Odd Nonlinearities Research Articles

Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler–Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator–Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.

Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities

In this paper we study the existence of homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities ü−L(t)u+Wu(t,u)=0, where L(t) and W(t,u) are not assumed to be periodic in t. We get, under certain assumptions on L and W, infinitely many homoclinic solutions for both subquadratic and superquadratic cases by using the fountain theorems in critical point theory.

Existence results for solutions to nonlinear Dirac equations on compact spin manifolds

We consider super-linear and sub-linear nonlinear Dirac equations on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals on a Hilbert space. For both cases, we establish existence results via Galerkin type approximations and linking arguments. For a particular case of odd nonlinearities, we prove the existence of infinitely many solutions.

Solutions with many mixed positive and negative interior spikes for a semilinear Neumann problem

We study the existence of sign-changing multiple interior spike solutions for the following Neumann problem $$\varepsilon^2\Delta v-v+f(v) = 0 \,\, {\rm in} \,\, \Omega, \quad \frac{\partial v}{\partial \nu} = 0 \,\, {\rm on} \,\, \partial \Omega,$$ where Ω is a smooth bounded domain of \({\mathbb {R}^N}\) , e is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. No symmetry on Ω is assumed. To our knowledge, only positive interior peak solutions have been obtained for this problem and it remains a question whether or not multiple interior peak solutions with mixed positive and negative peaks exist. In this paper we assume that Ω is a two-dimensional strictly convex domain and, provided that k is sufficiently large, we construct a (k + 1)-peak solutions with k positive interior peaks aligned on a closed curve near ∂Ω and 1 negative interior peak located in a more centered part of Ω.

Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient

We address the Gross–Pitaevskii equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works, we show that the intersite cubic nonlinear terms in the discrete nonlinear Schrödinger (DNLS) equation appear beyond the applicability of assumptions of the tight-binding approximation. Instead of these terms, for an even linear potential and an odd nonlinearity coefficient, the DNLS equation and other reduced equations have the quintic nonlinear term, which correctly describes bifurcation of gap solitons in the semi-infinite gap.

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.

Nodal Clustered Solutions for Some Singularly Perturbed Neumann Problems

In this paper we are concerned with the following Neumann problem where ϵ is a small positive parameter, f is an odd superlinear and subcritical nonlinearity, Ω is a bounded 𝒞4 domain in ℝ N without any symmetry assumption. Denoting by ℋ(P), P ∈ ∂Ω, the mean curvature of the boundary, it is known that this problem has positive multiple boundary peak solutions with each peak concentrating at a different critical point of ℋ or with all the peaks approaching a local minimum point of ℋ. In this paper we assume that ℋ has a nondegenerate maximum point P 0 ∈ ∂Ω and we show that there exists a ℓ-peak solution with mixed positive and negative peaks concentrating at P 0.

On superlinear [formula omitted]-Laplacian equations in [formula omitted

We consider the p ( x ) -Laplacian equations in R N . The nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. We obtain ground states of the equations, improving a recent result of Fan [X.L. Fan, p ( x ) -Laplacian equations in R N with periodic data and nonperiodic perturbations, J. Math. Anal. Appl. 341 (2008) 103–119]. We also establish a Bartsch–Wang type compact embedding theorem for variable exponent spaces. Then, a multiplicity result for the equations is proved for odd nonlinearity.

Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

We study the existence and asymptotic behavior of positive and sign-changing multipeak solutions for the equation $$ -\varepsilon^2\Delta v+V(x)v=f(v)\quad{\rm in}\,\,\,\mathbb{R}^N, $$ where e is a small positive parameter, f a superlinear, subcritical and odd nonlinearity, V a uniformly positive potential. No symmetry on V is assumed. It is known (Kang and Wei in Adv Differ Equ 5:899–928, 2000) that this equation has positive multipeak solutions with all peaks approaching a local maximum of V. It is also proved that solutions alternating positive and negative spikes exist in the case of a minimum (see D’Aprile and Pistoia in Ann Inst H. Poincare Anal Non Lineaire 26:1423–1451, 2009). The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of V.

On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates

An analytical approximation is developed for solving large amplitude nonlinear free vibration of simply supported laminated cross-ply composite thin plates. Applying Kirchhoff’s hypothesis and the nonlinear von Kármán plate theory, a one-dimensional nonlinear second-order ordinary differential equation with quadratic and cubic nonlinearities is formulated with the aid of an energy function. By imposing Newton’s method and harmonic balancing to the linearized governing equation, we establish the higher-order analytical approximations for solving the nonlinear differential equation with odd nonlinearity. Based on the nonlinear differential equation with odd and even nonlinearities, two new nonlinear differential equations with odd nonlinearity are introduced for constructing the analytical approximations to the nonlinear differential equation with general nonlinearity. The analytical approximations are mathematically formulated by combining piecewise approximate solutions from such two new nonlinear systems. The third-order analytical approximation with better accuracy is proposed here and compared with other numerical and approximate methods with respect to the exact solutions. In addition, the method presented herein is applicable to small as well as large amplitude vibrations of laminated plates. Several examples including large amplitude nonlinear free vibration of simply supported laminated cross-ply rectangular thin plates are illustrated and compared with other published results to demonstrate the applicability and effectiveness of the approach.

Generalized decomposition methods for singular oscillators

Generalized decomposition methods based on a Volterra integral equation, the introduction of an ordering parameter and a power series expansion of the solution in terms of the ordering parameter are developed and used to determine the solution and the frequency of oscillation of a singular, nonlinear oscillator with an odd nonlinearity. It is shown that these techniques provide solutions which are free from secularities if the unknown frequency of oscillation is also expanded in power series of the ordering parameter, require that the nonlinearities be analytic functions of their arguments, and, at leading-order, provide the same frequency of oscillation as two-level iterative techniques, the homotopy perturbation method if the constants that appear in the governing equation are expanded in power series of the ordering parameter, and modified artificial parameter – Linstedt–Poincaré procedures.

APPROXIMATE ANALYTICAL SOLUTIONS FOR THE RELATIVISTIC OSCILLATOR USING A LINEARIZED HARMONIC BALANCE METHOD

The analytical approximate technique developed by Wu et al. for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the relativistic oscillator. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation period and periodic solutions are constructed for this oscillator. The approximate periods obtained are valid for the complete range of oscillation amplitudes, A, and the discrepancy between the second approximate period and the exact one never exceeds 1.24%, and it tends to 1.09% when A tends to infinity. Excellent agreement of the approximate periods and periodic solutions with the exact ones are demonstrated and discussed.

Linearized Harmonic Balancing Approach for Accurate Solutions to the Dynamically Shifted Oscillator

The analytical approximate technique developed by Wu et al for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the dynamically shifted oscillator. This nonlinear oscillator is described by an equation of motion which includes a linear restoring force and an anti-symmetric, constant force which is a nonlinear force depending only upon the sign of the displacement. By combining Newton’s method with the method of harmonic balance, analytical approximations to the oscillation frequency and periodic solutions are constructed for this oscillator and the approximate periods obtained are valid for the complete range of oscillation amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed and the results reveal that this technique is very effective and convenient for solving this class of conservative nonlinear oscillatory systems.

Infinitely many solutions for cooperative elliptic systems with odd nonlinearity

By using the minimax technique in critical point theory, we establish the existence of infinitely many solutions for a class of cooperative elliptic systems with odd nonlinearities. Our results improve several known results in the literature. In contrast with the existing ones, our results only demand restrictions on the behavior of the nonlinearities at infinity.

Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities

This Letter applies He's parameter-expansion method to oscillators with smooth nonlinearities. The method does not depend upon small parameter assumption, hence it is very better than the perturbation method. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain. The method can be easily extended to other nonlinear oscillations.

An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities

An artificial parameter method for obtaining the periodic solutions of oscillators with smooth odd nonlinearities 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. The method 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 any minimization procedure; neither does it require the expansion of constants in terms of the artificial parameter. It is shown that the method presented here is also a decomposition technique and a homotopy perturbation method provided that in these techniques the unknown frequency of oscillation is expanded in terms of an artificial parameter and the nonsecularity condition is imposed at each order in the expansion procedure. It is also shown by means of six examples that the first approximation to the frequency of oscillation coincides with that obtained by means of harmonic balance methods, two- and three-level iterative techniques, and modified Linstedt–Poincaré procedures based on the expansion of the solution and constants that appear in the differential equation in terms of an artificial parameter.

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.

On the variational iteration method and other iterative techniques for nonlinear differential equations

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–Poincaré techniques. The method is also applied to two boundary-value problems.

An artificial parameter-decomposition method for nonlinear oscillators: Applications to oscillators with odd nonlinearities

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–Poincaré 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 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.