Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

Applications of cubic B‐splines collocation method for solving nonlinear inverse parabolic partial differential equations

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B‐splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B‐splines for spatial variable and derivatives, which produce an ill‐posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 88–104, 2017     

Finite‐region stabilization via dynamic output feedback for 2‐D Roesser models

Finite‐region stability (FRS), a generalization of finite‐time stability, has been used to analyze the transient behavior of discrete two‐dimensional (2‐D) systems. In this paper, we consider the problem of FRS for discrete 2‐D Roesser models via dynamic output feedback. First, a sufficient condition is given to design the dynamic output feedback controller with a state feedback‐observer structure, which ensures the closed‐loop system FRS. Then, this condition is reducible to a condition that is solvable by linear matrix inequalities. Finally, viable experimental results are demonstrated by an illustrative example.     

Exponential stability of positive neural networks in bidirectional associative memory model with delays

This paper is concerned with the problem of exponential stability of positive neural networks in bidirectional associative memory (BAM) model with multiple time‐varying delays and nonlinear self‐excitation rates. On the basis of a systematic approach involving extended comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive cone called the admissible set of initial conditions. In combination with the use of Brouwer's fixed point theorem and M‐matrix theory, we then derive conditions for the existence and global exponential stability of a unique positive equilibrium of the model. An extension to the case of BAM neural networks with proportional delays is also presented. The effectiveness of the obtained results is illustrated by a numerical example with simulations.     

Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.     

Discussion of “Virtual age,is it real?”

In this note, some point of views on virtual ages are presented in terms of the discussion paper written by Finkelstein and Cha, which include generalized stochastic order‐based virtual ages, system‐level virtual ages, virtual ages in Weibull distribution and repair degrees with virtual ages. Finally, some possible future researches on virtual ages are described.     

Finite‐time stability of a class of oscillating systems with two delays

In this paper, we study finite‐time stability of an oscillating system with 2 delays. To derive a bounded of state vector, we use a representation of explicit solution involving 2‐delayed matrix polynomial of 2 indices after deriving some fundamental estimates for such delayed matrix polynomial of 2 indices. A sufficient condition is given. Finally, an example is given to demonstrate the application of the main result.     

Disease‐selective predation may lead to prey extinction

In real world bio‐communities, predational choice plays a key role to the persistence of the prey population. Predator's ‘sense’ of choice for predation towards the infected and noninfected prey is an important factor for those bio‐communities. There are examples where the predator can distinguish the infected prey and avoids those at the time of predation. Based on the examples, we propose two mathematical models and observe the dynamics of the systems around biologically feasible equilibria. For disease‐selective predation model there is a high risk of prey extinction. On the other hand, for non‐disease selective predation both populations co‐exist. Local stability analysis and global stability analysis of the positive interior equilibrium are performed. Moreover, conditions for the permanence of the system are obtained. Finally, we conclude that strictly disease‐selective predation may not be acceptable for the persistence of the prey population. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.     

G‐invariant persistent homology

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self‐homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo‐distance d_G associated with a group G ? Homeo(X), we need to adapt persistent homology and consider G‐invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G‐invariant persistent homology with respect to the natural pseudo‐distance d_G. We also show how G‐invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self‐homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis of quaternion‐valued Cohen‐Grossberg neural networks

In this paper, by starting from basic quaternion algebra properties and algorithms, a quaternion‐valued Cohen‐Grossberg neural network was derived, subsequently, several new sufficient conditions are derived to ensure existence and global asymptotic stability (GAS) and global exponential stability (GES) of the equilibrium point (EP) for quaternion‐valued Cohen‐Grossberg neural networks. The obtained criteria can be checked easily in practice and have a distinguished feature from previous studies. Finally, we have numerical evidences that the mathematical system and the conclusions presented are validated.     

An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

A new BDF‐type scheme is proposed for the numerical integration of the system of ordinary differential equations that arises in the Method of Lines solution of time‐dependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability. The method proposed in this article is almost L‐stable and of algebraic order three. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable some PDE problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The spatial patterns through diffusion-driven instability in a predator-prey model

Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs,application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

Using self‐dissimilarity to quantify complexity

For many systems characterized as “complex” the patterns exhibited on different scales differ markedly from one another. For example, the biomass distribution in a human body “looks very different” depending on the scale at which one examines it. Conversely, the patterns at different scales in “simple” systems (e.g., gases, mountains, crystals) vary little from one scale to another. Accordingly, the degrees of self‐dissimilarity between the patterns of a system at various scales constitute a complexity “signature” of that system. Here we present a novel quantification of self‐dissimilarity. This signature can, if desired, incorporate a novel information‐theoretic measure of the distance between probability distributions that we derive here. Whatever distance measure is chosen, our quantification of self‐dissimilarity can be measured for many kinds of real‐world data. This allows comparisons of the complexity signatures of wholly different kinds of systems (e.g., systems involving information density in a digital computer vs. species densities in a rain forest vs. capital density in an economy, etc.). Moreover, in contrast to many other suggested complexity measures, evaluating the self‐dissimilarity of a system does not require one to already have a model of the system. These facts may allow self‐dissimilarity signatures to be used as the underlying observational variables of an eventual overarching theory relating all complex systems. To illustrate self‐dissimilarity, we present several numerical experiments. In particular, we show that the underlying structure of the logistic map is picked out by the self‐dissimilarity signature of time series produced by that map. © 2007 Wiley Periodicals, Inc. Complexity 12: 77–85, 2007     

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ_τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.