as a function of the noise correlation time exhibits a maximum and a minimum where the minimum for art is identified as the signature of the non gaussian noise enhanced stability of the foraging colonies and iv the time delay can enhance the stability of the foraging colonies in the steady state while the departure from gaussian noise can weaken it namely the time delay and departure from gaussian noise play opposite roles in art or ncf

invariant manifolds of dynamical systems close to a rotation transverse to the rotation axis abstract we consider one parameter families of vector fields depending on a parameter such that for 0 the system becomes a rotation of r 2 r n around 0 r n and such that for 0 the origin is a hyperbolic singular point of saddle type with say attraction in the rotation plane and expansion in the complementary space we look for a local subcenter invariant manifold extending the stable manifolds to 0 afterwards the analogous case for maps is considered in contrast with the previous case the arithmetic properties of the angle of rotation play an important role

global phase structure of the restricted isosceles three body problem with positive energy we study a restricted three body problem with special symmetries the restricted isosceles three body problem for positive energy the energy manifold is partially compactified by adding boundary manifolds corresponding to infinity and triple collision we use a new set of coordinates which are a variation on the mcgehee coordinates of celestial mechanics these boundary manifolds are used to study the global phase structure of this gradational system the orbits are classified by intersection number that is the number of times the infinitesimal body cross the line of syzygy before escaping to infinity

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

justification of input output methods for systems with unbounded control and observation the authors present definitions of feedback stabilizability and detectability that are weaker than those previously appearing in the literature stabilizability and detectability in this sense are sufficient to ensure an equivalence between internal and external stability for well posed systems furthermore stabilizability and detectability are shown to be preserved under feedback and other connections these results are linked to present a justification of input output controller design methods for systems with unbounded control and observation

corrigendum to oscillatory motions in restricted n body problems j differential equations 265 2018 779 803 abstract at the beginning of paper 1 there is an error that spreads along the rest of the work and the conclusions are not correct in their present form precisely in section 2 page 783 there is a contradiction related to the scaling in the paragraph before formula 6 it is said that t e 3 t but hamiltonian 6 is not scaled accordingly we have fixed the problem and after performing due changes the conclusions are obtained the existence of the manifolds at infinity is guaranteed theorem 3 1 and the transversal intersection of them is concluded in theorem 5 1 the applications in section 6 are also valid after adapting them to the new version of the theorems

invariant manifolds of parabolic fixed points i existence and dependence on parameters abstract in this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases using the parametrization method the parametrization method relies on a suitable approximate solution of a functional equation in the case of parabolic points if the manifolds have dimension two or higher in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in 4 assuming a sufficiently good approximation is found here we provide an a posteriori result which gives a true invariant manifold close to the approximate one in the differentiable case in some cases there is a loss of regularity we also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters examples are provided we apply our method to prove that in several situations namely related to the parabolic infinity in the elliptic spatial three body problem these invariant manifolds exist and do have polynomial approximations

parabolic orbits in the planar three body problem it is not known whether there exist oscillatory and capture orbits in the planar 3 body problem sitnikov 6 proved such orbits exist for the restricted 3 body problem alekseev i extended this work and related it to the existence of homoclinic orbits mcgehee and easton 3 studied a model problem closely related to the planar 3 body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3 sphere the goal of this paper is to construct an invariant 3 sphere at infinity in the planar 3 body problem and to prove that the stable and unstable manifolds of this sphere are lipschitz manifolds to complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3 sphere intersect each other this is a difficult question which is not treated here however using methods of melnikov and using one of the masses as a perturbation parameter such an investigation might be possible the stable manifold theorem which is proven here is not standard since the invariant 3 sphere is not hyperbolic geometric methods of mcgehee 4 are extended and applied to obtain the theorem a special case of the system of equations that we study has the form j x dy lx e x y l f 4 dy l x e x y l 0 1 4

coprime factorizations and well posed linear systems we study the basic notions related to the stabilization of an infinite dimensional well posed liner system in the sense of salamon and weiss we first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in h infty the converse is also true every function with a doubly coprime factorization in h infty is the transfer function of a jointly stabilizable and detectable well posed linear system we show further that a stabilizable and detectable system is stable if and only if its input output map is stable finally we construct a dynamic possibly non well posed stabilizing compensator the notion of stability that we use is the natural one for the quadratic cost minimization problem and it does not imply exponential stability

stabilizability representations and factorizations for time varying linear systems abstract we study the system representations factorizations and stabilizability of a discrete time time varying linear system in the framework of nest algebra we shown that every time varying linear system admits a normalized left factorization while it may possibly not have a right factorization the relationship between the system representation and factorization is studied and some new stabilizability criteria are established

coprime factorization and optimal control on the doubly infinite discrete time axis we study the problem of strongly coprime factorization over h infinity of the unit disc we give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete time axis in particular we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic riccati equation of an arbitrary realization have a solution in general unbounded and even nondensely defined and that a coupling condition involving these solutions is satisfied

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

stable manifolds associated to fixed points with linear part equal to identity abstract we consider maps defined on an open set of r n m having a fixed point whose linear part is the identity we provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map these maps arise naturally in some problems of celestial mechanics we apply the results to prove the existence of parabolic orbits of the spatial elliptic three body problem

coprime factorization and dynamic stabilization of transfer functions it is known that a matrix valued transfer function p has a stabilizing dynamic controller q i e begin smallmatrix i q p i end smallmatrix 1 in rm h infty iff p has a right or left coprime factorization we show that the same result is true in the operator valued case thus the standard youla bongiorno parameterization applies to every dynamically stabilizable function we then derive further equivalent conditions one of them being that p has a stabilizing controller with internal loop this and some others are new even in the scalar valued case we also establish certain related results for example we extend the classical results on coprime factorization and partial feedback measurement feedback stabilization to nonrational transfer functions all our results apply in both discrete and continuous time settings except that in the latter it is not clear whether the controller q can always be chosen so that it is continuous time proper holomorphic and bounded on a right half plane unless e g p z rightarrow0 as operatorname re z rightarrow infty

the existence of oscillatory and superhyperbolic motion in newtonian systems we establish the existence of solutions for the collinear newtonian four body problem where the maximum spacing between particles exceeds any given function of time as t as an example there exist solutions where the maximum spacing separates faster than any constant multiple of time also it is shown that the three and four body problems admit oscillatory solutions

stable manifolds under very weak hyperbolicity conditions abstract we present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions

an approach to mel nikov theory in celestial mechanics using a completely analytic procedure based on a suitable extension of a classical method we discuss an approach to the poincare mel nikov theory which can be conveniently applied also to the case of nonhyperbolic critical points and even if the critical point is located at the infinity in this paper we concentrate our attention on the latter case and precisely on problems described by kepler type potentials in one or two degrees of freedom in the presence of general time dependent perturbations we show that the appearance of chaos possibly including arnol d diffusion can be proved quite easily and in a direct way without resorting to singular coordinate transformations such as the mcgehee or blowing up transformations natural examples are provided by the classical gylden problem originally proposed in celestial mechanics but also of interest in different fields and by the general three body problem in classical mechanics

robustly stabilizing controllers with respect to left coprime factor perturbations for infinite dimensional linear systems we solve the problem of robust stabilization with respect to left coprime factor perturbations for a large class of irrational transfer functions with a realization in terms of unbounded operators the key condition is that the associated dynamical system and its dual should satisfy a finite cost condition so that two optimal cost operators exist we obtain an explicit parameterization of infinitely many robustly stabilizing controllers in terms of these optimal cost operators and the generating operators of the realization

the existence of transverse homoclinic points in the sitnikov problem abstract using melnikov s method we are able to prove the existence of transverse homoclinic orbits and therefore the existence of a horseshoe in a special restricted three body problem this analysis is an alternative to the one described by moser stable and random motions in dynamical systems princeton univ press princeton nj 1973 based on sitnikov s original work dokl akad nauk ussr 133 no 2 1960 303 306 where the task is accomplished using a more direct construction of the horseshoe

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

oscillatory motions in restricted n body problems the authors have received partial support from project grant red de cuerpos academicos de ecuaciones diferenciales sistemas dinamicos y estabilizacion promep 2011 sep mexico and from projects mtm 2014 59433 c2 1 p of the ministry of economy and competitiveness of spain and mtm 2017 88137 c2 1 p of the ministry of economy industry and competitiveness of spain

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

destruction of invariant curves in the restricted circular planar three body problem by using comparison of action the classical principle of least action says that orbits of mechanical systems extremize action an important subclass are those orbits that minimize action in this paper we utilize this principle along with aubry mather theory to construct birkhoff regions of instability for a certain three body problem given by a hamiltonian system of 2 degrees of freedom we believe that these methods can be applied to construct instability regions for a variety of hamiltonian systems with 2 degrees of freedom the hamiltonian model we consider describes dynamics of a sun jupiter comet system and under some simplifying assumptions we show the existence of instabilities for the orbit of the comet in particular we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity e 0 66 can increase in eccentricity up to e 0 96 in the sequels to this paper we extend the result to beyond e 1 and show the existence of ejection orbits such orbits are initially well within the range of our solar system this might give an indication of why most objects rotating around the sun in our solar system have relatively low eccentricity

alternative to the diffusion equation in population genetics since its inception by kimura in 1955 m kimura proc natl acad sci u s a 41 144 1955 the diffusion equation has become a standard technique of population genetics the diffusion equation is however only an approximation valid in the limit of large populations and small selection moreover useful quantities such as the fixation probabilities are not easily extracted from it and need the concomitant use of a forward and backward equation we show here that the partial differential equation governing the probability generating function can be used as an alternative to the diffusion equation with none of its drawbacks it does not involve any approximation it has well defined initial and boundary conditions and its solutions are finite polynomials we apply this technique to derive analytical results for the moran process with selection which encompasses the kimura diffusion equation