Bifurcation from a double eigenvalue in the unstirred chemostat

This article concerns a competition model in the unstirred chemostat. The bifurcation solution from a double eigenvalue is obtained. We see that this bifurcation solution connects the positive solution from the semitrivial solution (θ _a ,?0) with that from the other semitrivial solution (0,?θ _b ). Moreover, the asymptotic stability of the positive solution corresponding to this bifurcation is derived under certain conditions. The method we used here is based on spectral analysis, comparison principle, bifurcation theory and Lyapunov–Schmidt procedure.     

The kth Laplacian eigenvalue of a tree

Let λ_k(G) be the kth Laplacian eigenvalue of a graph G. It is shown that a tree T with n vertices has and that equality holds if and only if k < n, k|n and T is spanned by k vertex disjoint copies of , the star on vertices. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

This paper is devoted to the study of global bifurcation from infinity of nontrivial solutions of a nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. We prove the existence of two families of unbounded continua of nontrivial solutions to this problem, which emanate from bifurcation points in $ℝ\times \{\infty \}$ and possess oscillatory properties of eigenfunctions (and their derivatives) of the corresponding linear problem in some neighborhoods of these bifurcation points.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

On thek-th largest eigenvalue of the Laplacian matrix of a graph

In this paper, we give the upper bound and lower bound ofk-th largest eigenvalue λk of the Laplacian matrix of a graphG in terms of the edge number ofG and the number of spanning trees ofG. This research is supported by the National Natural Science Foundation of China (Grant No.19971086) and the Doctoral Program Foundation of State Education Department of China.     

The eigenvalue problem for the Laplacian equations

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω, u = 0, x ∈ (δ)Ω, where Ω (∩) Rn is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T.Yau et al.     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

Bounds for the second largest eigenvalue of a transition matrix

We find an upper bound, with general form, for the second largest eigenvalue of a transition matrix; special cases of which have previously been proposed as upper bounds and others which are new improvements.     

Global bifurcation theorem for a class of boundary conditions for ordinary differential equations of second order

In this paper we deal with boundary value problems where l : C¹([a, b], ?^k) → ?^k × ?^k is continuous, μ ≤ 0 and φ is a Caratheodory map. We define the class S of maps l, for which a global bifurcation theorem holds for the problem (+), with φ(t, x, y, λ) = λ(|x₁|, …, |x^k|) + o(|x| + |y|). We show that the class S contains Sturm‐Liouville boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the second largest eigenvalue of line graphs

In this paper all connected line graphs whose second largest eigenvalue does not exceed 1 are characterized. Besides, all minimal line graphs with second largest eigenvalue greater than 1 are determined. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 61–66, 1998     

Stability and bifurcation analysis in a system of four coupled neurons with multiple delays

A system of delay differential equations is studied which represent a model for four neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself. The linear stability and bifurcation of the system are studied in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. Meanwhile, the bifurcation set are drawn in the parameter space.     

A continuation method for a weakly elliptic two-parameter eigenvalue problem

We show that the continuation method can be used to solve aweakly elliptic two-parameter eigenvalue problem. We generalizethe continuation method for a nonsymmetric eigenvalue problemAx = x by T. Y. Li, Z. Zeng and L. Cong (1992 SIAM J. Numer.Anal. 29, 229–248) to two-parameter problems.     

DYNAMICAL BEHAVIORS FOR A THREE－DIMENSIONAL DIFFERENTIAL EQUATION IN CHEMICAL SYSTEM

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