Existence and Stability of Periodic Solution Related to Valveless Pumping

Valveless pumping, also known as Liebau effect, can be described as the unidirectional flow of liquid in a system without valves that is caused by the asymmetry of placing of the periodically working pump. Recently, the research in this field has been reevoked, partially due to its possible application in nanotechnologies. In this paper, a configuration of one pipe and one tank is considered from the mathematical point of view. Qualitative properties of a class of nonlinear differential equations that model the assumed system configuration are investigated. New sufficient conditions for the existence of positive T-periodic solutions are given. Correspondingly, exponential stability of periodic solution is treated. Presented results are new. They extend and complement earlier ones in the literature.

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Existence and Stability of Periodic Solution to Delayed Nonlinear Differential Equations

Abstract and Applied Analysis ◽

10.1155/2014/156948 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Xiang Gu ◽

Huicheng Wang ◽

P. J. Y. Wong ◽

Yonghui Xia

Keyword(s):

Periodic Solution ◽

Asymptotic Stability ◽

Differential Equations ◽

Lyapunov Functional ◽

Nonlinear Differential Equations ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Degree Theory ◽

Competition System ◽

Existence And Stability

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional.

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Existence and exponential stability of a periodic solution for fuzzy cellular neural networks with time-varying delays

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-011-0051-9 ◽

2011 ◽

Vol 21 (4) ◽

pp. 649-658 ◽

Cited By ~ 6

Author(s):

Qianhong Zhang ◽

Lihui Yang ◽

Daixi Liao

Keyword(s):

Neural Networks ◽

Periodic Solution ◽

Exponential Stability ◽

Differential Inequality ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Cellular Neural Networks ◽

Time Varying ◽

Fuzzy Cellular Neural Networks ◽

Inequality Technique

Existence and exponential stability of a periodic solution for fuzzy cellular neural networks with time-varying delays Fuzzy cellular neural networks with time-varying delays are considered. Some sufficient conditions for the existence and exponential stability of periodic solutions are obtained by using the continuation theorem based on the coincidence degree and the differential inequality technique. The sufficient conditions are easy to use in pattern recognition and automatic control. Finally, an example is given to show the feasibility and effectiveness of our methods.

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Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales

Advances in Artificial Neural Systems ◽

10.1155/2014/750532 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Jinxiang Cai ◽

Zhenkun Huang ◽

Honghua Bin

Keyword(s):

Periodic Solution ◽

Stability Analysis ◽

Exponential Stability ◽

Time Scales ◽

Continuous Time ◽

Lyapunov Functional ◽

Contraction Mapping ◽

Sufficient Conditions ◽

Contraction Mapping Principle ◽

Time Varying

We present stability analysis of delayed Wilson-Cowan networks on time scales. By applying the theory of calculus on time scales, the contraction mapping principle, and Lyapunov functional, new sufficient conditions are obtained to ensure the existence and exponential stability of periodic solution to the considered system. The obtained results are general and can be applied to discrete-time or continuous-time Wilson-Cowan networks.

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EXISTENCE AND GLOBAL EXPONENTIAL STABILITY OF PERIODIC SOLUTION OF A CLASS OF IMPULSIVE NETWORKS WITH INFINITE DELAYS

International Journal of Neural Systems ◽

10.1142/s0129065707000506 ◽

2007 ◽

Vol 17 (01) ◽

pp. 35-42 ◽

Cited By ~ 4

Author(s):

YONGHUI XIA ◽

JINDE CAO ◽

MUREN LIN

Keyword(s):

Periodic Solution ◽

Exponential Stability ◽

Lyapunov Functions ◽

Global Exponential Stability ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Degree Theory ◽

Mawhin’S Continuation Theorem ◽

Neuron Networks ◽

Infinite Delays

Sufficient conditions are obtained for the existence and global exponential stability of a unique periodic solution of a class of impulsive tow-neuron networks with variable and unbounded delays. The approaches are based on Mawhin's continuation theorem of coincidence degree theory and Lyapunov functions.

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Easy verifiable sufficient conditions for existence and exponential stability of periodic solution for delay cellular neural networks

2010 IEEE Fifth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA) ◽

10.1109/bicta.2010.5645214 ◽

2010 ◽

Author(s):

Zhihui Yang ◽

Zhaohui Wen

Keyword(s):

Neural Networks ◽

Periodic Solution ◽

Exponential Stability ◽

Sufficient Conditions ◽

Cellular Neural Networks

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Dynamic Analysis of Beddington–DeAngelis Predator-Prey System with Nonlinear Impulse Feedback Control

Complexity ◽

10.1155/2019/5308014 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Dezhao Li ◽

Huidong Cheng ◽

Yu Liu

Keyword(s):

Periodic Solution ◽

Feedback Control ◽

Control Period ◽

Sufficient Conditions ◽

Parameter Analysis ◽

Pest Population ◽

Predator Prey ◽

Prey System ◽

Threshold Conditions ◽

Existence And Stability

In this paper, a predator-prey system with pesticide dose-responded nonlinear pulse of Beddington–DeAngelis functional response is established. First, we construct the Poincaré map of the impulsive semidynamic system and discuss its main properties including the monotonicity, differentiability, fixed point, and asymptote. Second, we address the existence and globally asymptotic stability of the order-1 periodic solution and the sufficient conditions for the existence of the order-k(k ≥ 2) periodic solution. Thirdly, we give the threshold conditions for the existence and stability of boundary periodic solutions and present the parameter analysis. The results show that the pesticide dosage increases with the extension of the control period and decreases with the increase of the threshold. Besides, the state pulse feedback control can manage the pest population at a certain level and avoid excessive application of pesticides.

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PERIODIC SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS: VIA TOPOLOGICAL DEGREE THEORY AND MATRIX SPECTRAL THEORY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412501969 ◽

2012 ◽

Vol 22 (08) ◽

pp. 1250196 ◽

Cited By ~ 9

Author(s):

YONG-HUI XIA

Keyword(s):

Periodic Solution ◽

Spectral Theory ◽

Lyapunov Functional ◽

Nonlinear Differential Equations ◽

Coincidence Degree ◽

Degree Theory ◽

Topological Degree Theory ◽

Maximal Eigenvalue ◽

Existence And Stability ◽

Better Than

The main purpose of this article is to establish the existence and stability of a periodic solution of nonlinear differential equation connected with a problem from mathematical biology. The existence and stability conditions are given in terms of spectral radius of explicit matrices, which are better than conditions obtained by using classic norms. The approaches are based on Mawhin's coincidence degree theory, matrix spectral theory and Lyapunov functional. It should be noted that the new problem appears due to the introduction of Gilpin–Ayala effect. The standard methods used in the previous literature cannot be used to analyze the asymptotic stability of such systems. To handle this problem, two novel techniques should be employed. One is to rescale the system by [Formula: see text], not [Formula: see text]. The other is to analyze the maximal eigenvalue of matrix. In fact, the analytic technique is nontrivial. It is original and very interesting. Finally, some examples and their simulations show the feasibility of our results.

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Application of Mawhin's Coincidence Degree and Matrix Spectral Theory to a Delayed System

Abstract and Applied Analysis ◽

10.1155/2012/940287 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 2

Author(s):

Yong-Hui Xia ◽

Xiang Gu ◽

Patricia J. Y. Wong ◽

Syed Abbas

Keyword(s):

Periodic Solution ◽

Asymptotic Stability ◽

Spectral Theory ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Stability Conditions ◽

Predator Prey ◽

Predator Prey Model ◽

Existence And Stability ◽

Delayed System

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model withM-predators andN-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

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Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2015-0171 ◽

2017 ◽

Vol 18 (1) ◽

pp. 19-27 ◽

Cited By ~ 1

Author(s):

Li Yang ◽

Zhouhong Li ◽

Liyan Pang ◽

Tianwei Zhang

Keyword(s):

Neural Networks ◽

Periodic Solution ◽

Recurrent Neural Networks ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Degree Theory ◽

Almost Periodic Solution ◽

Almost Periodic ◽

Mawhin’S Continuation Theorem ◽

Existence And Stability

Abstract:By means of Mawhin’s continuation theorem of coincidence degree theory and Lyapunov function, some simple sufficient conditions are obtained for the existence and stability of a unique positive almost periodic solution of a delayed Lotka–Volterra recurrent neural networks. To a certain extent, the work in this paper corrects the defect of a recent paper. Finally, an example and simulations are given to illustrate the feasibility and effectiveness of the main result.

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Existence and stability of anti-periodic solutions for impulsive fuzzy Cohen-Grossberge neural networks on time scales

Mathematica Slovaca ◽

10.2478/s12175-013-0191-5 ◽

2014 ◽

Vol 64 (1) ◽

Author(s):

Qianhong Zhang ◽

Lihui Yang ◽

Jingzhong Liu

Keyword(s):

Neural Networks ◽

Exponential Stability ◽

Periodic Solutions ◽

Time Scales ◽

Lyapunov Functional ◽

Global Exponential Stability ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Existence And Stability

AbstractBy applying the method of coincidence degree and constructing suitable Lyapunov functional, some sufficient conditions are established for the existence and global exponential stability of anti-periodic solutions for a kind of impulsive fuzzy Cohen-Grossberg neural networks on time scales. Moreover an example is given to illustrate our results.

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