Evolving Systems of Stochastic Differential Equations

Journal of Theoretical Probability - We introduce Evolving Systems of Stochastic Differential Equations. This model generalizes the well-known stochastic differential equations with Markovian...     

动力系统的稳定化问题

In this paper, we are concerned with the problem of stabilization for autonomous dynamical systems. We use theories in Liapunov stability and Lasalle stability theory and show that system (H) is stabilizable.     

Global asymptotic stability analysis by the localization method of invariant compact sets

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

Evolution of Unit Fraction Conceptions in Two Fifth-Graders with a Learning Disability: An Exploratory Study

The literature seems limited in what is known about conceptual processes that underlie evolution of students with learning disabilities (SLD) conceptions of fractions. This exploratory study examines how a foundational scheme of unit fractions (1/n) may evolve through the mathematical activity of two fifth grade girls. We analyze data segments from episodes conducted during a teaching experiment grounded in the activity of iterating estimates of one person's equal share. Our findings include four distinct conceptual stages: (1) No Conception of the Nature of Adjustment to the Magnitude of a Unit Fraction, (2) Evolving Anticipation of the Nature of Adjustment but not of its Relative Amount, (3) Anticipation of the Nature of Adjustment with an Evolving Partial Amount, and (4) a Dual Anticipation of the Nature and Amount of Adjustment. Findings demonstrate each girl was able to use her constructed scheme to successfully solve and reason about novel problems. We discuss the need for more research to confirm the findings from this study, while offering a conjecture of the possibilities for more SLDs to advance their conceptions of fractions in future interventions.     

One‐parametric families of canard cycles: two explicitly solvable examples

Systems of singularly perturbed autonomous ordinary differential equations possessing in a parameter plane two intersecting bifurcation curves connected with the generation of limit cycles with large and small amplitude respectively, have a special class of limit cycles called canards or french ducks describing an exponentially fast transition from a small amplitude limit cycle to limit cycle with a large amplitude. We present two explicitly integrable examples of non‐autonomous singularly perturbed di.erential equations with canard cycles without a second parameter.     

Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.     

Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov’s functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system’s behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.     

Modelling a hormone-inspired controller for individual- and multi-modular robotic systems

For all living organisms, the ability to regulate internal homeostasis is a crucial feature. This ability to control variables around a set point is found frequently in the physiological networks of single cells and of higher organisms. Also, nutrient allocation and task selection in social insect colonies can be interpreted as homeostatic processes of a super-organism. And finally, behaviour can also represent such a control scheme. We show how a simple model of hormone regulation, inspired by simple biological organisms, can be used as a novel method to control the behaviour of autonomous robots. We demonstrate the formulation of such an artificial homeostatic hormone system (AHHS) by a set of linked difference equations and explain how the homeostatic control of behaviour is achieved by homeostatic control of the internal ‘hormonal’ state of the robot. The first task that we used to check the quality of our AHHS controllers was a very simple one, which is often a core functionality in controller programmes that are used in autonomous robots: obstacle avoidance. We demonstrate two implementations of such an AHHS controller that performs this task in differing levels of quality. Both controllers use the concept of homeostatic control of internal variables (hormones) and they extend this concept to also include the outside world of the robots into the controlling feedback loops: As they try to regulate internal hormone levels, they are forced to keep a homeostatic control of sensor values in a way that the desired goal ‘obstacle avoidance’ is achieved. Thus, the created behaviour is also a manifestation of the acts of homeostatic control. The controllers were evaluated using a stock-and-flow model that allowed sensitivity analysis and stability tests. Afterwards, we have also tested both controllers in a multi-agent simulation tool, which allowed us to predict the robots' behaviours in various habitats and group sizes. Finally, we demonstrate how this novel AHHS controller is suitable to control a multi-cellular robotic organism in an evolutionary robotics approach, which is used for self-programming in a gait-learning task. These examples shown in this article represent the first step in our research towards autonomous aggregation and coordination of robots to higher-level modular robotic organisms that consist of several joined autonomous robotic units. Finally, we plan to achieve such aggregation patterns and to control complex-shaped robotic organisms using AHHS controllers, as they are described here.     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

Higher index theory for certain expanders and Gromov monster groups,I

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum–Connes assembly map is injective, but not surjective, for the associated metric space X.Expanders with this girth property are a necessary ingredient in the construction of the so-called ‘Gromov monster’ groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum–Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum–Connes assembly map with these coefficients is an isomorphism.     

Existence and stability for some partial neutral functional differential equations with infinite delay

In this paper, we study a class of partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. A principle of linearized stability is also established in the autonomous case. To illustrate our abstract results, we conclude this work by an example.     

Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

Boundedness and stability of solutions to difference equations

This paper is concerned with the qualitative behaviour of solutions to difference equations. We focus on boundedness and stability of solutions and we present a unified theory that applies both to autonomous and nonautonomous equations and to nonlinear equations as well as linear equations. Our presentation brings together new, established, and hard-to-find results from the literature and provides a theory that is both memorable and easy to apply. We show how the theoretical results given here relate to some of those in the established literature and by means of simple examples we indicate how the use of Lipschitz constants in this way can provide useful insights into the qualitative behaviour of solutions to some nonlinear problems including those arising in numerical analysis.     

Using an analog of the Lyapunov function with sign-alternating derivative in the study of global asymptotic stability of equilibria

The paper deals with the problem of global asymptotic stability of the zero solution of a system of autonomous differential equations. A proposed method for studying this problem is based on the use of an auxiliary positive definite function whose derivative can be sign-alternating along solutions of the system.     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.     

The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay

In this paper, we consider an autonomous predator-prey Lotka-Volterra system in which individuals in the population may belong to one of two classes: the immatures and the matures, the age to maturity is represented by a time delay. By using eigenvalue analysis, principal term analyze method, reduction to absurdity, and iterative method, we obtain some simple conditions for global asymptotic stability of the unique positive equilibrium point. Moreover, a condition that the prey population in the system get extinction and the predator population in the system get permanence will be obtained. Meanwhile the theorems extend the corresponding conclusions in which there have no two stage structures.     

基于广义反射函数与自治系统等价的非自治系统

自治系统的研究比非自治系统要容易得多,以广义反射函数为前提,给出了等价于自治系统的非自治系统,从而将非自治系统的研究转化为自治系统的研究,并研究了该类系统的周期解及稳定性.     

Modulation of localized solutions for the Schrödinger equation with logarithm nonlinearity

We investigate the presence of localized analytical solutions of the Schrödinger equation with logarithm nonlinearity. After including inhomogeneities in the linear and nonlinear coefficients, we use similarity transformation to convert the nonautonomous nonlinear equation into an autonomous one, which we solve analytically. In particular, we study stability of the analytical solutions numerically.