Newton's method for the eigenvalue problem of a symmetric matrix

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction

This paper presents a proportional-derivative protocol for the consensus problem of a class of linear second-order multi-agent systems with local information transmission. The communication topology among the agents is switching and agents receive information within a critical bounded distance. As new observations, we show that the desired protocol system undergoes consensus and swarming behaviours when 1 is a simple eigenvalue of the adjacency matrix. In this case, both final velocity and final relative position are formulated. Simulation results show the effectiveness of the proposed protocol.

Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

A degenerate bifurcation from simple eigenvalue theorem

<abstract><p>A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.</p></abstract>

Path-connectedness in global bifurcation theory

<p style='text-indent:20px;'>A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.</p>

Elemental access to limit cycle existence in Biomath education.

This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on Liénard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a Liénard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.

On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets

We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.