Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations

In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.     

On one approach to finding the branching lines and bifurcation points of solutions of nonlinear integral equations whose kernels depend analytically on two spectral parameters

We consider iterative algorithms of finding the curves of eigenvalues and their bifurcation points of a nonlinear algebraic two-parameter spectral problem, which appears in the solution of the problem of synthesis of plane antenna arrays by a given amplitude directivity pattern. These algorithms are based on the numerical procedure of calculation of ordinary and partial derivatives of a matrix determinant and the algorithm of finding all eigenvalues in a given domain of change in the spectral parameters. We also present some results of numerical experiments.     

Smooth bifurcation for variational inequalities based on the implicit function theorem

We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which could be called simple eigenvalues of the homogenized variational inequality. If the bifurcation parameter is one-dimensional, the main difference between the case of equations and the case of variational inequalities (when the cone is not a linear subspace) is the following: For equations two smooth half-branches bifurcate, for inequalities only one. The proofs are based on scaling techniques and on the implicit function theorem. The abstract results are applied to a fourth order ODE with pointwise unilateral conditions (an obstacle problem for a beam with the compression force as the bifurcation parameter).     

Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction–diffusion model

This paper is concerned with an autocatalysis model subject to no-flux boundary conditions. The existence of Hopf bifurcation are firstly obtained. Then by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established. On the other hand, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to verify the analytical results.     

Local bifurcations of nonlinear viscoelastic panel in supersonic flow

The stability and bifurcation behaviors of a two-dimensional nonlinear viscoelastic panel in supersonic flow are investigated with analytical and numerical methods. One type of critical points for the bifurcation response equations is considered, which is characterized by a pair of purely imaginary eigenvalues and a pair of complex conjugate eigenvalues having negative real part. With the aid of computer language Maple and the normal form theory, Hopf bifurcation solution of the model is investigated. Finally, numerical simulations are shown, which agree with the theoretical analytical results.     

Bifurcation of Solutions for an Inverse Problem in Potential Theory

Local existence and global numerical continuation of some solution branches are studied in an inverse potential problem in the exterior of a sphere. It is shown how multiple solutions arise where the solution field is tangent to the sphere. This work provides a tractable example of bifurcation arising from the edge of a continuum of eigenvalues.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

Zur integrativen Behandlung der Bifurkation von einfachen Eigenwerten bei gewöhnlichen Differentialgleichungen

Zusammenfassung Für die numerische Behandlung der Bifurkation von einfachen Eigenwerten bei gewöhnlichen Differentialgleichungen wird eine neue Technik vorgestellt. Sie basiert auf einer Verallgemeinerung der analytischen Nekrassow-Methode und wird sowohl in einer iterativen als auch in einer RWP-Form formuliert.Zwei numerische Beispiele demonstrieren die Effizienz der entwickelten Algorithmen.

---

Summary A new numerical technique is presented to treat the bifurcation from simple eigenvalues in ordinary differential equations. The technique is based on a generalization of the analytical Nekrassow-method and is presented in an iterative form as well as in a boundary value problem form.Two sample problems demonstrate the efficiency of the new algorithms.

     

The Hopf Bifurcation in a Parabolic Free Boundary Problem with Double Layers

We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value.     

Stability and Bifurcation Analysis of a Spinning Space Tether

A detailed, geometrically exact bifurcation analysis is performed for a model of a power-generating tethered device of interest to the space industries. The structure, a short electrodynamic tether, comprises a thin, long rod that is spun in a horizontal configuration from a satellite in low Earth orbit, with a massive electrically conducting disk at its free end. The system is modelled using a Cosserat formulation leading to a system of Kirchhoff equations for the rod's shape as a function of position and time. Moving to a rotating frame, incorporating the effects of internal damping, intrinsic curvature due to the deployment method and novel force and moment boundary conditions at the contactor, the problem for steady rotating solutions is formulated as a two-point boundary value problem. Using numerical continuation methods, a bifurcation analysis is carried out varying rotation speeds up to many times the critical resonance frequency. Spatial finite differences are used to formulate the stability problem for each steady state and the corresponding eigenvalues are computed. The results show excellent agreement with earlier multibody dynamics simulations of the same problem.     

具有初始缺陷弹性板的稳定性分析

本文以Marguerre方程为基础,用奇异性理论研究了初始挠度缺陷以及横向载荷对弹性板屈曲后分叉解的影响。借助于普适开折的原理,在单特征值局部邻域内将该问题的失稳分析转化为三次代数方程的讨论,从而确定出分叉解的性态。同时绘出了在不同参数下的分叉解文,讨论了几何缺陷和横向载荷对特征值的影响。     

Remarks on bifurcation for elliptic operators with odd nonlinearity

On estimating the eigenvalues for a class of semilinear elliptic operators, we obtain bifurcation and comparison results concerning the eigenvalues of some related linear problem.     

Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

A bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered. The existence of global bifurcation between two eigenvalues of a certain type of the Laplacian is proved. For a class of abstract inclusions with compact multivalued mappings in a Hilbert space, it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single-valued problem, and the jump of the degree is proved. As a consequence, global bifurcation for such abstract inclusions is obtained. The weak formulation of the boundary value problem mentioned is a particular case.     

On spectral asymptotics and bifurcation for some elliptic equations of Kirchhoff-type with odd superlinear term

In this paper, from estimating the eigenvalues for Kirchhoff elliptic equations, we obtain spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.     

Computation of bifurcation branches using projection methods

Using a straightforward Newton's method argument, convergence properties of projection methods for the computation of bifurcation branches off simple eigenvalues for general operator equations and for the computation of Hopf bifurcation branches for ordinary differential equations are established.     

Bifurcation analysis on a diffusive Holling–Tanner predator–prey model

This paper is concerned with a diffusive Holling–Tanner predator–prey model subject to homogeneous Neumann boundary condition. By choosing the ratio of intrinsic growth rates of predator to prey λ as the bifurcation parameter, we find that spatially homogeneous and non-homogeneous Hopf bifurcation occur at the positive constant steady state as λ varies. The steady state bifurcation of simple and double eigenvalues are intensively investigated. The techniques of space decomposition and the implicit function theorem are adopted to deal with the case of double eigenvalues. Our results show that this model can exhibit spatially non-homogeneous periodic and stationary patterns induced by the parameter λ. Numerical simulations are presented to illustrate our theoretical results.     

一类互惠共存系统的定态分歧与稳定性

一类互惠共存系统的定态分歧与稳定性李大华，傅一平（华中理工大学数学系，武汉４３００７４）基金项目：国家自然科学基金资助项目．１）现在江西省九江师范专科学校工作，邮政编码３３２０００．１９９２年５月６日收到．一、引言在文［１］中Ｒ．Ｍ．Ｍａｙ提出了一个...     

数值离散化对环面分支结构的影响

Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.     

An eigenvalue problem arising in a patch formation model

A simple numerical scheme has been developed for the solution of the eigenvalue problem arising in a patch formation model given by Del Grosso et al. [1]. The scheme is based on finding bounds which separate the eigenvalues. The exact eigenvalues are obtained by solving an algebraic equation given by the corresponding regular Frobenius series solution. At the same time eigenfunctions may also be obtained from this series solution.     

Evans Function for Lax Operators with Algebraically Decaying Potentials

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.