twin positive solutions for higher order m point boundary value problems with sign changing nonlinearities a new fixed point theorem on double cones is applied to obtain the existence of at least two positive solutions for the 2nd order m point boundary value problemy 2 n t f t y t y t y 2 n 1 t 0

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

a stochastic dynamic model to evaluate the influence of economy and well being on unemployment control abstract this paper presents a stochastic dynamic mathematical model to study the evolution of the unemployment rate and other relevant related variables in a country this model is composed by three basic interrelated subsystems demographic economic and well being ones a key aspect of this model is that it considers three un well being variables simultaneously human development index gender empowerment index and gender differentiation index these variables involve key concepts for human development as health education economy and female labor with this model the most outstanding variables found in the literature in relation with unemployment control can be used to design strategies and scenarios to reduce the unemployment rate in the future the model has been fitted for the case of spain in the 2002 2014 period the largest one with information about all the variables involved in the model finally several tentative scenarios and strategies have been tested to reduce the unemployment rate in spain in the horizon of year 2025 and the corresponding forecast evolution of the gender differentiation index national income per capita public debt and the ratio between public debt and gross domestic product are shown

analogy between a 10d model for nonlinear wave wave interaction in a plasma and the 3d lorenz dynamics this paper investigates nonlinear wave wave interactions in a system that describes a modified decay instability and consists of three langmuir and one ion sound waves as a means to establish that the underlying dynamics exists in a 3d space and that it is of the lorenz type both continuous and discrete time multivariable global models were obtained from data these data were obtained from a 10d dynamical system that describes the modified decay instability obtained from zakharov s equations which characterise langmuir turbulence this 10d model is equivariant under a continuous rotation symmetry and a discrete order 2 rotation symmetry when the continuous rotation symmetry is modded out that is when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism it is shown that a 3d system may describe the underlying dynamics for certain parameter values the models obtained using global modelling techniques from three time series from the 10d dynamics with the continuous rotation symmetry modded out generate attractors which are topologically equivalent these models can be simulated easily and due to their simplicity are amenable for analysis of the original dynamics after symmetries have been modded out moreover it is shown that all of these attractors are topologically equivalent to an attractor generated by the well known lorenz system 2003 elsevier science b v all rights reserved

a generalized multi point boundary value problem for second order ordinary differential equations let f 0 1 x r 2 r be a function satisfying caratheodory s conditions and e t l 1 0 1 let x i t j 0 1 a i b j r i 1 2 m 2 j 1 2 n 2 0 x 1 x 2 x m 2 1 0 t 1 t 2 t n 2 1 be given this paper is concerned with the problem of existence of a solution for the generalized multi point boundary value problems x t f t x t x t e t 0

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

an observer based tracker for hybrid interval chaotic systems with saturating inputs the chaos evolutionary programming approach this paper presents a novel chaos evolutionary programming algorithm cepa which merges a modified chaotic optimization algorithm coa with a modified evolutionary programming algorithm epa due to the nature of chaotic variable i e pseudo randomness ergodicity and irregularity the cepa can effectively and quickly search many local minimum or maximum in parallel thereby enhancing the probability of finding the global one the cepa is then successfully applied to solve challenging non convex optimization problems and to obtain the best nominal dual rate observer based digital tracker for robust tracking a periodic solution embedded into a hybrid interval chaotic system with saturating inputs and not to track the strange attractor itself an illustrative example is presented to demonstrate the effectiveness of the proposed algorithm

global vector field reconstruction by using a multivariate polynomial l 2 approximation on nets a multivariate polynomial mathit l 2 approximation on nets is designed for global vector field reconstructions of time continuous dynamical systems the technique is tested by investigating standard forms of the r ossler band the lorenz mask and a chaotic attractor produced by a simple model of thermal lens oscillations

existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem abstract in this paper we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem the hamiltonian of the differential system is regularized by a levi civita type transformation and an appropriate scaling of time the initial condition of this orbit is shown to be the infimum of some well chosen set this existence proof is direct and surprisingly simple further a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group d 4 then roberts symmetry reduction method is applied to show the linear stability it turns out that the rhomboidal periodic orbit in the planar equal mass four body problem is linearly stable

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

multiple positive solutions of a m point boundary value problem for 2nth order singular integro differential equations in banach spaces abstract in this paper we obtain the existence of multiple positive solutions of a m point boundary value problem for 2 n th order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

a stochastic dynamical social model involving a human happiness index abstract this paper presents a new human happiness index built through five dimensions development freedom solidarity justice and peace these five dimensions are evaluated through quantitative variables obtained from the human development reports world data bank and eurostat the new happiness index has been built following the guidelines set by the human development reports of the un for the construction of quality indices and it has been compared on a set of 13 eu countries with the overall life satisfaction index which is used by the un moreover the new index has been included in a dynamic mathematical model through the demographic rates to study the evolution of the population the obtained model has been calibrated for the period 2004 2009 and validated for the period 2010 2015 for the case of spain finally the model has been used to maximize the happiness index in spain for the period 2016 2030 with the conclusion that to achieve this purpose it is necessary to invest in education research and development

multiple positive solutions of a boundary value problem for a class of 2 nth order m point singular integro differential equations in banach spaces in this paper we obtain the existence of multiple positive solutions of a boundary value problem for 2nth order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

dynamical effects of overparametrization in nonlinear models abstract this paper is concemed with dynamical reconstruction for nonlinear systems the effects of the driving function and of the complexity of a given representation on the bifurcation patter are investigated it is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns the complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters in this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes moreover some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels in order to precisely quantify the complexity of the reconstructions global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks

positive solutions of m point boundary value problems for higher order ordinary differential equations abstract in this paper we study m point boundary value problems for higher order ordinary differential equation y 2 n t f t y t y t y 2 n 1 t 0 t 1 y 2 i 0 0 y 2 i 1 j 1 m 2 k i j y 2 i j 0 i n 1 where f is allowed to change sign and 0 0 1 2 m 2 m 1 1 we show sufficient conditions for the existence of at least two positive solutions by applying a new fixed point theorem in cones and the associated green s function in particular the second positive solutions for the above problem is not concave

local false nearest neighbors and dynamical dimensions from observed chaotic data the time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension the minimum necessary global embedding dimension mathit d mathit e may still be larger than the actual dimension of the underlying dynamics mathit d mathit l the embedding theorem only guarantees that the attractor of the system is fully unfolded using mathit d mathit e greater than 2 mathit d mathit a with mathit d mathit a the fractal attractor dimension using the idea of local false nearest neighbors we discuss methods for determining the integer valued mathit d mathit l

a necessary and sufficient condition for 2nth order singular super linear m point boundary value problems abstract in this paper we study the existence of positive solutions for singular super linear m point boundary value problems of 2nth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 n 2 0 1 positive solutions as well as c 2 n 1 0 1 positive solutions is given by means of the fixed point theorems on cones

chaos in numerical analysis of ordinary differential equations abstract the discretisation of the ordinary nonlinear differential equation d y d t y 1 y by the entral difference scheme is studied for fixed mesh size in the usual numerical computation this method produces some ghost solution for the long range calculation regarding this discretisation as a dynamical system in r2 these pathological behaviors are shown to be a kind of chaos in the dynamical system for any mesh size moreover some combination of the central difference scheme and the euler s scheme is studied for the above equation it gives some motivation for henon s model the usual discretisation of a second order differential equation are studied also it gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by henon heiles

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

existence and uniqueness of positive solutions for an integral boundary value problem abstract this paper deals with the existence and uniqueness of positive solutions for the second order integral boundary value problem u f u 0 u 0 0 1 u d u 1 0 1 u d where f c r r is sign changing the main tools used in the proofs are the a priori estimate method and the leray schauder fixed point theorem

central difference scheme and chaos abstract the existence of chaotic orbits is proved for area preserving diffeomorphisms of the plane defined by polynomials of degree two derived from the discretized logistic equation

on the iteration of a rational function computer experiments with newton s method using newton s method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function this article describes the behavior of newton iteration for cubic polynomials after a change of variables these polynomials can be parametrized by a single complex parameter and the newton transformation has a single critical point other than its fixed points at the roots of the polynomial we describe the behavior of the orbit of the free critical point as the parameter is varied the julia set points where newton s method fail to converge is also pictured these sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle

two positive solutions for second order quasilinear differential equation boundary value problems with sign changing nonlinearities in this paper the second order quasilinear differential equation y q t f t y 0 0 1 subject to dirichlet boundary conditions and mixed boundary conditions is studied where f is allowed to change sign v v p 2v p 1 we show the existence of at least two positive solutions by using a new fixed point theorem in cones

a multistep transversal linearization mtl method in non linear dynamics through a magnus characterization a new form of a multi step transversal linearization mtl method is developed and numerically explored in this study for a numeric analytical integration of non linear dynamical systems under deterministic excitations as with other transversal linearization methods the present version also requires that the linearized solution manifold transversally intersects the non linear solution manifold at a chosen set of points or cross section in the state space however a major point of departure of the present method is that it has the flexibility of treating non linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system even though these linearized terms become explicit functions of time from this perspective the present development is closely related to the popular practice of tangent space linearization adopted in finite element fe based solutions of non linear problems in structural dynamics the only difference is that the mtl method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an fe framework the resulting time varying linearized system matrix is then treated as a lie element using magnus characterization w magnus on the exponential solution of differential equations for a linear operator commun pure appl math vii 1954 649 673 and the associated fundamental solution matrix fsm is obtained through repeated lie bracket operations or nested commutators an advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non linear problem yet another advantage of the transversal linearization lies in the non unique representation of the linearized vector field an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors a simple analysis of the formal orders of accuracy is provided within a finite dimensional framework only a limited numerical exploration of the method is presently provided for a couple of popularly known non linear oscillators viz a hardening duffing oscillator which has a non linear stiffness term and the van der pol oscillator which is self excited and has a non linear damping term

positive solutions for 2nth order singular sub linear m point boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order singular sub linear m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c2n 2 a b c2n a b and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c2n 2 0 1 as well as c2n 1 0 1 positive solutions our nonlinearity f t x1 x2 xn may be singular at xi 0 i 1 2 n t 0 and or t 1

uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with delay and advance a new class of fitted operator finite difference methods are constructed via non standard finite difference methods nsfdm s for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments the main idea behind the construction of our method s is to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results unlike other fofdms constructed in standard ways the methods that we present in this paper are fairly simple to construct and thus enrich the class of fitted operator methods by adding these new methods these methods are shown to be e uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration this paper further clarifies several doubts e g why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies finally we provide some numerical examples which illustrate the theoretical findings copyright 2005 john wiley sons ltd

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

displacement in the parameter space versus spurious solution of discretization with large time step in order to investigate a possible correspondence between differential and difference equations it is important to possess discretization of ordinary differential equations it is well known that when differential equations are discretized the solution thus obtained depends on the time step used in the majority of cases such a solution is considered spurious when it does not resemble the expected solution of the differential equation this often happens when the time step taken into consideration is too large in this work we show that even for quite large time steps some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered to reduce such a displacement a judicious choice of the discretization scheme should be made to this end a recent discretization scheme based on the lie expansion of the original differential equations proposed by monaco and normand cyrot will be analysed such a scheme will be shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo period of the underlying dynamics

a chaotic mapping that displays its own homoclinic structure abstract we describe some properties of a two dimensional non linear mapping which has been derived from a differential equation model of turbulence for certain parameter values the mapping displays chaotic behaviour owing to the peculiar way in which the phase space expands and contracts parts of the attractor follow the oscillatory invariant curves associated with homoclinic points in non integrable systems

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

positive solutions of a nonlinear m point boundary value problem let ai 0 for i 1 m 3 and am 2 0 let i satisfy 0 1 2 m 2 1 and m 2i 1 ai i 1 we study the existence of positive solutions to the boundary value problem where a c 0 1 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed point theorem in cones

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four body problem abstract we apply the symmetry reduction method of roberts to numerically analyze the linear stability of a one parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four body problem with a mass m 0 1 as the parameter this reduces the linear stability analysis to the computation of two eigenvalues of a 3 3 matrix for each m 0 1 obtained from numerical integration of the linearized regularized equations along only the first one eighth of each regularized periodic orbit the results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over 0 1 with linear instability for m close or equal to 0 01 and linear stability for m close or equal to 1

a separating surface for sitnikov like n 1 body problems abstract we consider the generalized sitnikov problem of newtonian mechanics for periodic planar configurations of n bodies which are symmetric under rotation by a fixed angle the z axis is invariant we consider the effect of placing a massless particle on the z axis the study of the motion of this particle can then be modeled as a time dependent hamiltonian system we give a geometric construction of a surface in the three dimensional phase space separating orbits for which the massless particle escapes to infinity from those for which it does not a procedure for removing the periodicity condition of the planar configuration is outlined the construction of the surface is demonstrated numerically in a few examples

algorithms for controlling chaotic motion application for the bvp oscillator the bvp oscillator is a biologically and physically important dynamical system exhibiting chaotic motion phaselocking phenomenon and so on in this paper we investigate the controlling of chaos in this system both when periodic and constant currents are present we use the various control algorithms and investigate their effectiveness in particular we study the controlling by the following mechanisms i adaptive control algorithm ii weak periodic perturbations iii ott grebogi yorke method iv weak feedback control and v addition of noise also we show the suppression of chaos by numerical models

modeling and synchronizing chaotic systems from time series data the problem of obtaining ordinary differential equations ode s from chaotic time series data is addressed the vector fields for the ode s are polynomials constructed from a basis set that is orthonormal on the data the method for constructing the model is similar to the integration of ode s using adams predictor corrector integration the method is compared to the usual euler model and is shown to be accurate for much larger sampling intervals in addition the method used to construct the model is capable of determining the optimal polynomial vector field for the given data finally we demonstrate that it is possible to synchronize in the sense of fujisaka and yamada prog theor phys 69 32 1983 as well as pecora and carroll phys rev lett 64 821 1990 phys rev a 44 2374 1991 the model to a time series synchronization is used as a nontrivial test to determine how close the model vector field is to the true vector field implications and possible applications of synchronization are discussed

positive solutions of an m point boundary value problem with sign changing nonlinearities let a i 0 for i 1 m 2 let x i satisfy 0 x 0 x 1 x 2 x m 2 x m 1 1 and m 2 i 1a email protected i 1 we study the existence of positive solutions to the boundary value problem u f t u 0 0

defect controlled numerical methods and shadowing for chaotic differential equations abstract it is well known that chaotic ode s are in the classical sense unstable or ill conditioned it is not clear that variable step size codes based on local error control can solve such problems in a useful way in particular it is usually difficult to show that the apparent chaos in the resulting solution is not a numerical artifact i show here that a defect controlled method gives useful solutions for chaotic problems a pragmatically modified definition of what it means for a dynamical system to be chaotic is also presented