Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.     

On bifurcation for semilinear elliptic Dirichlet problems and the Morse–Smale index theorem

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on star-shaped domains, where the bifurcation parameter is introduced by shrinking the domain. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.     

On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.     

Bifurcation of homoclinics of Hamiltonian systems

We obtain sufficient conditions for bifurcation of homoclinic trajectories of nonautonomous Hamiltonian vector fields parametrized by a circle, together with estimates for the number of bifurcation points in terms of the Maslov index of the asymptotic stable and unstable bundles of the linearization at the stationary branch.

     

An efficient algorithm for the determination of certain bifurcation points

Many practical problems require information about a branch of solutions of a system of nonlinear equations dependent upon a scalar parameter. We discuss some techniques for following such a branch through a turning point and describe an efficient method, with second order convergence, for finding the turning point. We also show that, if extra information is available about the solution branch, the method can be successfully applied to finding simple bifurcation points.     

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.     

Passage to diffusion chaos in a model of an ecological system

We carry out analytical and numerical analysis of a model of an ecological system described by a system of nonlinear partial differential equations of reaction-diffusion type. We find conditions for the bifurcation of periodic spatially homogeneous and inhomogeneous solutions from the thermodynamic branch of the system. We show that the passage to diffusion chaos in the model occurs, in complete agreement with the universal Feigenbaum-Sharkovskii-Magnitskii bifurcation theory, via a subharmonic cascade of bifurcations of stable limit cycles.     

Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

Generalizing previous results of M. Comte and P. Mironescu, it is shown that for degree d large enough (such that ), there is a bifurcation branch in the set of the solutions of the Ginzburg-Landau equation, emanating from the branch of radial solutions at the critical value _d of the parameter. Moreover, the solutions on the bifurcation branch admit exactly d zeroes, and the energy on the bifurcation branch is strictly smaller than the energy on the radial branch.     

Pressure and flow in two dimensional numerical bifurcation models

The finite element method has been used to solve the Navier-Strokes equations for steady flow conditions in bifurcations. The results are presented as pressure, velocity and streamline plots at different Reynolds number. The three bifurcations considered have rigid walls and bifurcation angles of 0°, 20° and 180°. For the bifurcation with branch angles 0° and 20° there is flow separation along the inner wall of the outlet branches and large spatial pressure variations, these phenomena being more pronounced at the higher Reynolds numbers. For the bifurcation with a branch angle of 180° the high pressure gradients occured at the outer corner and for the high Reynolds number a vortex formation developed downstream of this corner.     

A bifurcation analysis of the Ornstein-Zernike equation with hypernetted chain closure

Motivated by the large number of solutions obtained when applying bifurcation algorithms to the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure from liquid state theory, we provide existence and bifurcation results for a computationally-motivated version of the problem.We first establish the natural result that if the potential satisfies a short-range condition then a low-density branch of smooth solutions exists. We then consider the so-called truncated OZ HNC equation that is obtained when truncating the region occupied by the fluid in the original OZ equation to a finite ball, as is often done in the physics literature before applying a numerical technique.On physical grounds one expects to find one or two solution branches corresponding to vapour and liquid phases of the fluid. However, we are able to demonstrate the existence of infinitely many solution branches and bifurcation points at very low temperatures for the truncated one-dimensional problem provided that the potential is purely repulsive and homogeneous.     

Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

We give here a planar quadratic differential system depending on two parameters, λ, δ. There is a curve in the λ-δ space corresponding to a homoclinic loop bifurcation (HLB). The bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles. This is the first example of such a bifurcation in planar quadratic differential systems. We propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate Hopf bifurcation, and dying in a degenerate HLB. This model shows a deep duality between degenerate Hopf bifurcations and degenerate HLBs. We give a bound for the maximal number of cycles that can appear in certain simultaneous Hopf and homoclinic loop bifurcations. We also give an example of quadratic system depending on three parameters which has at one place a degenerate Hopf bifurcation of order 3, and at another place a Hopf bifurcation of order 2 together with a HLB. We characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop.     

Global bifurcation of solutions for a predator–prey model with prey‐taxis

We study pattern formations in a predator–prey model with prey‐taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth

We study the stationary Keller–Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing’s instability as the chemotaxis rate \({\chi}\) surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover, we show that solutions stay strictly positive in the continuum of each branch. The stabilities of these steady-state solutions are well studied when the creation and degradation rate of the chemical is assumed to be a linear function. Finally, we investigate the asymptotic behaviors of the monotone steady states. We construct solutions with interesting patterns such as a boundary spike when the chemotaxis rate is large enough and/or the cell motility is small.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

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.     

Local bifurcation from the second eigenvalue of the Laplacian in a square

In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point . We also compute the Morse index of the solutions in the four branches.

     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

Two new approaches to computing Hopf bifurcation problems

We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter ε; practical experience shows that one gets convergence for ε sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures.     

Diffusive logistic equation with constant yield harvesting, I: Steady States

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

     

计算立方体上Henon方程多个正解的分歧方法

本文首先应用分歧方法给出计算立方体上Henon方程边值问题D₄(3)对称正解的三种算法, 然后以Henon方程中的参数r为分歧参数, 在D₄(3)对称正解解枝上 用扩张系统方法求出对称破缺分歧点, 进而用解枝转接方法计算出其它具有不同对称性质的正解.