using triangular orthogonal functions for solving fredholm integral equations of the second kind the present work proposes a method for solving fredholm integral equations this is demonstrated by using a complementary pair of orthogonal triangular functions set derived from the well known block pulse functions set the operational matrices for integration product of two triangular functions and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of fredholm integral equation to the solution of algebraic equations illustrative examples are included to show the high accuracy of the estimation and to demonstrate validity and applicability of the method

periodic solutions with prescribed energy for singular conservative systems involving strong forces this paper establishes under appropriate conditions which are stated below the existence of periodic solutions of the equation ii t vv q t 0 1 1 where q t e rn n 1 2 and v i is the gradient of a real function i which has a singularity at the origin in the literature the problem of the existence of periodic solutions of 1 1 has been addressed following two different points of view i given t 0 find periodic solutions of 1 1 having t as period fixed period problem ii given h e r find periodic solutions of 1 1 such that for all t e r

chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

dynamics of conservative bykov cycles tangencies generalized cocoon bifurcations and elliptic solutions abstract this paper presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a conservative bykov cycle where trajectories turn in opposite directions near the two saddle foci we show that within the class of divergence free vector fields that preserve the cycle tangencies of the invariant manifolds of two hyperbolic saddle foci densely occur the global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with non uniformly hyperbolic suspended horseshoes a generalized version of the cocoon bifurcations for conservative systems is obtained

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

new approach for numerical solution of hammerstein integral equations in this paper we use a numerical procedure for solving a class of nonlinear integral equations of the hammerstein type using the globally defined sinc basis functions properties of the sinc procedure are utilized to reduce the computation of the hammerstein equations to some algebraic equations we used some numerical examples to illustrate the accuracy and the implementation of the method

numerical implementation of adomian decomposition method for linear volterra integral equations of the second kind in this paper we propose new ideas to implement adomian decomposition method to solve volterra integral equations numerical examples are presented to illustrate the method for linear volterra integral equations of the second kind

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

an expansion method to treat integral equations this paper introduces a method for treating integral equation of the second and first kind of both types fredholm or volterra the method is very useful in treating integral equation of the first kind the method is tested by various numerical examples

dense heteroclinic tangencies near a bykov cycle abstract this article presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a bykov cycle where trajectories turn in opposite directions near the two nodes we say that the nodes have different chirality we show that in the set of vector fields defined on a three dimensional manifold there is a class where tangencies of the invariant manifolds of two hyperbolic saddle foci occur densely the class is defined by the presence of the bykov cycle and by a condition on the parameters that determine the linear part of the vector field at the equilibria this has important consequences the global dynamics is persistently dominated by heteroclinic tangencies and by newhouse phenomena coexisting with hyperbolic dynamics arising from transversality the coexistence gives rise to linked suspensions of cantor sets with hyperbolic and non hyperbolic dynamics in contrast with the case where the nodes have the same chirality we illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three dimensional sphere

on the effect of invisibility of stable periodic orbits at homoclinic bifurcations a b s t r a c t we study bifurcations of a homoclinic tangency to a saddle focus periodic point we show that the stability domain for single round periodic orbits which bifurcate near the homoclinic tangency has a characteristic comb like structureanddependsstronglyonthesaddlevalue i e onthearea contracting properties of the map at the saddle focus in particular when the map contracts two dimensional areas we have a cascade of periodic sinks in any one parameter family transverse to the bifurcation surface that corresponds to the homoclinic tangency however when the area contraction property is broken while three dimensional volumes are still contracted the cascade of single round sinks appears with probability zero only thus if three dimensional volumes are contracted chaos associated with a homoclinictangencytoasaddle focusisalwaysaccompaniedbystabilitywindows howevertheviolation of the area contraction property can make the stability windows invisible in one parameter families

application of sinc collocation method for solving a class of nonlinear fredholm integral equations in this paper the numerical solution of nonlinear fredholm integral equations of the second kind is considered by two methods the methods are developed by means of the sinc approximation with the single exponential se and double exponential de transformations these numerical methods combine a sinc collocation method with the newton iterative process that involves solving a nonlinear system of equations we provide an error analysis for the methods so far approximate solutions with polynomial convergence have been reported for this equation these methods improve conventional results and achieve exponential convergence some numerical examples are given to confirm the accuracy and ease of implementation of the methods

harmonic wavelet method towards solution of the fredholm type integral equations of the second kind periodic harmonic wavelets phw were applied as basis functions in solution of the fredholm integral equations of the second kind two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions it is proved that phw satisfy the properties of the multiresolution analysis

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

using rationalized haar wavelet for solving linear integral equations the main purpose of this paper is to demonstrate that using rationalized haar wavelet for solving linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

global bifurcations close to symmetry abstract heteroclinic cycles involving two saddle foci where the saddle foci share both invariant manifolds occur persistently in some symmetric differential equations on the 3 dimensional sphere we analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1 dimensional connection the saddle foci have the same chirality when part of the symmetry is broken the 2 dimensional invariant manifolds intersect transversely creating a heteroclinic network of bykov cycles we show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics there are n pulse heteroclinic tangencies trajectories that follow the original cycle n times around before they arrive at the other node each n pulse heteroclinic tangency is accumulated by a sequence of n 1 pulse ones this coexists with the suspension of horseshoes defined on an infinite set of disjoint strips where the first return map is hyperbolic we also show how as the system approaches full symmetry the suspended horseshoes are destroyed creating regions with infinitely many attracting periodic solutions

periodic solutions of a class of singular hamiltonian systems soit r n 0 n n n 3 et soit v c 1 rx r on etudie l existence des solutions periodiques t u t avec une periode prescrite du systeme suivant de n equations differentielles d ordre 2 u v t u

solution of a volterra integral equation by the sinc collocation method in this paper a collocation procedure is developed for a linear volterra integral equation of the second kind by using the sinc basis functions the approximate solutions are given so that the auxiliary basis functions satisfy the four different end conditions it is shown that the sinc procedure converges to the solution at an exponential rate of o e k n numerical results are included to confirm the efficiency and accuracy of the method

global generic dynamics close to symmetry abstract our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in s 3 the cycle involves two saddle foci of different type and is structurally stable within the class of z 2 z 2 symmetric vector fields the cycle contains a two dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry breaking perturbations gradually breaking the symmetry in a two parameter family we get a wide range of dynamical behaviour an attracting periodic trajectory other heteroclinic trajectories homoclinic orbits n pulses suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of shilnikov type we also show that generically the coexistence of linked homoclinic orbits at the two saddle foci has codimension 2 and takes place arbitrarily close to the symmetric cycle

numerical solutions for the system of fredholm integral equations of second kind by a new approach involving semiorthogonal b spline wavelet collocation method in this paper a new approach based on linear b spline wavelet method has been developed to approximate the solutions of system of linear fredholm integral equations of second kind compactly supported linear semi orthogonal b spline scaling functions and wavelet functions together with their dual functions are applied to approximate the solutions of linear fredholm integral equations system of second kind this method reduces the system of integral equations to a linear system of algebraic equations that can be solved easily with any of the usual methods the numerical results obtained by the present method have been compared with those obtained by adaptive method based on trapezoidal rule numerical examples are presented to illustrate the accuracy of the method

spiralling dynamics near heteroclinic networks abstract there are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically we construct explicitly a two parameter family of vector fields on the three dimensional sphere s 3 whose flow has a spiralling attractor containing the following two hyperbolic equilibria heteroclinic trajectories connecting them transversely and a non trivial hyperbolic invariant and transitive set the spiralling set unfolds a heteroclinic network between two symmetric saddle foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols coexisting with newhouse phenomena the vector field is the restriction to s 3 of a polynomial vector field in r 4 in this article we also identify global bifurcations that induce chaotic dynamics of different types

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

quasiattractors and homoclinic tangencies recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented such systems cover a large variety of dynamical models known from natural applications and it is established that so called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or lorenz like attractors for instance the impossibility of giving a finite parameter complete description of dynamics and bifurcations of the quasiattractors is shown besides it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive lyapunov exponents if the dimension of a phase space is not too low greater than four for flows and greater than three for maps it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors keywords attractor homoclinic bifurcations dynamical models

numerical solution of some classes of integral equations using bernstein polynomials this paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using bernstein polynomials as basis the integral equations considered are fredholm integral equations of second kind a simple hypersingular integral equation and a hypersingular integral equation of second kind the method is explained with illustrative examples also the convergence of the method is established rigorously for each class of integral equations considered here

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials abstract we study the existence of periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials we found periodic solutions lying between two prescribed spheres

new proof and generalizations of the demidowitsch schneider criterion a new proof of the demidowitsch schneider criterion on the absence of closed trajectories of r3 is given the new proof is generalized and applied in several directions rn vector fields n 3

two methods for solving integral equations the objective of this work is to introduce two efficient strategies that produce the accurate solutions to linear and nonlinear volterra and fredholm integral equations of the second kind for illustration several linear nonlinear and integro differential examples have been tested and the results showed improvements over existing techniques

collocation method for linear and nonlinear fredholm and volterra integral equations a collocation procedure is developed for the linear and nonlinear fredholm and volterra integral equations using the globally defined b spline and auxiliary basis functions the solution is collocated by cubic b spline and then the integral equation is approximated by the 5 points gauss turan quadrature formula with respect to the legendre weight function combination of these two approaches is the main idea of this paper to reduce the cost of computation and complexity the error analysis of proposed numerical method is studied theoretically numerical results are given to illustrate the efficiency of the proposed method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

closed orbits of fixed energy for singular hamiltonian systems this paper deals with the existence of periodic solutions of q v q 0 such that 1 2 q 2 v q h where q r n h is a given number v c 2 r n 0 r has a singularity at x 0 and v denotes the gradient of v

non simultaneous collision periodic solutions forn body problems abstract in this paper we prove the existence of a non simultaneous collision t periodic solution for any given t for a class of n body type problems

cubic b spline approximation for linear stochastic integro differential equation of fractional order abstract in this paper the cubic b spline collocation method is used for solving the stochastic integro differential equation of fractional order we show that stochastic integro differential equation of fractional order is equivalent to a modified stochastic integral equation then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation using this method the problem solving turns into a linear system solution of equations also the convergence analysis of this numerical approach has been discussed in the end examples are given to test the accuracy and the implementation of the method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

unique solvability of nonlinear volterra equations in weighted spaces abstract we investigate integral equations of the form x t g t t f t s x s ds in general this equation is history dependent so one needs to give an initial condition on 0 in order to obtain a unique solution by introducing a weight function on r we can single out a class of admissible solutions and give conditions for the unique solvability of in this restricted class we also study some fredholm equations on these weighted spaces in addition we also treat a class of equations of the first kind for which similar conclusions can be drawn

numerical solution of linear integral equations by using sinc collocation method a collocation procedure is developed for linear fredholm integral equations of the second kind using sinc basis functions the convergence rate of the method is o e k m numerical results are included to confirm the efficiency and accuracy of the method

strange attractors and wandering domains near a homoclinic cycle to a bifocus abstract in this paper we explore the three dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence if the invariant manifolds of the bifocus satisfy a non degeneracy condition a sequence of hyperbolic suspended horseshoes arises near the cycle with one expanding and two contracting directions we extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits under an additional hypothesis about the coexistence of two heteroclinically related periodic points one without dominated splitting into one dimensional sub bundles the heteroclinic tangencies can be slightly modified in order to satisfy tatjer s conditions for a generalized tangency of codimension two this configuration may be seen as the organizing center by which one can obtain bogdanov takens bifurcations and therefore strange attractors infinitely many sinks and non trivial contracting wandering domains

using lin s method to solve bykov s problems abstract we consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria the constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions and called t points in this context we study codimension two t points and their unfoldings in r n in our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria in doing so we establish lin s method as a unified approach to re gain and extend results of bykov s seminal studies and related works to a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane

multiple closed orbits of fixed energy forn body type problems with gravitational potentials abstract using the equivariant ljusternik schnirelmann theory and the estimates of the upper bound and lower bound of the critical values for the variational functional we obtain results in general concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planarn body type problems

solving nonlinear integral equations in the urysohn form by newton kantorovich quadrature method the newton kantorovich method is a well known method for solving nonlinear integral equations this method attempts to solve a sequence of linear integral equations in this article we develop a new method which is a combination of the newton kantorovich and quadrature methods the new method solves the nonlinear integral equations of the urysohn form in a systematic procedure some numerical examples are provided and the obtained numerical approximations are compared with the corresponding exact solutions

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type

periodic solutions to some problems of n body type we prove the existence of at least one t periodic solution to a dynamical system of the type m i ddot u i sum limits j 1 j ne i n triangledown v ij u i u j text t 1 where the potentials v ij are t periodic in t and singular at the origin u i e r k i 1 n and k 3 we also provide estimates on the h 1 norm of this solution the proofs are based on a variant of the ljusternik schnirelman method the results here generalize to the n body problem some results obtained by bahri rabinowitz on the 3 body problem in 6