Classification of Simple Higher-Order Symmetry-Breaking Bifurcations and Their Computation

This paper deals with the classification of the simple higher-order symmetry-breaking bifurcation in multiparameter nonlinear problems with Z2-symmetry. The regular extended systems for computing the simple higher-order symmetry-breaking bifurcation points with different singularities are proposed. An etficient algorithm for solving the extended systems is given. Finally, some numerical examples are shown to demonstrate the efficiency of the algorithm.

THE COMPUTATION OF SYMMETRY－BREAKING BIFURCATION POINTS IN Z_2×Z_2－SYMMETRIC NONLINEAR PROBLEMS

This paper is mainly concerned with corank-2 and corank-3 symmetrybreaking bifurcation point in Z2×Z2-symmetric nonlinear problems. Regular extended systems are used to compute corank-2 and corank-3 symmetry--breaking bifurcation points. Two numerical examples are given. In addition, we show that there exist three quadratic pitchfork bifurcation point curves passing through corank-2 symmetry breaking bifurcation point.

A NEW APPROACH FOR THE COMPUTATION OF HOPF BIFURCATION POINTS

A new approach is proposed to compute Hopf bifurcation points. The method could produce small extended systems and therefore could reduce the computational effort and storage. One numerical example is presented to demonstrate that the method is efficient.

Discrete Conley index and bifurcation points

In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have the similar bifurcation points is given.

Stability Analysis of a Single-Degree-of Freedom Mechanical Model with Distinct Critical Points: I. Bifurcation Theory Approach

The buckling and post-buckling response of a single-degree-of-freedom mechanical model is re-examined in this work, within the context of nonlinear stability and bifurcation theory. This system has been reported in pioneer as well as in more recent literature to exhibit all kinds of distinct critical points. Its response is thoroughly discussed, the effect of all parameters involved is extensively examined, including imperfection sensitivity, and the results obtained lead to the important conclusion that the model is possibly associated with the butterfly singularity, a fact which will be validated by the contents of a companion paper, based on catastrophe theory.

BIFURCATION OF SINGULAR POINTS NEAR A DOUBLE FOLD POINT INZ_2 -SYMMETRIC NONLINEAR EQUATIONS

We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.

A NEW APPROACH FOR THE COMPUTATION OF DOUBLE SYMMETRY-BREAKING TURNING POINTS

We will consider an extension of a direct method due to Griewank and Reddien for the computation of double symmetry-breaking turning points which arise in the two-parameter dependent nonlinear problem of the form f(x,λ,μ)= 0. The method proposed could produce an extended system which does not introduce the null vectors as variables, and could therefore reduce the computational effort and storage. One numerical example is presented to demonstrate that the method is efficient.

DOUBLE HIGH ORDER S－BREAKING BIFURCATION POINTS AND THEIR NUMERICAL DETERMINATION

We consider double high order S-breaking bifurcation points of two-Parameter dependent nonlinear problems with Z_2×Z_2-symmetry. Because of the underlying symmetry we could propose some regular extended systems to determine double high order S-breaking bifurcation points. and we could also show that there exist two quadratic pitchfork bifurcation point paths passing through the point being considered.

Explicit FE wrinkling simulation and method to catch critical bifurcation point in tube bending process

The wrinkling has become the main defect in the thin-walled tube NC bending process. In the study, a dynamic explicit FE model for aluminum alloy thin-walled tube NC bending process is developed to predict the wrinkling by using FE code ABAQUS/Explicit. Attention was paid to the influences of mass scaling, loading rate scaling, mesh density and element type on accurate wrinkling prediction. So the wrinkling modes and mechanism are revealed based on the reliable FE model. Then a two step strategy is proposed to capture the critical bifurcation point for the optimal design process. The results show: 1) The boundary conditions determine the tube materials response greatly so that the frequency analysis is meaningless to the simulation. It is the contact conditions that make the effect of the mass scaling and loading rate less significant.2) There are two wrinkling modes in the tube bending process. One refers to that local ripples occur initially in the straight regions contacted with wiper die and mandrel; the other refers to that local wrinkles occur in the curved regions due to the relative slipping between tube and clamp die. 3) Both the difference of the in-plane compressive stresses and the relative slipping distance are chosen to be the quantitative indexes to represent the critical point and wrinkling tendency. The experiment of aluminum alloy (5052 O) tube bending was carried out to verify whether the above wrinkle modes exist and the indexes proposed are reasonable to catch the critical bifurcation point. The results may help better understanding of the wrinkling mechanism and the process optimization of the tube bending.

Some Problems in Nonlinear Dynamic Instability and Bifurcation Theory for Engineering Structures

In civil engineering, the nonlinear dynamic instability of structures occurs at a bifurcation point or a limit point. The instability at a bifurcation point can be analyzed with the theory of nonlinear dynamics, and that at a limit point can be discussed with the theory of elastoplasticity. In this paper, the nonlinear dynamic instability of structures was treated with mathematical and mechanical theories. The research methods for the problems of structural nonlinear dynamic stability were discussed first, and then the criterion of stability or instability of structures, the method to obtain the bifurcation point and the limit point, and the formulae of the directions of the branch solutions at a bifurcation point were elucidated. These methods can be applied to the problems of nonlinear dynamic instability of structures such as reticulated shells, space grid structures, and so on. Key words nonlinear dynamic instability - engineering structures - non-stationary nonlinear system - bifurcation point - instability at a bifurcation point - limit point MSC 2000 74K25 Project supported by the Science Foundation of Shanghai Municipal Commission of Education (Grant No. 02AK04), the Science Foundation of Shanghai Municipal Commission of Science and Technology (Grant No. 02ZA14034)

HOPF BIFURCATION FOR A ECOLOGICAL MATHEMATICAL MODEL ON MICROBE POPULATIONS

The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase space. The qualitative property and stability of equilibrium points were analysed. The conditions under which the positive equilibrium point exists and becomes and O+ attractor are obtained. The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.

EXTENDED SYSTEM, BIFURCATION PROBLEM AND HIGH ORDER SINGULARITIES

In this paper, we consider two extended systems. When using them for the two parameter bifurcation problems, the simple bifurcation point with regard to lambda on turn into the simple turning point with. regard to mu. Simple high orde bifurcation point is first studied without using the symmetry condition.

BIFURCATION SOLUTION BRANCHES AND ITS NUMERICAL ANALYSIS OF A SEMI-LINEAR ELLIPTIC PROBLEM

In this paper the bifurcation solution branches of a semi-linear elliptic problem are studied, and its extended systems are constructed.

Stability and boundary equilibrium bifurcations of modified Chua's circuit with smooth degree of 3

Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.

Bifurcation method for solving multiple positive solutions to boundary value problem of p-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.

Uncertainty Principle and Bifurcations in the SU(2) Nonlinear Semiquantum Dynamics

In this paper, a nonlinear semiquantum Hamiltonian associated to the special unitary group SU(2) Lie algebra is studied so as to analyze its dynamics. The treatment here applied allows for a reduction in: 1) the system’s dimension, as well as 2) the number of system’s parameters (to only three). We can now discern clear patterns in: 1) the complete characterization of the system’s fixed points and 2) their stability. It is shown that the parameter associated to the uncertainty principle, which constitutes a very strong constraint, is the key one in determining the presence of fixed points and bifurcation curves in the parameter’s space.

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.

Homoclinic Bifurcation Properties near Eight-figure Homoclinic Orbit

In this paper we investigate the homoclinic bifurcation properties near an eight-figure homoclinic orbit of co-dimension two of a planar dynamical system. The corresponding local bifurcation diagram is also illustrated by numerical computation.

The Dynamical Properties of a Class of Discrete Smith Diffusion Model

In this paper, the dynamical properties of Smith type diffusion model with Dirichlet boundary conditions are studied. The properties of hyperbolic fixed points and non-hyperbolic fixed points of the model are analyzed. By using the central manifold theorem, the bifurcation phenomenon of the model is studied. The results show that flip, transcritical, pitchfork and Fold-flip bifurcations exist at non-hyperbolic fixed points.