Double Eigenvalue Research Articles

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z2. The Schrodinger operator H(k1, k2) of the system for k1 = k2 = π, where k = (k1, k2) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π − 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β2 and also an explicit form of the eigenfunctions of H(π, π −2β) for these eigenvalues.

Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie–Gower predator–prey model

We consider a diffusive Leslie–Gower predator–prey model subject to the homogeneous Neumann boundary condition. Treating the diffusion coefficient d as a parameter, the Hopf bifurcation and steady-state bifurcation from the positive constant solution branch are investigated. Moreover, the global structure of the steady-state bifurcations from simple eigenvalues is established by bifurcation theory. In particular, the local structure of the steady-state bifurcations from double eigenvalues is also obtained by the techniques of space decomposition and implicit function theorem.

Mode jumping analysis of thin film secondary wrinkling

In this paper, a mathematic expression is presented to describe the mode jumping and the equilibrium path reversal characteristics of thin film secondary wrinkling based on the non-linear bifurcation buckling theory. With the usage of variable transformation and the introduction of dimensionless parameters, the non-linear Von-Karman equilibrium equation considering the two-direction loading characteristics, is transformed into a non-linear boundary value problem with zero trivial solution. Based on the bifurcation theory, the eigenvalue problem is addressed through the linearization of non-linear boundary value problem. An approach to critical load prediction of thin film secondary wrinkling is proposed by introducing a critical aspect ratio parameter, since the mechanism of thin film secondary wrinkling is double eigenvalues splitting. Furthermore, this paper presents an analysis in detail on the critical load of the rectangular thin film secondary wrinkling under shear, indicating that the critical load is linearly proportional to the initial stretching displacement, while its relationship with the size of the free edge is nonlinear. The validity of this critical load prediction approach is verified via the digital image correlation experiment. The results provide strong supports for the controlling and tuning of the wrinkles for high precision pre-tensioned thin film structures.

New surface order reconstruction induced by electric field application in a nanoconfined HAN cell with a topological defect

In accordance with the 2D Landau–de Gennes tensorial formalism, we investigated the influence of an applied electric field E parallel to the defect line on the position and structure of a nematic line defect with topological charge M = − 1/2 in a hybrid alignment nematic cell with different cell gaps d. A new type of surface order reconstruction occurs in the cell as E is increased. Regardless of d, two biaxial layers can be achieved near the top and bottom substrates of the cell with different E values. This process involves double eigenvalue exchange across the cell. However, the structural transition processes vary for different d values.

Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion

A diffusive one-prey and two-competing-predators system under homogeneous Dirichlet boundary conditions is studied. First, we obtain sufficient conditions for the extinction and existence of global attractor of the time-dependent system by means of the comparison principle. Second, we discuss the existence and nonexistence of coexistence states, and give sufficient conditions for the existence of coexistence states by using the fixed point index theory. In addition, we investigate the bifurcation from a double eigenvalue by virtue of space decomposition and implicit function theorem. Finally, some numerical simulations are made to verify and complement the theoretical analysis.

The Spectrum of a Nonlocal Difference Operator with Variable Coefficients

The spectrum of a non-symmetric matrix A is studied, which resulting from difference approximation of the differential operators of the second order with nonlocal boundary conditions and variable coefficients. Such matrices can be both simple and double real eigenvalue and a complex conjugate pairs of eigenvalues. The original matrix A is mapped to the so-called associated matrix L --- symmetric tridiagonal matrix, little differing from A . It is known that all eigenvalues of the matrix L are real and different. They can be found by the bisection method using a sequence of the Sturm's polynomials. In the paper on the basis of numerous examples are compared the properties of the spectra of matrices L and A . The method offers of approximate calculating of eigenvalues of the matrix A .

Turing patterns in a reaction–diffusion model with the Degn–Harrison reaction scheme

In this paper, we consider a reaction–diffusion model with Degn–Harrison reaction scheme. Some fundamental analytic properties of nonconstant positive solutions are first investigated. We next study the stability of constant steady-state solution to both ODE and PDE models. Our result also indicates that if either the size of the reactor or the effective diffusion rate is large enough, then the system does not admit nonconstant positive solutions. Finally, we establish the global structure of steady-state bifurcations from simple eigenvalues by bifurcation theory and the local structure of the steady-state bifurcations from double eigenvalues by the techniques of space decomposition and implicit function theorem.

Steady-state solutions and stability for a cubic autocatalysis model

A reaction-diffusion system, based on thecubic autocatalytic reaction scheme, with the prescribedconcentration boundary conditions is considered. The linearstability of the unique spatially homogeneous steady state solutionis discussed in detail to reveal a necessary condition for thebifurcation of this solution. The spatially non-uniform stationarystructures, especially bifurcating from the double eigenvalue, arestudied by the use of Lyapunov-Schmidt technique and singularitytheory. Further information about the multiplicity and stability of the bifurcationsolutions are obtained. Numerical examples are presented to supportour theoretical results.

Computing Dark Energy and Ordinary Energy of the Cosmos as a Double Eigenvalue Problem

We compute the dark energy and ordinary energy density of the cosmos as a double Eigenvalue problem. In addition, we validate the result using two different theories. The first theory is based on Witten’s 11 dimensional spacetime and the second is based on ‘tHooft’s fractal renormalization spacetime. In all cases, the robust result is E(O) = mc2/22 for ordinary energy and E(D) = mc2(21/22) for dark energy. Adding E(O) to E(D) we obtain Einstein’s famous equation which confirms special relativity, although it adds a quantum twist to its interpretation. This new interpretation is vital because it brings relativity theory in line with modern cosmological measurements and observations. In particular, we replace calculus by Weyl scaling in all computation which is essentially transfinite discrete.

Secondary bifurcation of a compressible rod with spring-supports

We consider the problem of determining the stability boundary and postbuckling behavior of an elastic rod with spring supports at clamped ends. The rod is loaded by a compressive force and the constitutive equations of the rod take into account the compressibility of the rod axis. Using the Liapunov–Schmidt procedure local bifurcation analysis is performed. The spring stiffness is chosen to be in the neighborhood of the critical one corresponding to a double eigenvalue. Due to the splitting of eigenvalues secondary bifurcations occur. The results show that the type of primary bifurcation depends on the slenderness ratio and that there are three groups of bifurcation diagrams . Also, asymptotic expansions of postbuckling states are constructed. • Postbuckling behavior of a compressible rod near a double eigenvalue is analyzed. • The existence and type of primary bifurcations depend on compressibility. • Secondary bifurcations depend on spring stiffness and compressibility. • There are three groups of bifurcation diagrams depending on the slenderness ratio. • The asymptotic expansions of postbuckling states are constructed.

Instabilities and propagation properties in a fourth-order reaction–diffusion equation

In this paper we investigate the instability and the propagation properties of a class of reaction–diffusion equations of fourth order. Two examples are introduced, the extended Fisher Kolmogorov equation (EFK), and the Swift–Hohenberg equation (SH). Both have been studied before by related methods (see for example, Peletier and Rottschafer, 2004 [19]; Van Saarloos, 2003 [24]) but the analysis here will support the introduced linear mechanism in front selection. These two equations support a patterned front solutions, and the double eigenvalue mechanism is used to provide evidence for that and to determine a minimal front speed.

Casimir-Like Energy as a Double Eigenvalues of Quantumly Entangled System Leading to the Missing Dark Energy Density of the Cosmos

Starting from a quantumly entangled system we derive the dark energy and ordinary energy density of the cosmos as a double Eigenvalue problem. In addition we validate the result using two different theories. The first theory is based on Witten’s 11 dimensional spacetime and the second is based on ‘tHooft’s fractal renormalization spacetime. In all cases the robust result is E(O) = mc2/22 for ordinary energy and E(D) = mc2(21/22) for the endophysical dark energy. Adding E(O) to E(D) we obtain Einstein’s famous equation which confirms special relativity although it adds a quantum twist to its interpretation. This new interpretation is vital because it brings relativity theory in line with modern cosmological measurements and observations. Wider technological aspects of the new insights are discussed in the light of E(D) = mc2/(21/22) being related to a Casimir-like energy.

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Exact solution for the fractional cable equation with nonlocal boundary conditions

Abstract The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.

Some Aspects of Research on Mechanics of Thin Elastic Rod

Some aspects of research wok on the mechanics of thin elastic rod based on Kirchhoff-Cosserat's model were summarized. The analytical mechanics with arc-coordinate s and time t as double variables was established to formulate the motion of elastic rod. In stability analysis the difference and relationship between Lyapunov's stability and Euler's stability were discussed. The first approximate stability was determined by the characteristic equation with double eigenvalues in different domains, one of which can be determined by geometric conditions in static analysis. The Lyapunov's and Euler's stability conditions of the rod in space domain are the necessary conditions of Lyapunov's stability in time domain. As applications of the Kirchhoff's rod in molecular biology, the explanations of nucleosome structure and the chromosome coiling of DNA were given. Concerning the application in engineering the shape of a hanging rod under gravity and the coiling and stretching process of an extendable space mast were discussed. The motion of an axial moving beam with constant velocity and axial extensive force was discussed as an example of exact Cosserat's rod, the special case of small deformation is the Timoshenko's beam.

Bifurcation from double eigenvalue for nonlinear equation with third-order nondegenerate singularity

In this paper, the steady state bifurcation from a double eigenvalue for nonlinear equation with singularity being second-order fully degenerate and third-order nondegenerate is investigated. By the normalized Lyapunov–Schmidt reduction method, the precise criteria for the existence and nonexistence of bifurcation and the topological properties of regular bifurcated branches are obtained.

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.

Dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping

In this paper, stability and bifurcations in a simply supported rotating shaft are studied. The shaft is modeled as an in-extensional spinning beam with large amplitude, which includes the effects of nonlinear curvature and inertia. To include the internal damping, it is assumed that the shaft is made of a viscoelastic material. In addition, the torsional stiffness and external damping of the shaft are considered. To find the boundaries of stability, the linearized shaft model is used. The bifurcations considered here are Hopf and double zero eigenvalues. Using center manifold theory and the method of normal form, analytical expressions are obtained, which describe the behavior of the rotating shaft in the neighborhood of the bifurcations.

Multiple Solutions for a Class of Dirichlet Double Eigenvalue Quasilinear Elliptic Systems Involving the (p1, …, pn)‐Laplacian Operator

Existence results of three weak solutions for a Dirichlet double eigenvalue quasilinear elliptic system involving the (p 1,…, p n)-Laplacian operator, under suitable assumptions, are established. Our main tool is based on a recent three-critical-point theorem obtained by Ricceri. We also give some examples to illustrate the obtained results.