Bifurcation method for solving multiple positive solutions to boundary value problem of <Emphasis Type="Italic">p</Emphasis>-Henon equation on the unit disk

In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O（2） symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.     

Two-dimensional bifurcations of stokes waves

Bifurcation techniques are used to obtain a new class of small amplitude water waves of permanent form. This calculation illustrates an approach which can be applied to nonlinear waves of various types to generate new steady solutions from old.Stokes waves are used as a starting point, and the critical value of steepness at which bifurcation can occur is computed for various choices of modulation wavelength and angular orientation. It is found that, for two-dimensional surfaces, bifurcation can occur at small values of wave steepness.Second-order corrections to the wave amplitude, modulation, frequency, and speed, which apply when one moves off the bifurcation point onto a new branch of solutions, are also computed. Two types of new solutions are found, one symmetric with respect to the carrier wave propagation direction, and one asymmetric.The nonlinear Schrödinger equation is used to model water waves, and thus the calculation can be applied rather directly to other systems governed by the nonlinear Schrödinger equation.     

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

I. INTRODUCTION In practice, cavity formulation in materials is recognized as precursors to failure. Thus void nucleationand growth in solid materials have a great in?uence on failure mechanism. Gent and Lindley[1] have observed experimentally the phe…     

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

Global Hopf Bifurcation Analysis of a Nicholson’s Blowflies Equation of Neutral Type

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays.     

O(2)‐symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

The steady, axisymmetric laminar flow of a Newtonian fluid past a centrally‐located sphere in a pipe first loses stability with increasing flow rate at a steady O(2)‐symmetry breaking bifurcation point. Using group theoretic results, a number of authors have suggested techniques for locating singularities in branches of solutions that are invariant with respect to the symmetries of an arbitrary group. These arguments are presented for the O(2)‐symmetry encountered here and their implementation for O(2)‐symmetric problems is discussed. In particular, how a bifurcation point may first be detected and then accurately located using an ‘extended system’ is described. Also shown is how to decide numerically if the bifurcating branch is subcritical or supercritical. The numerical solutions were obtained using the finite element code ENTWIFE. This has enabled the computation of the symmetry breaking bifurcation point for a range of sphere‐to‐pipe diameter ratios. A wire along the centerline of the pipe downstream of the sphere is also introduced, and its effect on the critical Reynolds number is shown to be small. Copyright © 2000 John Wiley & Sons, Ltd.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Bifurcation Corresponding to the Onset of Asymmetric Periodic Structures and a Self-Induced Pressure Gradient in a Modified Taylor Flow

With reference to the example of a modified Taylor flow, the bifurcation of the loss of flow symmetry with the onset of a self-induced pressure gradient is studied theoretically and numerically. A linear analysis shows that the bifurcation is supercritical. It is necessarily accompanied by the appearance of a longitudinal pressure gradient and takes place at values of the parameters for which the solution of the linear system for the perturbations satisfies the condition of zero mass flow. It is established that, as a result of the bifurcation, two asymmetric solutions with oppositely directed pressure gradients are simultaneously generated. In the supercritical region, the symmetric branch of the solutions is also retained but becomes unstable. Bifurcation of the loss of symmetry and a self-induced pressure gradient can occur only in nonlinear systems.     

Hopf bifurcation in general Brusselator system with diffusion

The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results.     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincaré section and its symmetric section for the symmetric wheel-rail system are established. Then the ??resultant bifurcation diagram?? method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.     

An Alternative Approach to Study Bifurcation from a Limit Cycle in Periodically Perturbed Autonomous Systems

The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov–Schmidt method.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.     

温盐双扩散系统的多平衡稳定性

采用数值试验的方法研究了温盐双扩散系统中的多平衡稳定性在系统中引入非线性状态方程、温通量边界条件和变强迫因子在数值模拟中引入边界拟会坐标系,并按照温盐浮力比λ大于1,等于1和小于1将流场分为温度控制场、温盐均势场和盐度控制场得到了四种稳定平衡状态,特别是,大强迫因子和强非线性影响下的四胞环流对称状态．在对称的定常强迫边条件作用下的流场、温度场和盐度场均为对称分布引入温通量边条件后,对称流场在小干扰下失去对称性,发展为非对称分布,并因为干扰位置的不同而造成不同的非对称性,出现了从对称状态到非对称状态的音叉分叉,得到了a-Ras的稳定曲线及分叉图,并解释了分叉的本质     

Stability and folds

It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds.The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Émden-Fowler type. An analysis concerning an elastic rod is also presented.     

Pointwise Growth and Uniqueness of Positive Solutions for a Class of Sublinear Elliptic Problems where Bifurcation from Infinity Occurs

In this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary‐value problem of general sublinear type with a weight function multiplying the nonlinearity. When this function vanishes on some subdomain, the problem exhibits a bifurcation from infinity. In this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs. In this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite Dirichlet boundary conditions. This behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs. Also, we solve the problem using spectral collocation methods coupled with path‐following techniques to show how the main uniqueness result is optimal. Throughout the paper the mathematical analysis aids the numerical study, and the numerical study confirms and illuminates the analysis. (Accepted February 17, 1998)     

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Stability,bifurcation and post-critical behavior of a homogeneously deformed incompressible isotropic elastic parallelepiped subject to dead-load surface tractions

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.