Quasi-periodic solutions for perturbed autonomous delay differential equations

This work discusses the persistence of quasi-periodic solutions for delay differential equations. We prove that the perturbed system possesses a quasi-periodic solution under appropriate hypotheses if an unperturbed linear system has quasi-periodic solutions. We extend some well-known results on ordinary differential equations to delay differential equations.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

On quasiperiodic solutions of semilinear systems of differential equations

We prove existence theorems for analytic quasi-periodic solutions for analytic systems of differential equations in a Banach space by the method of accelerated convergence. The results obtained are new even in the finite-dimensional case. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Hopf-transcritical bifurcation in retarded functional differential equations

Firstly, we analyze a codimension-two unfolding for the Hopf-transcritical bifurcation, and give complete bifurcation diagrams and phase portraits. In particular, we express explicitly the heteroclinic bifurcation curve, and obtain conditions under which the secondary bifurcation periodic solutions and the heteroclinic orbit are stable. Secondly, we show how to reduce general retarded functional differential equation, with perturbation parameters near the critical point of the Hopf-transcritical bifurcation, to a 3-dimensional ordinary differential equation which is restricted on the center manifold up to the third order with unfolding parameters, and further reduce it to a 2-dimensional amplitude system, where these unfolding parameters can be expressed by those original perturbation parameters. Finally, we apply the general results to the van der Pol’s equation with delayed feedback, and obtain the existence of stable or unstable equilibria, periodic solutions and quasi-periodic solutions.     

带Poisson跳和Markovian调制的中立型随机延迟微分方程的数值解的收敛性

本文研究带Poisson跳和Markovian调制的中立型随机微分方程的数值解的收敛性质.用数值逼近方法求此微分方程的解,并证明了Euler近似解在此线性增长条件和全局Lipschitz条件更弱的条件下仍均方收敛于此方程的解析解.     

轴对称圆板(含叠层板)的三维非线性分析

本文提出了轴对称固支圆板(含叠层板)受均布横向载荷作用下的三维非线性摄动解答.文中所考虑的是一种中等大挠度的几何非线性,并采用一种发展的摄动方法对复杂的三维非线性平衡微分方程进行求解.该方法的基本思想是以二维解答为基础,对板的厚度参数进行摄动而求得相应的三维解答.文中给出了一般板及叠层板的三维非线性理论结果及数值结果,并图示出了各个应力的分布情况.而且,该三维非线性结果能退化为完全一致的相应的二维板理论非线性结果.结果表明,该方法对板的三维非线性分析是一种行之有效的方法.     

Quasiperiodic solutions for nonlinear differential equations of second order with symmetry

In this paper, we study the existence of quasi-periodic solutions and the boundedness of solutions for a wide class nonlinear differential equations of second order. Using the KAM theorem of reversible systems and the theory of transformations we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.     

Boundedness in Asymmetric Quasi-periodic Oscillations

In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ ax~+-bx~-+φ(x) = p(t), where a≠b are two positive constants and φ(s), p(t) are real analytic functions. Moreover, the p(t) is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions,the quasi-periodic oscillator has the Lagrange stability.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

Asymptotic expansion for the solution of singularly perturbed delay differential equations

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley-Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.     

Quasi-periodic solutions of forced isochronous oscillators at resonance

We deal with the existence of quasi-periodic solutions of forced isochronous oscillators with a repulsive singularity, the nonlinearity is a bounded perturbation. Using a variant of Moser's twist theorem of invariant curves, due to Ortega [R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999) 381-413], we show that there are many quasi-periodic solutions and the boundedness of all solutions.     

On Quasi-Periodic Solutions of Differential Equations with Piecewise Constant Argument

We study the existence of quasi-periodic solutions to differential equations with piecewise constant argument (EPCA, for short). It is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. The appearance of quasi-periodic rather than periodic solutions is due to the piecewise constant argument. This new phenomenon illustrates a crucial difference between ODE and EPCA. The results are extended to nonlinear equations.     

Effective Computation of the Dynamics Around a Two-Dimensional Torus of a Hamiltonian System

The purpose of this paper is to give an explicit analysis of the nonlinear dynamics around a two-dimensional invariant torus of an analytic Hamiltonian system. The study is based on high-order normal form techniques and the computation of an approximated first integral around the torus. One of the main novel aspects of the current work is the implementation of the symplectic reducibility of the quasi-periodic time-dependent variational equations of the torus. We illustrate the techniques in a particular example that is a quasi-periodic perturbation of the well-known Restricted Three Body Problem. The results are useful for describing the neighborhood of the triangular points of the Sun-Jupiter system.     

Stable phase-locked periodic solutions in a delay differential system

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory.     

Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems

Definitions for pseudospectra and stability radii of an analytic matrix function are given, where the structure of the function is exploited. Various perturbation measures are considered and computationally tractable formulae are derived. The results are applied to a class of retarded delay differential equations. Special properties of the pseudospectra of such equations are determined and illustrated.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.