notes on a conservative nonlinear oscillator the amplitude frequency relationship is an important mathematical property for a nonlinear oscillator he s amplitude frequency formulation and the max min approach are used to handle the conservative nonlinear oscillator email protected email protected 2 x 0 for the amplitude frequency relationship the obtained result is compared with those in the open literature revealing the effectiveness of the used methods

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

global conservative solutions of a modified two component camassa holm shallow water system abstract we first prove the existence of global conservative solutions to the cauchy problem for a modified two component camassa holm shallow water system then we show that these global solutions which depend continuously on the initial date construct a semigroup

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

a convenient technique for evaluating angular frequency in some nonlinear oscillations in this paper a convenient technique for evaluating angular frequency in some nonlinear oscillations is proposed it is well known that once the restoring force function is given beforehand the period of motion can be determined by an integral the angular frequency has a relation with the period of motion as well as the integral one makes a little modification for the integrand in the integral and a change of variable it is found that if the three divisions are chosen on the integration interval and the trapezoid quadrature rule is used a higher accurate result for the angular frequency can be achieved for the restoring force being an odd function three numerical examples are presented the eardrum type oscillation is studied as well higher accurate results for the angular frequency are obtained in all those examples

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

persistence properties for a family of nonlinear partial differential equations abstract we examine the persistence of decay properties for a family of dispersive nonlinear partial differential equations we show that certain decay properties of the initial data persist for as long as the solution exists on the other hand for a subset of the family certain decay rates are possible only for the trivial solution for example the only solution that remains with compact support for any further time is the trivial solution

global weak solutions to the novikov equation abstract the existence of global weak solutions to the cauchy problem for the novikov equation is established in the space c 0 r l 0 h 1 r without the sign condition on the initial value the limit of viscous approximations for the equation is used to establish the existence of the global weak solution the key elements in our analysis include a new one sided super bound and a new higher norm estimate on the first order derivatives of the solution

approximating small and large amplitude periodic orbits of the oscillator x 1 x 2 x 0 abstract it is shown that the oscillator x 1 x 2 x 0 is conservative and scaling laws for the period for small and large amplitude vibrations are derived analytical approximations of the periodic orbits are also constructed and these show excellent agreement with numerical solutions

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

blow up phenomena and local well posedness for a generalized camassa holm equation with cubic nonlinearity abstract in this paper we consider the cauchy problem for a generalized camassa holm equation with cubic nonlinearity in besov spaces we first establish the local well posedness of the equation in the besov space b p r s by using the littlewood paley theory then under a sign condition we reach the sign preserved property and a precise blow up criterion applying this precise criterion we present a blow up result and the precise blow up rate for strong solutions to the equation

local well posedness and blow up criteria for a two component novikov system in the critical besov space abstract in this paper we mainly investigate the cauchy problem of a two component novikov system we first prove the local well posedness of the system in besov spaces b p r s 1 b p r s with p r 1 s max 1 1 p 3 2 by using the littlewood paley theory and transport equations theory then by virtue of logarithmic interpolation inequalities and the osgood lemma we establish the local well posedness of the system in the critical besov space b 2 1 1 2 b 2 1 3 2 moreover we present two blow up criteria for the system by making use of the conservation laws

pure odd order oscillators with constant excitation abstract in this paper the excited vibrations of a truly nonlinear oscillator are analyzed the excitation is assumed to be constant and the nonlinearity is pure without a linear term the mathematical model is a second order nonhomogeneous differential equation with strong nonlinear term using the first integral the exact value of period of vibration i e angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained the closed form solution has the form of gamma function the period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term for the case of pure odd order oscillators the approximate solution of differential equation is obtained in the form of trigonometric function the solution is based on the exact value of period of vibration for the case when additional small perturbation of the pure oscillator acts the so called cveticanin s averaging method for a truly nonlinear oscillator is applied two special cases are considered one when the additional term is a function of distance and the second when damping acts to prove the correctness of the method the obtained results are compared with those for the linear oscillator example of pure cubic oscillator with constant excitation and linear damping is widely discussed comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement the investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems for example such mechanical systems are seats in vehicles supports for machines cutting machines with periodical motion of the cutting tools presses etc the examples can be find in electronics electromechanical devices like micro actuators and micro oscillators in music instruments hammers in piano in human voice producing folds voice cords etc

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

homotopy perturbation method for a conservative x1 3 force nonlinear oscillator the homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x 1 3 we find this method works very well for the whole range of initial amplitudes excellent agreement of the approximate frequencies with the exact ones has been demonstrated and discussed only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0 60 for small and large values of oscillation amplitude while this relative error is as low as 0 050 for the second iteration comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators

global dissipative solutions of a modified two component camassa holm shallow water system by introducing a new set of independent variables we first transform a modified two component camassa holm shallow water system into a semilinear system to obtain a dissipative solution we modify the corresponding system into a discontinuous system then we map the solution of system to the dissipative solution of original equation moreover we prove that these global dissipative solutions construct a semigroup

a perturbation technique that works even when the non linearity is not small abstract we extend the lindstedt poincare small non linearity based perturbation scheme to strongly non linear systems the extended technique starts from a physically non existent neighboring linear system this system is created by adding an optimal linear spring term to the system side of each linear differential equation starting at the zeroth level and balancing that term by adding an equal one to the forcing side one level down unlike the ritz method which also works for strongly non linear systems results can be obtained beyond the first level without resorting to numerical techniques thus the extended method keeps the intuition guiding straightforwardness of the perturbation method while giving accurate beyond first level hand derivable numerical results examples are presented for free vibrations with cubic and antisymmetric quadratic non linearities and for harmonic solutions of the undamped duffing equation the forms of the solutions allow the regions of validity of the expansions to be estimated an internal method of estimating accuracy of the perturbation solutions is developed and checked with the free vibration examples the exact solutions of which are available this internal method is then used to estimate the accuracy of the new solutions to the duffing equation these new duffing equation solutions give accurate results in all regions where stable harmonic solutions are known to exist in particular they give accurate results at and near the vertical tangent to the resonance curve a region where hayashi s amplitude expansion based perturbation solution necessarily breaks down

on prime ideals in hereditary pi rings abstract it is shown that if p is a prime ideal of a right hereditary ring r satisfying a polynomial identity then r p is a hereditary noetherian ring

global weak solutions for the novikov equation in the paper we mainly study the existence and uniqueness of global weak solutions for the novikov equation we first recall some results and definitions on strong solutions and weak solutions for the equation then we show that the equation has smooth solutions which exist globally in time provided the initial data satisfy certain sign conditions finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions

on the determination of approximate periodic solutions of some non linear ode abstract we propose a numerical symbolic method for the approximation of periodic solutions of a type of non linear ode the efficiency of our method is contrasted with the harmonic balance method and with another one which combines the differential transformation method with pade approximants on a non trivial example the relativistic oscillator it is shown that our method is computationally more reliable

on the cauchy problem for a two component b family system with high order nonlinearity abstract this paper is devoted to the cauchy problem of two component b family system with high order nonlinearity which includes the well known camassa holm equation degasperis procesi equation novikov equation and their two component extension systems as special cases firstly we establish the local well posedness of the system in the non homogeneous besov space by using the littlewood paley decomposition and the transport equation theory secondly we investigate the blow up of strong solutions to the system thirdly we study the gevrey regularity and analyticity of the solutions to the system in the gevrey sobolev spaces by using the generalized ovsyannikov theorem moreover we find a lower bound of the lifespan and prove the continuity of the data to solution mapping

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

solutions of a quadratic nonlinear oscillator iteration procedure a modified iteration procedure is applied to a quadratic nonlinear oscillator qno when the solutions of the two auxiliary equations are available we obtain the first and second analytical approximate solutions to the qno our second approximation result significantly improves on the accuracy of the approximate solution obtained by using the first order harmonic balance method

a successive integration technique for the solution of the duffing oscillator equation with damping and excitation abstract in this paper a successive integration technique is suggested for solving the duffing oscillator equation with damping and excitation in the technique one performs integrations from some initial values over a wider range some of them may even be divergent however some of them will give a convergent result such that the periodic condition of motion is satisfied in fact the convergent result represents a stable periodic solution for the motion if a convergent result or a stable periodic solution is obtained the stability test is passed a harmonic balance method in conjunction with the successive integration technique abbreviated as hbmsit is also suggested in the method the initial values are obtained from the harmonic balance method therefore the hbmsit belongs to the successive integration techniques many examples with calculated results are presented

the residue harmonic balance for fractional order van der pol like oscillators when seeking a solution in series form the number of terms needed to satisfy some preset requirements is unknown in the beginning an iterative formulation is proposed so that when an approximation is available the number of effective terms can be doubled in one iteration by solving a set of linear equations this is a new extension of the newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series when fourier series is employed the method is called the residue harmonic balance in this paper the fractional order van der pol oscillator with fractional restoring and damping forces is considered the residue harmonic balance method is used for generating the higher order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator the highly accurate solutions to angular frequency and limit cycle of the fractional order van der pol equations are obtained analytically the results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives the influence of the fractional order on the geometry of the limit cycle is investigated for the first time

iterative solution of some nonlinear differential equations we propose an iterative method to solve some non linear ordinary differential equations comparing on the mathieu van der pol and hill equation of fourth order we see that this method is much more efficient than the well known methods by lyapunov or picard

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

several numerical solution techniques for nonlinear eardrum type oscillations three numerical solution techniques for nonlinear eardrum type oscillations with an even function in the restoring force are studied the first suggested technique is named the target function technique in the technique the second zeros of the target function is the circular frequency of motion the second suggested technique is named the multiple parameter technique and the involved parameters are evaluated from the governing equation the initial conditions and properties of motion the third suggested technique is named the direct integration technique all techniques depend on the computer computation significantly and provide very accurate numerical results finally numerical examples are given

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

the cauchy problem for the integrable novikov equation in this paper we consider the cauchy problem for the integrable novikov equation by using the littlewood paley decomposition and nonhomogeneous besov spaces we prove that the cauchy problem for the integrable novikov equation is locally well posed in the besov space b p r s with 1 max 1 1 p 3 2 in particular when u 0 is an element of b p r s boolean and h l with 1 max 1 1 p 3 2 for all t is an element of 0 t we have that vertical bar vertical bar u t vertical bar vertical bar h l vertical bar vertical bar u 0 vertical bar vertical bar h l we also prove that the local well posedness of the cauchy problem for the novikov equation fails in b 2 3 2 infinity c 2012 elsevier inc all rights reserved

an artificial parameter decomposition method for nonlinear oscillators applications to oscillators with odd nonlinearities abstract a method for obtaining series solutions of nonlinear second order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well known adomian s decomposition technique the method is formulated in both integral and differential forms for the determination of the limit cycle of oscillators with odd nonlinearities two differential forms and one integral form of the artificial parameter method are presented these versions are based on introducing a linear stiffness term with an unknown frequency and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator the three formulations provide identical results and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified linstedt poincare methods however the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation it is also shown that two and three level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined

a modified mickens iteration procedure for nonlinear oscillators based on the mickens procedure a new iteration scheme is proposed for nonlinear oscillators different from the existing mickens procedures in the proposed procedure the algebraic equation governing the frequency of oscillation deduced at each iteration stage is always linear taking the duffing equation as an illustrative example excellent approximations can be easily obtained furthermore the proposed procedure is extended to obtain the limit cycle of the van der pol equation additionally a computational disadvantage of the mickens procedure is eliminated

an analytical technique for solving a class of strongly nonlinear conservative systems abstract recently an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms usually a set of nonlinear algebraic equations is solved in this method however analytical solutions of these algebraic equations are not always possible especially in the case of a large oscillation previously such algebraic equations for the duffing equation were solved in powers of a small parameter but the solutions measure desired results when the amplitude is an order of 1 in this article different parameters of the same nonlinear problems are found for which the power series produces desired results even for the large oscillation moreover two or three terms of this power series solution measure a good result when the amplitude is an order of 1 besides these a suitable truncation formula is found in which the solution measures better results than existing solutions the method is mainly illustrated by the duffing oscillator but it is also useful for many other nonlinear problems

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

comments on investigation of the properties of the period for the nonlinear oscillator x 1 x 2 x 0 this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp006 007

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

construction of approximate analytical solutions to a new class of non linear oscillator equation the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire by assuming that harmonic balance will hold solutions are devised for a steady state limit cycle and or limit point motion a method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective fourier expansions the latter technique is actually a generalization of the method proposed by kryloff and bogoliuboff 1943

asymptotic representations of the period for the nonlinear oscillator this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp006 007

comments on the general dynamics of the nonlinear oscillator x 1 x 2 x 0 abstract velocity dependent frequency vdf oscillators are represented for one degree of freedom systems by the differential equation x f x x 0 f 0 0 however the standard methods for determining approximations to the periodic solutions may breakdown in their corresponding first order calculations in particular the frequencies may become singular for finite values of the amplitude a technique combining averaging and first order iteration is shown to resolve this difficulty

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

analytic upper and lower bounds for the period of nonlinear oscillators we show that analytic expressions for the frequency of second order nonlinear oscillations can be obtained from dual variational principles at each amplitude two analytic expressions can be constructed which constitute upper and lower bounds to the exact value of the frequency the results are accurate at small and large amplitude and compare well with the perturbative approach

investigation of the properties of the period for the nonlinear oscillator x 1 x 2 x 0 abstract the mathematical properties of a nonlinear oscillator having the square of the angular frequency depend quadratically on the velocity is presented the major conclusion is that the exact period t a is well defined for all values of the amplitude where the initial conditions are taken to be x 0 a and x 0 0 an approximate expression is derived for t a

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

solutions of nonlinear oscillators with fractional powers by an iteration procedure a modified iteration procedure is applied to nonlinear oscillations with fractional powers with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

blow up phenomena and local well posedness for a generalized camassa holm equation in the critical besov space abstract in this paper we mainly study the cauchy problem for a generalized camassa holm equation in a critical besov space first by using the littlewood paley decomposition transport equations theory logarithmic interpolation inequalities and osgood s lemma we establish the local well posedness for the cauchy problem of the equation in the critical besov space b 2 1 1 2 then we give a new blow up criterion for the cauchy problem of the equation finally we present a new blow up result and the exact blow up rate of strong solutions to the equation by making use of the conservation law and the obtained blow up criterion

a classical iteration procedure valid for certain strongly nonlinear oscillators a classical iteration procedure is applied to nonlinear oscillations of conservative single degree of freedom systems with odd nonlinearity with the procedure the analytical approximate frequency and the corresponding periodic solution valid for small as well as large amplitudes of oscillation can be obtained two examples are given to illustrate the accuracy and effectiveness of the method another advantage of this classical iteration approach is that it also works if the linear part of restoring force is zero

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

solutions of the duffing harmonic oscillator by an iteration procedure a modified iteration procedure is applied to the duffing harmonic oscillator with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

on the picard lindelof method for nonlinear second order differential equations we consider a second order nonlinear ordinary differential equation which satisfies a lipschitz continuity condition and show that the method of variation of parameters allows us to write the solution as a volterra integral equation that includes the initial values of the solution and its first order derivative we also show that this integral formulation can be written as an iterative method which converges uniformly to the unique solution of the problem by making use of an integral identity the initial conditions can be eliminated from the integral equation and a new integral formulation that depends on the second order derivative of the solution is obtained it is shown that this integral formulation is exactly the same as that of the variational iteration method but does not require a variational principle neither does it require constrained variations for nonlinear terms we also show that the two integral formulations can be written in differential form and correspond to a two level iterative method by writing the second order ordinary differential equation as a two equation system of first order ordinary differential equations three different iterative procedures are developed but only one of them coincides with that derived from the two integral equations for the solution

global existence and blow up phenomena for an integrable two component camassa holm shallow water system abstract this paper is concerned with global existence and blow up phenomena for an integrable two component camassa holm shallow water system a new global existence result and several new blow up results of strong solutions to the system are presented our obtained results for the system are sharp and improve considerably earlier results

on the small amplitude approximation to the differential equation x 1 x 2 x 0 abstract we obtain the small amplitude expansion for the period of the nonlinear oscillator x 1 x 2 x 0 with the initial conditions x 0 a and x 0 0 calculate its radius of convergence and show that the inverted perturbation series appears to converge smoothly from below

he s parameter expanding methods for strongly nonlinear oscillators ji huan he s work on asymptotic techniques is briefly reviewed and his parameter expanding methods including modified lindstedt poincare method and bookkeeping parameter method are discussed in detail some remarkable virtues of the methods are exploited and their applications are illustrated

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

on linstedt poincare techniques for the quintic duffing equation abstract an artificial parameter method for the quintic duffing equation is presented the method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter and results in linear ordinary differential equations at each order in the parameter by imposing the nonsecularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation the method does not require that a small parameter be present in the governing equation and its results are compared with those of linstedt poincare modified linstedt poincare harmonic balance and galerkin techniques it is shown that these four techniques predict the same first order approximation to the frequency of oscillation as the artificial parameter method presented in this paper and the latter introduces higher order corrections at second order whereas similar corrections are introduced by the linstedt poincare and modified linstedt poincare methods at orders equal to and higher than three

a split frequency harmonic balance method for nonlinear oscillators with multi harmonic forcing abstract a new harmonic balance method hbm is presented for accurately computing the periodic responses of a nonlinear sdof oscillator with multi harmonic forcing and non expansible nonlinearities the presence of multi harmonic forcing requires a large number of solution harmonics with a substantial increase in computational demand for either the conventional or the incremental hbm in this method the oscillator equation error is first defined in terms of two functions originally proposed for obtaining free vibration periods in r e mickens iteration procedure for determining approximate solutions to nonlinear oscillator equations journal of sound and vibration 116 1987 185 187 and more recently r e mickens a generalised iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillations journal of sound and vibration 287 2005 1045 1051 a fourier series solution is assumed in which the total number of harmonics is fixed by the chosen discrete time interval this series is split into two partial sums nominally associated with either low frequency or high frequency harmonics by exploiting a convergence property of the equation error functions the total solution is obtained in a new iterative scheme in which the low frequency components are computed via a conventional hbm using a small number of algebraic equations whereas the high frequency components are obtained in a separate step by updating by gradually increasing the number of harmonics in the low frequency group the equation error can be progressively reduced efficient use is made of fft based algebraic equation generation which allows an important class of non expansible nonlinearities to be handled the proposed method is tested on a duffing type oscillator and an oscillator with a non expansible 7th power stiffness term where in both cases up to 24 component multi harmonic forcing is applied as a comparison a conventional hbm is also used on the duffing model in which the algebraic equations are generated in symbolic form to totally avoid errors from entering the formulation through complicated expansion of the cubic stiffness term as in i senjanovic harmonic analysis of nonlinear oscillations of cubic dynamical systems journal of ship research 38 3 1994 225 238 and in a raghothama s narayanan periodic response and chaos in nonlinear systems with parametric excitation and time delay nonlinear dynamics 27 2002 341 365 the paper shows that in obtaining period 1 solutions the computational accuracy and efficiency of the proposed method is very good