Bifurcation at multiple eigenvalues and stability of bifurcating solutions

If K is a bounded linear operator from the real Banach space U into the real Banach space V and $\text{?:U×}$$\text{R}$→V has the value zero at (0, 0), the existence and linear stability of the equilibrium solutions of the dynamical system

$\text{K}\text{du}\text{dt}\text{= ?(u, α)}$

which are close to the origin in U×$\text{R}$ are studied. It is assumed that $\text{?}{}_{\mathrm{u}}\text{(0, 0): U → V}$ is a Freholm operator of index zero. The only restriction on the dimension of the null space of $\text{?}{}_{\mathrm{u}}\text{(0, 0)}$ and the order of vanishing, at (0, 0), of ? restricted to the null space of $\text{D?(0,0)}$:U×$\text{R}$→V, is that they both be finite positive integers. The main result gives conditions under which the equation, which determines the equilibrium solutions in a neighborhood of the origin, also determines the stability of these equilibrium solutions.     

Stability of solutions bifurcating from multiple eigenvalues

Let X and Y be real Banach spaces and G:X × $\text{R}$ be a twice continuously differentiate function which is not necessarily linear. Suppose G(u₀, α₀) = 0 and the dimension of the null space of G_u(u₀, α₀) is m, where 1 ? m < ∞. Usually, S = {(u, α):G(u, α) = 0}, in a neighborhood of (u₀, α₀), consists of a finite number of curves emanating from (u₀, α₀). We will determine the stability of points, (u, α), in S (i.e., the maximum of the real parts of the spectrum of G_u(u, α) for each (u, α) ∈ S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations.     

Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of nonconstant 2π-periodic solutions of the Hamilton equation

     

Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.     

Genericity of simple eigenvalues

Simple eigenvalues are shown to be generic in several senses for regular ordinary differential operators.     

On bifurcation of eigenvalues along convex symplectic paths

We consider a continuously differentiable curve $t?\gamma (t)$ in the space of $2n\times 2n$ real symplectic matrices, which is the solution of the following ODE:

$\frac{\mathrm{d}\gamma }{\mathrm{d}t}(t)=J_{2n}A(t)\gamma (t),\gamma (0)\in \mathrm{Sp}(2n,\mathbb{R}),$

where $J=J_{2n}\stackrel{\text{def}}{=}\left[\begin{array}{cc}\hfill 0\hfill & \hfill {\mathrm{Id}}_n\hfill \\ \hfill ?{\mathrm{Id}}_n\hfill & \hfill 0\hfill \end{array}\right]$ and $A:t?A(t)$ is a continuous path in the space of $2n\times 2n$ real matrices which are symmetric. Under a certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t\in \mathbb{R}$), we investigate the dynamics of the eigenvalues of $\gamma (t)$ when t varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein–Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: $\{t\in \mathbb{R}:\gamma (t)\text{ has a Krein indefinite eigenvalue of modulus }1\}$ is a discrete set.     

Stability of solutions bifurcating from a multiple eigenvalue

The asymptotic form of the Titchmarsh-Weyl m-coefficient for the differential equation ?(py′)′+q j = λwy on [a, b). (′≡d/dx), as |λ|→∞, is obtained under general conditions on the real-valued coefficients p, q and w, with p≧0 and w≧0. Examples are given to show that the result is near to but possible, although there are many interesting cases for which the asymptotic form of the m-coefficient is still not known.     

Multiple solutions for semilinear elliptic equations near resonance at higher eigenvalues

Using the minimax methods in critical point theory and a generalized Landesman-Lazer type condition, we obtain two solutions for a class of semilinear elliptic equations near resonance at higher eigenvalues.     

Multiplicity of solutions for gradient systems with strong resonance at higher eigenvalues

In this paper we establish existence and multiplicity of solutions for an elliptic system which has strong resonance at higher eigenvalues. We describe the resonance using an eigenvalue problem with indefinite weights. In all results we use Variational Methods and the Morse theory.     

Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay

In this paper, the van der Pol equation with a time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem.