Stability of Periodic Solutions Generated by Hopf Points Emanating from a Z＿2－symmetry－breaking Takens－Bogdanov Point

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EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.     

Numerical stability of one method of asymptotic stabilization

A problem of numerical projection onto a stable manifold in a neighborhood of a fixed point of hyperbolic type is considered. It is shown that a nonlinear iterative method of projection construction is stable with respect to errors occurring in solutions of intermediate problems. A formulation of the corresponding assertion and results of numerical simulations for solution of the problem on asymptotic stabilization over initial data for an equation of barotropic vortex type on a semisphere are presented.     

Travelling Waves for Reaction Diffusion Equations

In this paper the travelling waves for the reaction diffusion equation in most general case is considered. The existence of travelling wave solutions is proved under very weak conditions, which are also necessary for the nonlinear term. A difference method is suggested and Leray-Scbauder fixed point theorem is used to prove the existence of discrete travelling waves. Then the convergence is shown and so the solution for the differential equation is obtained.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

Solvability of fractional order differential systems with general nonlocal conditions

In this paper, fractional order nonlinear differential systems with general nonlocal conditions are investigated. The Lipschitz condition, linear and nonlinear growth conditions on the nonlinear terms are divided into two parts on two intervals; fixed point methods, the techniques on matrix and vector norms are used. Existence results for the solutions are derived under the weaken conditions.     

Nonlinear Stability of General Linear Methods

This paper considers the nonlinear stability of general linear methods. The diagonal matrix of an algebraically stable method is shown to be fixed. Readily testable necessary and sufficient conditions are obtained for algebraic stability and, more generally, for nonlinear stability in closed disk regions of the complex plane. It is shown that the latter criteria are satisfied by some explicit methods. It is also shown that certain methods, including some that are L-stable, suffer from nonautonomous instability along the negative real line near zero. A loose classification of methods is given according to nonlinear stability properties.     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

θ方法解滞时微分方程的动力学性质

本文研究求解滞时微分方程的θ-方法数值解的渐近性和方程真实解的关系。首先，我们把数值方法看成以步长为参数的动力系统，考察非线性滞时微分方程θ-方法的数值稳定性。并且证明了A－稳定的θ-方法是NP－稳定的。其次我们证明了θ-方法没有伪不动点，还研究了伪周期2解的存在性。最后我们给出一个例子说明了滞时微分方程θ-方法产生的伪周期2解是不稳定的。     

How to impose physically coherent initial conditions to a fractional system?

In this paper, it is shown that neither Riemann–Liouville nor Caputo definitions for fractional differentiation can be used to take into account initial conditions in a convenient way from a physical point of view. This demonstration is done on a counter-example. Then the paper proposes a representation for fractional order systems that lead to a physically coherent initialization for the considered systems. This representation involves a classical linear integer system and a system described by a parabolic equation. It is thus also shown that fractional order systems are halfway between these two classes of systems, and are particularly suited for diffusion phenomena modelling.     

Stability switches and bifurcation in a two-degrees-of-freedom nonlinear quarter-car with small time-delayed feedback control

In this paper, we investigate a two-degrees-of-freedom nonlinear quarter-car model with time-delayed feedback control. It is well known that a time delay has destabilizing effects in mathematical models. However, delays are not necessarily destabilizing. In this work we explore a system where a time delay can be both stabilizing and destabilizing. Using the generalized Sturm criterion, the critical control gain for the delay-independent stability region and critical time delays for stability switches are derived. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo stability switches with the variation of the time delay. These stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Properties of Hopf bifurcation such as direction and stability of bifurcating periodic solutions are determined by using the normal form theory and centre manifold theorem. Numerical simulations are provided to support the theoretical analysis. The critical conditions can provide a theoretical guidance for the design of vehicles with significant reduction of vibration in order to increase passengers ride comfort.     

Secondary Bifurcation in Nonlinear Diffusion Reaction Equations

A set of two coupled nonlinear diffusion reaction equations is studied and the existence of secondary bifurcation is shown. Using the method of two-timing, it is found that diffusion reaction equations of this type can exhibit an exchange of stability between distinct nontrivial solutions. This exchange can provide either a smooth or discontinuous transition between stable solutions, and the nontrivial solutions can be either steady or temporally periodic. This analysis is applied to the model biochemical reaction of Prigogine and the types of secondary bifurcation which occur in this model are classified.     

Regularization techniques in interior point methods

Regularization techniques, i.e., modifications on the diagonal elements of the scaling matrix, are considered to be important methods in interior point implementations. So far, regularization in interior point methods has been described for linear programming problems, in which case the scaling matrix is diagonal. It was shown that by regularization, free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded. Regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill-posed linear programming problems. In this paper, we study the factorization of the augmented system arising in interior point methods. In our investigation, we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite, but not diagonal. We show that regularization techniques may be applied beyond the linear programming case.     

An inverse problem for a linear diffusion equation with non-linear boundary condition

In this paper we consider the determination of an unknown radiation term in the non-linear boundary condition of a linear diffusion equation from an overspecified condition. It is shown that the solutions of this inverse problem is unique and stable.     

Complex nonlinear dynamics in subdiffusive activator-inhibitor systems

In this article we analyze the linear stability of nonlinear time-fractional reaction-diffusion systems. As an example, the reaction-subdiffusion model with cubic nonlinearity is considered. By linear stability analysis and computer simulation, it was shown that fractional derivative orders can change substantially an eigenvalue spectrum and significantly enrich nonlinear system dynamics. A overall picture of nonlinear solutions in subdiffusive reaction-diffusion systems is presented.     

Periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system

The periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system is considered in this paper. By using of Brouwer fixed point theorem and constructing a suitable Liapunov function, under some appropriate conditions, the system has a unique periodic solution which is globally stable.     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.