The delay systemx·(t)=Ax(t)+Bx(t-r)is considered. The necessary and sufficient conditions of the existence of a kind of Liapunov functional for the system are given.
In this paper, some kinds of Liapunov functionals are constructed. By these constructions we have obtained some criteria for the uniform asymptotic stability of zero solution of several RFDE systems. These criteria ar...In this paper, some kinds of Liapunov functionals are constructed. By these constructions we have obtained some criteria for the uniform asymptotic stability of zero solution of several RFDE systems. These criteria are in concise forms, easily checked and applicable.展开更多
It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function...It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.展开更多
By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established re...By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.展开更多
We present and discuss the partial oscillation with respect to equilibrium state ofm-dimensional Logistic delay ecologic models, and obtain some simple criteria.
The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solut...The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.展开更多
We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
A class of non-autonomous two species Lotka-Volterra competitive system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, periodic solution and global attractivity...A class of non-autonomous two species Lotka-Volterra competitive system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, periodic solution and global attractivity of the system are established by means of the comparison method and Liapunov functional.展开更多
In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a su...In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a suitable Liapunov function, the partial stability of the system can be achieved. Finally, two examples are given to illustrate the application of the results.展开更多
Two families of Liapunov functions are employed to study the global stability and bound- edness of functional differential systems. New stability and boundedness theorems are obtained. Ap- plications of these theorems...Two families of Liapunov functions are employed to study the global stability and bound- edness of functional differential systems. New stability and boundedness theorems are obtained. Ap- plications of these theorems to some nonlinear differential systems with infinite delay are discussed.展开更多
This paper considers two differential infectivity(DI) epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium ...This paper considers two differential infectivity(DI) epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.展开更多
By constructing Liapunov functions and building a new inequality, we obtain two kinds of sufficient conditions for the existence and global exponential stability of almost periodic solution for a Hopfield-type neural ...By constructing Liapunov functions and building a new inequality, we obtain two kinds of sufficient conditions for the existence and global exponential stability of almost periodic solution for a Hopfield-type neural networks subject to almost periodic external stimuli. Irt this paper, we assume that the network parameters vary almost periodically with time and we incorporate variable delays in the processing part of the network architectures.展开更多
The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserve...The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserved quantity of the mechanical system or a part of the system is obtained by using the Noether theory. It is possible that the conserved quantity becomes a Liapunov function and the stability of the system is proved by the Liapunov theorem.展开更多
The periodic motion and stability for a class of two-degree-of-freedom nonlinear oscillating systems are studied by using the method of Liapunov function. The sufficient conditions which guarantee the existence, uniqu...The periodic motion and stability for a class of two-degree-of-freedom nonlinear oscillating systems are studied by using the method of Liapunov function. The sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions are obtained.展开更多
Without assuming the smoothness,monotonicity and boundedness of the activation functions, some novel criteria on the existence and global exponential stability of equilibrium point for delayed bidirectional associativ...Without assuming the smoothness,monotonicity and boundedness of the activation functions, some novel criteria on the existence and global exponential stability of equilibrium point for delayed bidirectional associative memory (BAM) neural networks are established by applying the Liapunov functional methods and matrix_algebraic techniques. It is shown that the new conditions presented in terms of a nonsingular M matrix described by the networks parameters,the connection matrix and the Lipschitz constant of the activation functions,are not only simple and practical,but also easier to check and less conservative than those imposed by similar results in recent literature.展开更多
Generalized synchronization between two continuous dynamical systems is discussed. By exploring the Liapunov stability theory and constructing appropriately unidirectional coupling term, a sufficient condition for det...Generalized synchronization between two continuous dynamical systems is discussed. By exploring the Liapunov stability theory and constructing appropriately unidirectional coupling term, a sufficient condition for determining the generalized synchronization between continuous systems is proved. Two examples are used to show the effectiveness of this result.展开更多
In this paper, we directly use the tirear norm Liapunov function to investigate the stability of the linear discrete large-scale systems and obtain some criteria for the asymptotic stability of such a system.
This paper presents an existence theorem of the optimal solution for the infinite-horizon variational problem with the inclusion constraints on the state variables and velocity variables, give the estimation of the op...This paper presents an existence theorem of the optimal solution for the infinite-horizon variational problem with the inclusion constraints on the state variables and velocity variables, give the estimation of the optimal solution. Under some conditions, the authors obtain the local stability and global stability of the stationary solution.展开更多
This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e f...This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e form with negative semi--definite derivative (i.e. V≤0).展开更多
文摘The delay systemx·(t)=Ax(t)+Bx(t-r)is considered. The necessary and sufficient conditions of the existence of a kind of Liapunov functional for the system are given.
文摘In this paper, some kinds of Liapunov functionals are constructed. By these constructions we have obtained some criteria for the uniform asymptotic stability of zero solution of several RFDE systems. These criteria are in concise forms, easily checked and applicable.
基金This project is supported by the National Natural Science Foundation of China
文摘It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.
文摘By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.
文摘We present and discuss the partial oscillation with respect to equilibrium state ofm-dimensional Logistic delay ecologic models, and obtain some simple criteria.
文摘The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.
文摘We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
文摘A class of non-autonomous two species Lotka-Volterra competitive system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, periodic solution and global attractivity of the system are established by means of the comparison method and Liapunov functional.
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021) and the Doctoral Program Foun dation of Institution of Higher Education, China (Grant No 20040007022).
文摘In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a suitable Liapunov function, the partial stability of the system can be achieved. Finally, two examples are given to illustrate the application of the results.
基金Project supported by the National Science Foundation of China Under Grants 69871005
文摘Two families of Liapunov functions are employed to study the global stability and bound- edness of functional differential systems. New stability and boundedness theorems are obtained. Ap- plications of these theorems to some nonlinear differential systems with infinite delay are discussed.
文摘This paper considers two differential infectivity(DI) epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.
基金The Soft Project (B30145) of Science and Technology of Hunan Province.
文摘By constructing Liapunov functions and building a new inequality, we obtain two kinds of sufficient conditions for the existence and global exponential stability of almost periodic solution for a Hopfield-type neural networks subject to almost periodic external stimuli. Irt this paper, we assume that the network parameters vary almost periodically with time and we incorporate variable delays in the processing part of the network architectures.
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021) and the Doctoral Program Foundation of Institutions of Higher Education of China (Grant No 20040007022).
文摘The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserved quantity of the mechanical system or a part of the system is obtained by using the Noether theory. It is possible that the conserved quantity becomes a Liapunov function and the stability of the system is proved by the Liapunov theorem.
文摘The periodic motion and stability for a class of two-degree-of-freedom nonlinear oscillating systems are studied by using the method of Liapunov function. The sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions are obtained.
文摘Without assuming the smoothness,monotonicity and boundedness of the activation functions, some novel criteria on the existence and global exponential stability of equilibrium point for delayed bidirectional associative memory (BAM) neural networks are established by applying the Liapunov functional methods and matrix_algebraic techniques. It is shown that the new conditions presented in terms of a nonsingular M matrix described by the networks parameters,the connection matrix and the Lipschitz constant of the activation functions,are not only simple and practical,but also easier to check and less conservative than those imposed by similar results in recent literature.
基金Project supported by the National Natural Science Foundation of China(Nos.10672093,10372054 and 70431002)
文摘Generalized synchronization between two continuous dynamical systems is discussed. By exploring the Liapunov stability theory and constructing appropriately unidirectional coupling term, a sufficient condition for determining the generalized synchronization between continuous systems is proved. Two examples are used to show the effectiveness of this result.
文摘In this paper, we directly use the tirear norm Liapunov function to investigate the stability of the linear discrete large-scale systems and obtain some criteria for the asymptotic stability of such a system.
文摘This paper presents an existence theorem of the optimal solution for the infinite-horizon variational problem with the inclusion constraints on the state variables and velocity variables, give the estimation of the optimal solution. Under some conditions, the authors obtain the local stability and global stability of the stationary solution.
文摘This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e form with negative semi--definite derivative (i.e. V≤0).