Political and economic rationality leads to velcro bifurcation

In this paper political and economic rationality is modelled regarding an economic problem with a four-dimensional dynamical system taking into consideration the information about the problem spread among the people who support the political alternatives. Under special parameter conditions velcro bifurcation occurs, which destabilizes the equilibrium points when information is going to spread. The last stable equilibrium point is related to the economically rational equilibrium point.     

Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.     

群逆的扰动界及其在奇异线性系统中的应用

本文建立了群逆的扰动界,此界基于矩阵A的Jordan标准形和P-范数,其中P是非异矩阵满足 是非异上双对角阵且 当矩阵A和A+E有相同的秩且 较小时,得到了 较好的估计.在相同的条件下,研究了相容的奇异线性系统Aχ=b的扰动,给出了χopt=A#b扰动的上界,其中A#是A的群逆,χopt是最小P-范数解.     

The generalized HSS method for solving singular linear systems

For the singular, non-Hermitian, and positive semidefinite linear systems, we propose an alternating-direction iterative method with two parameters based on the Hermitian and skew-Hermitian splitting. The semi-convergence analysis and the quasi-optimal parameters of the proposed method are discussed. Moreover, the corresponding preconditioner based on the splitting is given to improve the semi-convergence rate of the GMRES method. Numerical examples are given to illustrate the theoretical results and the efficiency of the generalized HSS method either as a solver or a preconditioner for GMRES.     

On semi-convergence of modified HSS method for a class of complex singular linear systems

In this paper, we discuss the semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex singular linear systems. Some semi-convergence theories of the MHSS iteration method are established and are weaker than those appeared in previously published works.     

Existence of multiple positive solutions to singular elliptic systems involving critical exponents

This paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms. By exploiting the effect of the coefficient of the critical nonlinearity and a variational method, we establish the main result which is based on the argument of the compactness.     

On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling

Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.     

Dynamic soft variable structure control of singular systems

The dynamic soft variable structure control (VSC) of singular systems is discussed in this paper. The definition of soft VSC and the design of its controller modes are given. The stability of singular systems with the dynamic soft VSC is proposed. The dynamic soft variable structure controller is designed, and the concrete algorithm on the dynamic soft VSC is given. The dynamic soft VSC of singular systems which was developed for the purpose of intentionally precluding chattering, achieving high regulation rates and shortening settling times enhanced the dynamic quality of the systems. It is illustrated the feasibility and validity of the proposed strategy by a simulation example, and an outlook on its auspicious further development is presented.     

Hopf bifurcation for tearing instability in magnetohydrodynamics, theory and numerical results

Fusion plasmas inside a Tokamak may be represented using magnetohydrodynamic equations. An equilibrium solution consists in a stationary solution of these equations, generally depending only on some radial coordinate.

In Tokamak experiments, we try to obtain such equilibrium solutions; however, some instabilities may appear and destroy such an equilibrium configuration. For instance, the so-called tearing instability, which destroys the equilibrium magnetic configuration, is really observed in experiments. Above a critical value of some physical parameter, the instability appears first as a stationary solution; then, increasing again some physical parameter, oscillatory behaviour can be found, before more complicated states and exponential growth of the solutions.

These physically observed phenomena have also been numerically computed (new results using the DEMA code concern the so-called double-tearing instability).

The mathematical framework of bifurcation theory is well suited for such a study; we first recall the existence of a stationary branch of bifurcated solutions; then, using mainly a Center Manifold Theorem, we prove existence of Hopf bifurcation. Most of the results are obtained with given diffusion coefficients (viscosity, resistivity) but new results are also obtained when these coefficients depend nonlinearly of the unknowns.

We also give a new mathematical justification of the decomposition of the velocity and magnetic fields used in the previously cited DEMA code. Finally, we also recall some results concerning the existence of a global attractor (in dimension 2).     

A note on the singular linear system of the generalized finite element methods

We characterize the kernel of the global stiffness matrix in the singular linear system of the generalized finite element methods (GFEM) which uses the classical finite element (FE) shape functions and local approximation space of harmonic polynomials.     

A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.     

The center conditions and bifurcation of limit cycles at the infinity for a cubic polynomial system

In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system.     

线性方程组的矩阵求解算法

设计的算法是 ,在约当消元法的基础上 ,只需对行最简形矩阵进行删除行和列、增加行、交换行等运算即可得到方程组的通解 .本算法的独特之处是 ,消元过程结束后 ,不需指定自由变量和非自由变量 ,不需写出由自由变量表示非自由变量的具体表达式 ,利用行最简形矩阵求通解时 ,再不需进行乘法和加法运算 ,因而不会增加算法的精度损失量 ,并且由于算法中的运算对象只有矩阵 ,因而算法简单 ,易于实现     

A singular value decomposition algorithm based on solving hyperplane constrained nonlinear systems

A new algorithm for singular value decomposition (SVD) is presented through relating SVD problem to nonlinear systems whose solutions are constrained on hyperplanes. The hyperplane constrained nonlinear systems are solved with the help of Newton’s iterative method. It is proved that our SVD algorithm has the quadratic convergence substantially and all singular pairs are computable. These facts are also confirmed by some numerical examples.     

Positive periodic solutions of singular systems of first order ordinary differential equations

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

Homoclinic bifurcation and chaos control in MEMS resonators

The chaotic dynamics of a micromechanical resonator with electrostatic forces on both sides are investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including basin of attraction, and bifurcation diagram confirm the analytical prediction and reveal the effect of parametric excitation amplitude on the system transition to chaos. The main result of this paper indicates that it is possible to reduce the electrostatically induced homoclinic and heteroclinic chaos for a range of values of the amplitude and the frequency of the parametric excitation. Different active controllers are applied to suppress the vibration of the micromechanical resonator system. Moreover, a time-varying stiffness is introduced to control the chaotic motion of the considered system. The techniques of phase portraits, time history, and Poincare maps are applied to analyze the periodic and chaotic motions.     

Eigenvalue and stability of singular differential delay systems

The eigenvalue and the stability of singular differential systems with delay are considered. Firstly we investigate some properties of the eigenvalue, then give the exact exponential estimation for the fundamental solution, and finally discuss the necessary and sufficient condition of uniform asymptotic stability.     

Convergence criterion of Newton's method for singular systems with constant rank derivatives

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187-209].