Both the global exponential stability and the existence of periodic solutions for a class of recurrent neural networks with continuously distributed delays （RNNs） are studied. By employing the inequality α∏k=1^m ...Both the global exponential stability and the existence of periodic solutions for a class of recurrent neural networks with continuously distributed delays （RNNs） are studied. By employing the inequality α∏k=1^m bk^qk≤1/r ∑qkbk^r＋1/rα^r（α≥0,bk≥0,qk〉0,with ∑k=1^m qk=r-1,r≥1, constructing suitable Lyapunov r k=l k=l functions and applying the homeomorphism theory, a family of simple and new sufficient conditions are given ensuring the global exponential stability and the existence of periodic solutions of RNNs. The results extend and improve the results of earlier publications.展开更多
In this paper, it is discussed the model of a kind of nonlinear differential, equation d s d t=1-s-x 1s 0δQ 2(m 1s 0sk 1+s 0s-k) d x 1 d t=x 1Q(m 1s 0sk 1+s 0s-k)-x 1-x 2m 2x 1/Qk 2+x 1/Q...In this paper, it is discussed the model of a kind of nonlinear differential, equation d s d t=1-s-x 1s 0δQ 2(m 1s 0sk 1+s 0s-k) d x 1 d t=x 1Q(m 1s 0sk 1+s 0s-k)-x 1-x 2m 2x 1/Qk 2+x 1/Q d x 2 d t=x 2Q m 2x 1/Qk 2+x 1/Q-x 2.It is proved that the system is exist at least one stable periodic solution on under the following condition:m 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2】m 1δk 1(k 2+Q 2λ 2) 2.Furthermore, ifm 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2【m 1δk 1(k 2-Q 2λ 2) 2mold true them equilibrium point (s *,x * 1,x * 2)∈ set Ω is global asymptotic stable.展开更多
In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability,and show that there are not any periodic solutions in some a neibou...In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability,and show that there are not any periodic solutions in some a neibourhood of the equilibrium points of the dynamical systems.展开更多
In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability, and show that there are not any periodic solutions in some a neibo...In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability, and show that there are not any periodic solutions in some a neibourhed of the equilibrium points of the dynamical systems.展开更多
The global exponentially stability and the existence of periodic solutions of a class of Hopfield neural networks with time delays are investigated. By constructing a novel Lyapunov function, new criteria are provided...The global exponentially stability and the existence of periodic solutions of a class of Hopfield neural networks with time delays are investigated. By constructing a novel Lyapunov function, new criteria are provided to guarantee the global exponentially stability of such systems. For the delayed Hopfield neural networks with time-varying external inputs, new criteria are also acquired for the existence and exponentially stability of periodic solutions. The results are generalizations and improvements of some recent achievements reported in the literature on networks with time delays.展开更多
In this paper,we consider the dynamical system which are from general Hemilton systems under a disturbance,we use theories in Liapunov stability and show that there are not any periodic solutions in some a neighborhoo...In this paper,we consider the dynamical system which are from general Hemilton systems under a disturbance,we use theories in Liapunov stability and show that there are not any periodic solutions in some a neighborhood of the equilibrium points of the dynamical systems.展开更多
The paper is devoted to periodic attractor of delayed Hopfield neural networks with time-varying. By constructing Lyapunov functionals and using inequality techniques, some new sufficient criteria are obtained to guar...The paper is devoted to periodic attractor of delayed Hopfield neural networks with time-varying. By constructing Lyapunov functionals and using inequality techniques, some new sufficient criteria are obtained to guarantee the existence and global exponential stability of periodic attractor. Our results improve and extend some existing ones in [13-14]. One example is also worked out to demonstrate the advantages of our results.展开更多
This paper studies the stability of the periodic solutions of the second order Hamiltonian systems with even superquadratic or subquadratic potentials. The author proves that in the subquadratic case, there exist infi...This paper studies the stability of the periodic solutions of the second order Hamiltonian systems with even superquadratic or subquadratic potentials. The author proves that in the subquadratic case, there exist infinite geometrically distinct elliptic periodic solutions, and in the superquadratic case, there exist infinite geometrically distinct periodic solutions with at most one instability direction if they are half period non-degenerate, otherwise they are elliptic.展开更多
It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally pe...It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Korteweg-de Vries （KdV） equation and the damped Benjamin-Bona-Mahony （BBM） equation. In this paper we intend to show the same results hold for the pure KdV equation （without the damping terms） posed on a finite domain. Consideration is given to the initial-boundary-value problem {ut＋ux＋uux＋uxxx=0, u（x,0）=φ（x）, 0〈x〈1, t〉0,u（0,t）=h（t）, u（1,t） = 0, ux（1,t） = 0, t〉0.It is shown that if the boundary forcing h is periodic with small ampitude, then the small amplitude solution u of （＊） becomes eventually time-periodic. Viewing （＊） （without the initial condition） as an infinite-dimensional dynamical system in the Hilbert space L^2（0, 1）, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system （＊） admits a （locally） unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.展开更多
Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W...Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.展开更多
In this paper, we study the existence and global asymptotic stability of positive periodic solutions of a delayed periodic predator-prey system with Hoffing Ⅱ type functional response. By use of the continuation theo...In this paper, we study the existence and global asymptotic stability of positive periodic solutions of a delayed periodic predator-prey system with Hoffing Ⅱ type functional response. By use of the continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions are obtained.展开更多
The uniform permanence and global asymptotic stability of a class of almost periodic Lotka-Volterra type N-species competitive systems with diffusion and delays are investigated. It is shown that the system is uniform...The uniform permanence and global asymptotic stability of a class of almost periodic Lotka-Volterra type N-species competitive systems with diffusion and delays are investigated. It is shown that the system is uniformly persistent under some appropriate conditions, and new sufficient conditions are obtained for the global asymptotic stability of the unique positive almost periodic solution of the system.展开更多
Integrated pest management （IPM） is a long-term management strategy and has been proved to be more effective in pest control. To well-understand the mechanism and effect of the action of IPM, the geometric theory of...Integrated pest management （IPM） is a long-term management strategy and has been proved to be more effective in pest control. To well-understand the mechanism and effect of the action of IPM, the geometric theory of the involved semi-continuous dynamic systems is becoming more and more important. In this work, a geometric approach is applied to analyze the stability of the positive order-one periodic solution in semi-continuous dynamic systems. A stability criterion to test the stability of the order-one periodic solution is established. As an application, a stage-structure model involved chemical control is presented to show the efficiency of the proposed method. The sufficient conditions to insure the existence of the periodic solution are provided. In addition, the number and the stability of the periodic solutions are discussed accordingly. The simulations are carried out to verify the results.展开更多
The asymptotic behavior of an almost periodic competitive system is investigated. By using differential inequality, the module containment theorem and the Lyapunov function, a good understanding of the existence and g...The asymptotic behavior of an almost periodic competitive system is investigated. By using differential inequality, the module containment theorem and the Lyapunov function, a good understanding of the existence and global asymptotic stability of posi- tive almost periodic solutions is obtained. Finally, an example and numerical simulations are performed for justifying the theoretical results.展开更多
This paper is concerned with a host-macroparasite difference model. By applying the con- traction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investiga...This paper is concerned with a host-macroparasite difference model. By applying the con- traction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investigate the exponential stability of Mmost periodic solution by means of Lyapunov functional.展开更多
In this pape, almost periodic solution of a n-species Lotka-Volterra competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions anti Schauder fixed point the...In this pape, almost periodic solution of a n-species Lotka-Volterra competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions anti Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions under which the strictly positive space homogeneous almost perilodic solution of the system is globally asymptotically stable. Moreover, some numerical simulations are given to validate our theoretical analysis.展开更多
文摘Both the global exponential stability and the existence of periodic solutions for a class of recurrent neural networks with continuously distributed delays （RNNs） are studied. By employing the inequality α∏k=1^m bk^qk≤1/r ∑qkbk^r＋1/rα^r（α≥0,bk≥0,qk〉0,with ∑k=1^m qk=r-1,r≥1, constructing suitable Lyapunov r k=l k=l functions and applying the homeomorphism theory, a family of simple and new sufficient conditions are given ensuring the global exponential stability and the existence of periodic solutions of RNNs. The results extend and improve the results of earlier publications.
文摘In this paper, it is discussed the model of a kind of nonlinear differential, equation d s d t=1-s-x 1s 0δQ 2(m 1s 0sk 1+s 0s-k) d x 1 d t=x 1Q(m 1s 0sk 1+s 0s-k)-x 1-x 2m 2x 1/Qk 2+x 1/Q d x 2 d t=x 2Q m 2x 1/Qk 2+x 1/Q-x 2.It is proved that the system is exist at least one stable periodic solution on under the following condition:m 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2】m 1δk 1(k 2+Q 2λ 2) 2.Furthermore, ifm 2x * 2(k 1δ+s 0δ-Qλ 2-x * 2) 2【m 1δk 1(k 2-Q 2λ 2) 2mold true them equilibrium point (s *,x * 1,x * 2)∈ set Ω is global asymptotic stable.
文摘In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability,and show that there are not any periodic solutions in some a neibourhood of the equilibrium points of the dynamical systems.
文摘In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability, and show that there are not any periodic solutions in some a neibourhed of the equilibrium points of the dynamical systems.
基金the Science Foundation of Guangdong Province in China
文摘The global exponentially stability and the existence of periodic solutions of a class of Hopfield neural networks with time delays are investigated. By constructing a novel Lyapunov function, new criteria are provided to guarantee the global exponentially stability of such systems. For the delayed Hopfield neural networks with time-varying external inputs, new criteria are also acquired for the existence and exponentially stability of periodic solutions. The results are generalizations and improvements of some recent achievements reported in the literature on networks with time delays.
文摘In this paper,we consider the dynamical system which are from general Hemilton systems under a disturbance,we use theories in Liapunov stability and show that there are not any periodic solutions in some a neighborhood of the equilibrium points of the dynamical systems.
基金Foundation item： Supported by the National Science Foundation of Hunan Provincial Education Department （06C792 07C700）
文摘The paper is devoted to periodic attractor of delayed Hopfield neural networks with time-varying. By constructing Lyapunov functionals and using inequality techniques, some new sufficient criteria are obtained to guarantee the existence and global exponential stability of periodic attractor. Our results improve and extend some existing ones in [13-14]. One example is also worked out to demonstrate the advantages of our results.
文摘This paper studies the stability of the periodic solutions of the second order Hamiltonian systems with even superquadratic or subquadratic potentials. The author proves that in the subquadratic case, there exist infinite geometrically distinct elliptic periodic solutions, and in the superquadratic case, there exist infinite geometrically distinct periodic solutions with at most one instability direction if they are half period non-degenerate, otherwise they are elliptic.
文摘It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Korteweg-de Vries （KdV） equation and the damped Benjamin-Bona-Mahony （BBM） equation. In this paper we intend to show the same results hold for the pure KdV equation （without the damping terms） posed on a finite domain. Consideration is given to the initial-boundary-value problem {ut＋ux＋uux＋uxxx=0, u（x,0）=φ（x）, 0〈x〈1, t〉0,u（0,t）=h（t）, u（1,t） = 0, ux（1,t） = 0, t〉0.It is shown that if the boundary forcing h is periodic with small ampitude, then the small amplitude solution u of （＊） becomes eventually time-periodic. Viewing （＊） （without the initial condition） as an infinite-dimensional dynamical system in the Hilbert space L^2（0, 1）, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system （＊） admits a （locally） unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10825207, 11032009)by Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0968)
文摘Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.
基金This work is supported by Scientific Research Fund of ShanDong Agricultural University
文摘In this paper, we study the existence and global asymptotic stability of positive periodic solutions of a delayed periodic predator-prey system with Hoffing Ⅱ type functional response. By use of the continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions are obtained.
基金This research is supported by the National Natural Science Foundation of China(10171056).
文摘The uniform permanence and global asymptotic stability of a class of almost periodic Lotka-Volterra type N-species competitive systems with diffusion and delays are investigated. It is shown that the system is uniformly persistent under some appropriate conditions, and new sufficient conditions are obtained for the global asymptotic stability of the unique positive almost periodic solution of the system.
文摘Integrated pest management （IPM） is a long-term management strategy and has been proved to be more effective in pest control. To well-understand the mechanism and effect of the action of IPM, the geometric theory of the involved semi-continuous dynamic systems is becoming more and more important. In this work, a geometric approach is applied to analyze the stability of the positive order-one periodic solution in semi-continuous dynamic systems. A stability criterion to test the stability of the order-one periodic solution is established. As an application, a stage-structure model involved chemical control is presented to show the efficiency of the proposed method. The sufficient conditions to insure the existence of the periodic solution are provided. In addition, the number and the stability of the periodic solutions are discussed accordingly. The simulations are carried out to verify the results.
文摘The asymptotic behavior of an almost periodic competitive system is investigated. By using differential inequality, the module containment theorem and the Lyapunov function, a good understanding of the existence and global asymptotic stability of posi- tive almost periodic solutions is obtained. Finally, an example and numerical simulations are performed for justifying the theoretical results.
基金The author thanks the referees for their valuable comments and suggestions in improving the presentation of the manuscript. This work is supported by Natural Science Foundation of Education Department of Anhui Province （KJ2014A043）.
文摘This paper is concerned with a host-macroparasite difference model. By applying the con- traction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investigate the exponential stability of Mmost periodic solution by means of Lyapunov functional.
基金This work is supported by Science and Technology Project of Chongqing Municipal Education Committee （Grant No. KJ 110501） of China, Natural Science Foundation Project of CQ CSTC （Grants No. CSTC2012jjA20016） of China and the NSFC （Grant Nos. 51005264, 11101298, 40801214） of China.
文摘In this pape, almost periodic solution of a n-species Lotka-Volterra competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions anti Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions under which the strictly positive space homogeneous almost perilodic solution of the system is globally asymptotically stable. Moreover, some numerical simulations are given to validate our theoretical analysis.
