Lyapunov vector functions and partial boundedness of solutions with partially controlled initial conditions

On the basis of the method of Lyapunov vector functions, we obtain a sufficient test for the uniform partial boundedness of solutions with partially controlled initial conditions. We introduce the notions of partial equiboundedness, partial equiboundedness in the limit, and partial uniform boundedness in the limit of solutions with partially controlled initial conditions. By the method of Lyapunov vector functions, we obtain sufficient tests for the partial equiboundedness of solutions and for the partial uniform boundedness in the limit and partial equiboundedness in the limit of solutions with partially controlled initial conditions.     

Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.

     

Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notion of partial uniform boundedness of solutions with partially controlled initial conditions in the general case, that is, the case in which the part of variables with respect to which the boundedness of solutions is studied is a subset of the part of variables with respect to which the initial conditions are controlled. We obtain a criterion for the partial uniform boundedness of solutions with partly controlled initial conditions. We introduce the notion of partial total boundedness of solutions with partly controlled initial conditions. We obtain a sufficient condition for the partial total boundedness of solutions with partly controlled initial conditions.     

Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions

Mathematical Notes - Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives...     

Lyapunov Functions,Krasnosel’skii Canonical Domains,and the Existence of Poisson Bounded Solutions

Mathematical Notes - The notions of Poisson boundedness and Poisson partial boundedness of solutions of systems are introduced. Based on the Lyapunov function method and...     

Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notions of equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables for solutions with partly controlled initial conditions. We obtain sufficient conditions for the equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables of solutions with partly controlled initial conditions. We introduce the notions of equiboundedness and uniform boundedness with respect to part of the variables for solutions of systems with partly controlled initial conditions. We obtain sufficient conditions for the equiboundedness and uniform boundedness with respect to part of the variables of solutions with partly controlled initial conditions.     

Vector Lyapunov Functions and Ultimate Poisson Boundedness of Solutions of Systems of Differential Equations

Various forms of uniform-ultimate Poisson boundedness of solutions and of ultimate Poisson equiboundedness of solutions are introduced. Sufficient conditions for various forms of uniform-ultimate Poisson boundedness and of ultimate Poisson equiboundedness of solutions are obtained by using the method of vector Lyapunov functions.     

Boundedness results for a certain third order nonlinear differential equation

Sufficient conditions for the existence of solutions to boundedness and ultimate boundedness problems associated to a certain third order nonlinear differential equation are given by means of the Lyapunov’s second method. The appropriate Lyapunov function is given explicitly. Our results complement some well known results on the third order differential equations in the literature.     

Uniform Boundedness in the Sense of Poisson of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce several generalizations of the properties of equiboundedness and uniform boundedness of solutions of ordinary differential systems, which are united by the common names of equiboundedness in the sense of Poisson and uniform boundedness in the sense of Poisson. For each of the above-introduced properties, we use the method of Lyapunov vector functions to obtain sufficient criteria for the system to have a certain property. In terms of the upper Dini derivative of the Lyapunov function given by a system, several criteria are established for the solutions of this system to have the relevant type of uniform boundedness in the sense of Poisson.     

Formation of phase-locked states in a population of locally interacting Kuramoto oscillators

In this paper, we present an asymptotic formation of phase-locked states from the ensemble of Kuramoto oscillators with a symmetric and connected interaction topology. For a limited interaction topology that does not have an all-to-all interaction, Lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness. Thus, we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates. Our synchronization estimation method consists of two parts. First we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase-locked states using ?ojasiewicz gradient inequality. Second, we show that for the initial configurations lying in the half circle, the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear Gronwall?s differential inequality for the total phase variance.     

无限时滞的随机泛函微分方程解的渐近性质

本文研究了无限时滞随机泛函微分方程解的存在唯一性,矩有界性的问题.利用Lyapunov函数法以及概率测度的引入得到了确保方程解在唯一、矩有界、时间平均矩有界同时成立的一个新的条件.推广了Khasminskii-Mao定理的相关结果.     

Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equationa*

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicity of solutions.     

Higher-Order Derivatives of Lyapunov Functions And Ultimate Boundedness in the Sense of Poisson of Solutions to Systems of Differential Equations

Using the higher-order derivatives of Lyapunov functions, we obtain sufficient tests of various types for the uniform-ultimate boundedness in the sense of Poisson of solutions and various types of equiultimate boundedness of solutions in the sense of Poisson.     

Large‐time behaviour and blow up of solutions for Gierer–Meinhardt systems

The purpose of this paper is to give a proof of global existence of solutions for Gierer–Meinhardt systems with homogeneous Neumann boundary conditions. Our technique is based on Lyapunov functional argument that yields the uniform boundedness of solutions. The asymptotic behaviour of the solutions under suitable conditions is also studied. Moreover, under reasonable conditions on the exponents of the nonlinear term, we show the blow up in finite time of the solutions for the considered system. These results are valid for any positive initial data in , without any differentiability conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Uniform ultimate boundedness of nonlinear nonstationary systems

A certain class of nonlinear, nonstationary systems of differential equations is studied. It is assumed that the right-hand sides of the equations under consideration are homogeneous functions of order smaller than one with respect to the phase variables. The purpose of this paper is to obtain sufficient conditions for the uniform ultimate boundedness of systems of this form. A method for constructing nonstationary Lyapunov functions is suggested and applied to prove that the asymptotic stability of the zero solution of the corresponding averaged system implies the uniform ultimate boundedness of the initial nonstationary system. Classes of perturbations that do not violate uniform ultimate boundedness, even in the case where the order of the perturbations exceeds the homogeneity order of the unperturbed equations, are described. Unlike in previous works, where the results are based on the averaging method, the presence of a small parameter on the right-hand sides of the equations under examination is not assumed. Dissipativity is ensured at the expense of homogeneity orders.     

On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.     

Uniform boundedness in part of the variables of solutions to systems of differential equations with partially controllable initial conditions

The paper deals with the development of the theory of boundedness in part of the variables of solutions to systems of differential equations, which is in fact a version of the Lyapunov direct method in the theory of stability.     

Analysis of the set of trajectories of nonlinear dynamics: Stability and boundedness of motions

For a set of differential equations with the Hukuhara derivative, we obtain sufficient conditions for various types of boundedness of trajectories and stability of the set of stationary solutions. To this end, we use scalar and vector Lyapunov functions constructed on the basis of an auxiliary matrix-valued function.