classification of rotational special weingarten surfaces of minimal type in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r in this paper we classify the complete rotational special weingarten surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r i e rotational surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r whose mean curvature h and extrinsic curvature ke satisfy h f h2 ke for some function f in mathcal c 1 0 infty such that f 0 0 and 4x f x 2 1 for any x 0 furthermore we show the existence of non complete examples of such surfaces

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

extension and improvement to the krylov bogoliubov methods using elliptic functions abstract it is shown that the krylov bogoliubov methods that give the approximate oscillatory solution of the equation in terms of jacobi elliptic functions are applicable not only when c 1 0 and c 3 0 but also when c 1 0 and c 3 0 in particular the most precise of these methods the christopher brocklehurst method is discussed in detail its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters

on singly periodic minimal surfaces with planar ends the spaces of nondegenerate properly embedded minimal surfaces in quotients of r3 by nontrivial translations or by screw motions with nontrivial rotational part fixed finite topology and planar type ends are endowed with natural structures of finite dimensional real analytic manifolds this nondegeneracy is defined in terms of jacobi functions riemann s minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces we also give natural immersions of these spaces into certain complex euclidean spaces which turn out to be lagrangian immersions with respect to the standard symplectic structures

index one minimal surfaces in spherical space forms abstract we prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two this confirms a conjecture of r schoen for an infinite class of 3 manifolds

on the growth of supersolutions of nonlinear pde s on exterior domains abstract we obtain a comparison principle on annuli with catenoid like functions for supersolutions of non linear elliptic pdes over exterior domains in a non positively curved manifold with a pole this result is applied to get an upper estimate on the growth of such supersolutions and in particular of exterior graphs of non negative mean curvature

uniqueness of the riemann minimal examples we prove that a properly embedded minimal surface in r 3 of genus zero with infinite symmetry group is a plane a catenoid a helicoid or a riemann minimal example we introduce the language of hurwitz schemes to understand the underlying moduli space of surfaces in our setting

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

on ground state solutions for singular and semi linear problems including super linear terms at infinity abstract we establish a result concerning the existence of entire positive classical and bounded solutions which converge to zero at infinity for the semi linear equation u f x u x r n where f r n 0 0 is a suitable function and 0 is a real parameter this result completes the principal theorem of a mohammed a mohammed ground state solutions for singular semi linear elliptic equations nonlinear analysis 2008 doi 10 1016 j na 2008 11 080 mainly because his result does not address the super linear terms at infinity penalty arguments lower upper solutions and an approximation procedure will be used

frequency analysis of finite beams on nonlinear kelvin voight foundation under moving loads abstract the vibration of an euler bernoulli beam resting on a nonlinear kelvin voight viscoelastic foundation traversed by a moving load is studied in the frequency domain the objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response the parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient the solution is obtained by using the galerkin method in conjunction with the multiple scales method msm the governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations subsequently the solution is calculated for different harmonics by using the msm as one of the powerful perturbation techniques the steady state responses of the main harmonic as well as its two super harmonics are then obtained as a case study a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics moreover a thorough stability analysis of the system is carried out

minimal surfaces with helicoidal ends we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends in this way we are able to construct a family of complete minimal surfaces with helicoidal ends that contains the singly periodic genus one helicoid of hoffman karcher and wei

constant mean curvature graphs on exterior domains of the hyperbolic plane we prove an existence result for non rotational constant mean curvature ends in mathbb h 2 times mathbb r where mathbb h 2 is the hyperbolic real plane the value of the curvature is h in big 0 frac 1 2 big we use schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of mathbb h 2 we also prove a fine property of the asymptotic behavior of the rotational ends introduced by sa earp and toubiana

analytical methods for solving strongly non linear differential equations abstract in this paper various approximate analytical methods for obtaining solutions for strongly non linear differential equations in a complex function are developed the methods are based on the solution of the generating differential equation with a cubic complex term the method of harmonic balance the method of krylov bogoliubov and the elliptic perturbation method are adopted for solving strongly non linear differential equations in a complex function three examples are analyzed they describe the vibrations of a rotor with non linearity

an elliptic perturbation method for certain strongly non linear oscillators abstract an elliptic perturbation method is presented for calculating periodic solutions of strongly non linear oscillators of the formx c1x c3x3 f x x in which the jacobian elliptic functions are employed instead of usual circular functions in the conventional perturbation procedure three types of duffing van der pol equation withf x x 1 x2 xare studied in detail the present method can give a second approximate solution and therefore its accuracy is much higher than that of other methods in which the solutions are only first approximations

periodic solutions of strongly quadratic non linear oscillators by the elliptic lindstedt poincare method abstract the elliptic lindstedt poincare method is used employed to study the periodic solutions of quadratic strongly non linear oscillators of the form x c 1 x c 2 x 2 e f x x in which the jacobian elliptic functions are employed instead of the usual circular functions in the classical lindstedt poincare method the generalized van de pol equation with f x x 0 1 x 2 x 2 is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

non existence and existence of entire solutions for a quasi linear problem with singular and super linear terms abstract we establish results concerning non existence and existence of entire positive solutions for the nonlinear elliptic problem p u a x u m b x u n in r n u 0 in r n u x x 0 where m p 1 n with 1 p n a b 0 a b 0 are locally holder continuous functions and 0 is a real parameter the main purpose of this paper in short is to complement the principal theorem of b xu and z yang entire bounded solutions for a class of quasi linear elliptic equations boundary value problems 2007 art id 16407 8 pp showing existence and non existence of solutions for the above problem for 0 appropriately

ground state solutions for a semilinear elliptic problem with a convection term abstract by a sub supersolution method and a perturbed argument we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem u p x u q f x u u 0 x r n lim x u x 0 where q 1 2 p x c loc r n for some 0 1 p x 0 x r n and f r n 0 0 is a locally holder continuous function which may be singular at zero

the global structure of constant mean curvature surfaces in a recent ground breaking paper kapouleas constructed this century s first examples of complete finite topology properly embedded surfaces with non zero constant mean curvature ka l geometrically these surfaces are quite easy to visualize as we explain below the purpose of the present paper is to characterize the extent to which kapouleas examples are typical of all embedded constant mean curvature surfaces first and throughout this paper we scale space so that the constant mean curvature the trace of the second fundamental form of our surface l is identically one and we call such a surface an mc1 surface the only complete embedded mc1 surfaces of finite topology known before kapouleas work arise from the 1parameter family of axially symmetric and periodic examples found by delaunay in 1841 using the maximum axial radius r of such a surface as a parameter this family begins with the cylinder r 1 and evolves through embedded periodic surfaces until the family degenerates to a linear chain of touching spheres r 2 the delaunay family continues for all r 2 as periodic immersions kapouleas construction of embedded surfaces begins with a piecewise linear graph in r 3 having the property that edges are either finite segments with length a multiple of 4 or half infinite rays and with the further property that at each vertex some linear combination of their direction vectors with positive coefficients adds to zero see fig 1 he constructs an approximate solution to the problem by centering radius r 2 spheres at all vertices and connecting these smoothly by delaunay segments having r 2 but near 2 with axes along the graph edges since the periods of the delaunay surfaces are slightly longer than 4 he must assume also a flexibility condition on the graph to guarantee that such a gluing

a new approach for solving a complex valued differential equation abstract in this paper two analytical approximate solving procedures for a complex valued differential equation are developed one of the methods represents the generalization of the krylov bogolubov method for a strong differential equation with complex function the second method is based on the first integrals of the system the differential equation is transformed introducing the perturbed first integrals and the polar co ordinates the solution is obtained applying the straightforward series expansion the solution for the special case of without impact initial conditions is considered the method is applied on the system with strong cubic non linearity the small gyroscopic function and damping function are introduced the analytical approximate solution is compared with numerical exact one and shows a good agreement

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

the second order expansion of solutions to a singular dirichlet boundary value problem abstract in this paper we mainly study the second order expansion of classical solutions in a neighborhood of to the singular dirichlet problem u b x g u a x f u u 0 x u 0 where is a bounded domain with smooth boundary in r n 0 the weight functions b a c loc are positive in and both may be vanishing or be singular on the boundary the function g c 1 0 0 satisfies lim t 0 g t and f c 0 0 we show that the nonlinear term a x f u does not affect the second order expansion of solutions in a neighborhood of to the problem for some kinds of functions b and a

a uniqueness theorem for the singly periodic genus one helicoid the singly periodic genus one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature the celebrated hoffman karcher wei s genus one helicoid the objective of this paper is to give a uniqueness theorem for the singly periodic genus one helicoid provided the existence of one symmetry

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

a singular boundary value problem for a degenerate elliptic pde abstract we consider a singular boundary value problem associated with the laplacian and having nonseparable inhomogeneous term we establish the existence of ground state solutions to such a problem in bounded domains as well as in the whole euclidean space the investigation is based on subsolution and supersolution methods and existence of principal eigenfunctions to the eigenvalue problem of dirichlet type corresponding to the laplacian

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

global error minimization method for solving strongly nonlinear oscillator differential equations a modified variational approach called global error minimization gem method is developed for obtaining an approximate closed form analytical solution for nonlinear oscillator differential equations the proposed method converts the nonlinear differential equation to an equivalent minimization problem a trial solution is selected with unknown parameters next the gem method is used to solve the minimization problem and to obtain the unknown parameters this will yield the approximate analytical solution of the nonlinear ordinary differential equations odes this approach is simple accurate and straightforward to use in identifying the solution to illustrate the effectiveness and convenience of the suggested procedure a cubic duffing equation with strong nonlinearity is considered comparisons are made between results obtained by the proposed gem method the exact solution and results from five recently published methods for addressing duffing oscillators the maximal relative error for the frequency obtained by the gem method compared with the exact solution is 0 0004 which indicates the remarkable precision of the gem method

a schoen theorem for minimal surfaces in h 2 r abstract in this paper we prove that a complete minimal surface immersed in h 2 r with finite total curvature and two ends each one asymptotic to a vertical geodesic plane must be a horizontal catenoid moreover we give a geometric description of minimal ends of finite total curvature in h 2 r we also prove that a minimal complete end e with finite total curvature is properly immersed and that the gaussian curvature of e is locally bounded in terms of the geodesic distance to its boundary

riemann bilinear relations on minimal surfaces from a physical point of view minimal surfaces in r are objects submitted to a balanced force system consisting in the forces associated to non zero onedimensional homology classes in the surface several geometric properties can be studied in terms of those forces as embeddedness symmetries and deformations leading up to uniqueness and non existence results see 11 13 more precisely each closed curve in a minimal surface s r carries a force that expresses the stress produced by an unit conormal vector field along on the whole surface the action of this conormal field provides a tendency of translation or linear momentum f and another one of rotation around an axis or angular momentum m the first one is given by the force vector

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

generalized fourier series for the study of limit cycles the approximate solution to first order of non linear differential equations is studied using the method of harmonic balance with generalized fourier series and jacobian elliptic functions as an interesting use of the series very good analytic approximations to the limit cycles of lienard s ordinary differential equation ode x g x f x x are presented specifically it is shown that contrary to an opinion given in a well known textbook on non linear oscillations g x not only modifies the period but influences the topology in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 for 0 1 the presence of zero one or three limit cycles is found to depend on the value of ab

complex normal form for strongly non linear vibration systems exemplified by duffing van der pol equation abstract we extend the normal form method to study the asymptotic solutions of strongly non linear oscillatorsu 2u f u u wheref u u contains only linear and cubic non linear terms the novel contribution is the ansatzu u i 1 where 1is to be determined allowing for the change of the fundamental frequency during the course of vibration rather than usingu u i as suggested by nayfeh with the present method not only the stability of the periodic solutions but also the asymptotic expressions for the periodic solutions can be obtained easily the results obtained by the method presented coincide very well with the results obtained by numerical integration for the duffing van der pol oscillator withf u u 1 u2 u u3 when 1 nayfeh s method givers qualitatively different results from the numerical integration while our method works well even when 1 3 since nayfeh s method is based on weak non linearities and 1 3 is beyond the valid range of assumption

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

the space of complete embedded maximal surfaces with isolated singularities in the 3 dimensional lorentz minkowski space we prove that a complete embedded maximal surface in 3 dx12 dx22 dx32 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane and so it is asymptotic to a spacelike plane or a half catenoid we show that the moduli space of entire maximal graphs over x3 0 in with n 1 2 singular points and vertical limit normal vector at infinity is a 3n 4 dimensional differentiable manifold the convergence in means the one of conformal structures and weierstrass data and it is equivalent to the uniform convergence of graphs on compact subsets of x3 0 moreover the position of the singular points in 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology we also introduce the moduli space of marked graphs with n 1 singular points a mark in a graph is an ordering of its singularities which is a n 1 sheeted covering of we prove that identifying marked graphs differing by translations rotations about a vertical axis homotheties or symmetries about a horizontal plane the corresponding quotient space is an analytic manifold of dimension 3n 1 this manifold can be identified with a spinorial bundle associated to the moduli space of weierstrass data of graphs in

the classification of complete minimal surfaces with total curvature greater than 12 we classify complete orientable minimal surfaces with finite total curvature 87r introduction the classification of complete minimal surfaces with finite total curvature in r3 has been an important problem in the classical differential geometry some basic properties of these surfaces were studied by r osserman see 8 9 who showed the first nontrivial result about this subject concretely he characterized the catenoid and enneper surface as the unique complete orientable minimal surfaces of total curvature 47 however until recent years no more relevant results have been obtained w h meeks 6 gave the classification of nonorientable complete minimal surfaces with total curvature greater than 87 this paper is concerned with the total classification of orientable complete minimal surfaces with total curvature 87 chen and gackstatter 1 discovered the first example of a complete minimal surface properly of genus 1 see theorem 1 the picture of chen gackstatter surface is obtained by joining a handle on enneper s surface this genus one minimal surface has total curvature 87 and no other examples of such surfaces were found so it is expected that no other genus one orientable minimal surface of total curvature 87 does exist in this paper we give a proof of this fact more precisely we prove that chen gackstatter surface is the only genus one orientable complete minimal surface with finite total curvature 87 of course it is not difficult to find genus zero minimal surfaces with total curvature 8i a geometrically interesting example described by jorge and meeks 5 is the trinoid this surface has three embedded catenoid ends moreover its normal vectors at these ends are placed symmetrically in an equator of s2 received by the editors april 27 1990 and in revised form july 20 1990 1980 mathematics subject classification 1985 revision primary 53a10 research partially supported by dgicyt grant ps87 01 15 c03 02

a boundary value problem for normalized finsler infinity laplacian equations with singular nonhomogeneous terms abstract we study boundary value problems of finsler infinity laplacian equations with nonhomogeneous terms that may exhibit singularity when solutions vanish on the boundary

a remark on the existence of entire solutions of a singular semilinear elliptic problem it is proved that the singular semilinear elliptic equation u p x g u 0 p x x rn 2 n lims 0 g s andg c1 0 0 which is strictly decreasing in 0 has a unique positivec2 loc rn solution that decays to zero near provided 0 t t dt where t imax i x tg x

periodic solutions of strongly quadratic non linear oscillators by the elliptic perturbation method abstract the elliptic perturbation method is applied to the study of the periodic solutions of strongly quadratic non linear oscillators of the formx c1x c 2x2 f x x in which the jacobian elliptic functions are employed the generalized van der pol equation withf x x 0 1x 2x2is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

periodic solutions of strongly non linear oscillators by the multiple scales method abstract the multiple scales method developed for the systems with small non linearities is extended to the case of strongly non linear self excited systems two types of non linearities are considered quadratic and cubic the solutions are expressed in terms of jacobian elliptic functions higher order approximations of solution as well as modulations of amplitude and phase are derived comparisons to numerical simulations are provided and discussed