一类高次自催化耦合反应扩散系统的分歧和斑图

考虑了一类由于自催化剂的耦合而发生的反应扩散系统的空间结构．利用线性化理论讨论了平衡态解的稳定性并且证明了在非耦合系统中空间非一致解出现分歧的必要条件．进一步,利用弱非线性理论讨论了分歧点并且给出了弱耦合系统的图灵分歧解的振幅方程及其性质．     

On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3 solution and chaotic movement.     

Global actions of Lie symmetries for the nonlinear heat equation

By restricting to a natural class of functions, we show that the Lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding Lie group. Remarkably, in most of the cases, the action turns out to be linear.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.     

大脑皮层信号作用下人体步态节律运动的探讨

中枢模式发生器可产生节律性运动．目前的中枢模式发生器（CPG）建模研究可以很好地表现CPG的自激行为,但对于人脑信号的调节作用没有讨论．为了体现大脑皮层信号对于CPG网络的调控性,基于Matsuoka神经振荡器的CPG模型,对原有模型中输入刺激与网络内部参数的关联进行了复杂构建,使得模型本身各参数随输入信号的变化而变化,增强了输入信号对于网络自身的影响,令CPG网络不仅仅产生自激状态,同时能够产生自我调节的运动形式,从而体现出大脑信号的调控作用．数值模拟计算结果表明,修正后的模型随着输入刺激的变化可以产生不同模式及不同频率的运动形式,且各不同形式之间可以相互转换,从而在理论上很好地反映出大脑信号在步态节律运动过程中对步态的模式和频率起到了一定的调节作用,实现了各种步态运动之间的行为转换及恢复的功能,从理论上实现了自发节律与大脑调节性节律运动的共存性,做到大脑信号与CPG模型的统一．     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stability and transition for the Navier-Stokes-alpha model

The main aim of this paper is to investigate the stability and transition of the Navier-Stokes-alpha model. By using the continued-fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non-dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio-temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid-axis of the channel.     

Asymptotic bifurcation points, and global bifurcation of nonlinear operators and its applications

Using the cone theory and lattice structure, we discuss the existence of asymptotic bifurcation points and the global bifurcation of nonlinear operators which are not assumed to be cone mappings and may not be Frechet differentiable at points at infinity. As an application, the structure of the set of solutions of the superlinear Sturm-Liouville problems is investigated.     

Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation

The nonlinear wave equation utt=(c²x(u)ux) arises in various physical applications. Ames et al. [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] did the complete group classification for its admitted point symmetries with respect to the wave speed function c(u) and as a consequence constructed explicit invariant solutions for some specific cases. By considering conservation laws for arbitrary c(u), we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations. We use existing work on such related linear systems to extend the known symmetry classification in [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] to include nonlocal symmetries. Moreover, we find sets of c(u) for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation.     

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

In this work, we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.     

Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reaction-diffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.     

On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations

Original asymptotic solutions are determined for two autonomousdifferential equations. The application of initial conditionsfor the energy, wave number and phase shift proves to be lesscomplicated than in previous work. For the damped simple pendulum,explicit solutions demonstrate the dependence on the initialconditions. For strongly nonlinear wave packets of the Klein–Gordonequation, asymptotic solutions are compared. In both cases,the phase shift is shown to be highly sensitive to small perturbationsin the initial conditions.     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem

This paper is concerned with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation methods to obtain our results.

     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.     

POSITIVE SOLUTIONS AND BIFURCATION OF FULLY NONLINEAR ELLIPTIC EQUATIONS INVOLVING SUPER－CRITICAL SOBOLEV EXPONENTS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＡＮＤＢＩＦＵＲＣＡＴＩＯＮＯＦＦＵＬＬＹＮＯＮＬＩＮＥＡＲＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳＩＮＶＯＬＶＩＮＧＳＵＰＥＲ－ＣＲＩＴＩＣＡＬＳＯＢＯＬＥＶＥＸＰＯＮＥＮＴＳ￥ＱＵＣＨＡＮＧＺＨＥＮＧ（屈长征）（Ｉｎｓ...     

二维耦合Logistic映射的动力学性质及在流密码生成中的应用

本文研究了带二次耦合项的二维Logistic映射的性质和分岔行为,数值模拟了混沌的生成过程．若控制一个参数值近似为1,则产生近乎满的混沌区．这种混沌区产生的随机序列所生成的流密码具有很好的0-1分布、高线性复杂性、密钥敏感性等．最后给出了用于保密通信的模型．     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.