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Stability of stochastic switched epidemic systems with discrete or distributed time delay

Journal of Systems Engineering and Electronics 2014年第4期25卷 660-670页

作者： Xiaoying Meng Feiqi Deng Institute of Systems Engineering South China University of Technology School of Statistics and Mathematics Zhongnan University of Economics and Law

Stochastic switched epidemic systems with a discrete or distributed time delay are constructed and investigated. By the Lyapunov method and lto＇s differential rule, the existence and uniqueness of global positive solution of each system is proved. And stability conditions of the disease-free equilibrium of the systems are obtained. Numerical simulations are presented to illustrate the results.

关键词： stochastic switched epidemic system discrete timedelay distributed time delay stochastically stable Lyapunov functional.

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Global asymptotic stability of an SIR epidemic model with distributed time delay

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2001年第6期47卷 4107-4115页

作者： Beretta, E Hara, T Ma, WB Takeuchi, Y Shizuoka Univ Fac Engn Dept Syst Engn Hamamatsu Shizuoka 4328561 Japan Univ Urbino Ist Biomatemat I-61029 Urbino Italy Univ Osaka Prefecture Fac Engn Dept Math Sci Sakai Osaka 5998531 Japan

The SIR epidemic model with distributed time delay that specifies the numbers of the population susceptible to disease was analyzed. The global asymptotic stability of the disease free equilibrium E₀ and the endemic equilibrium E₊ was considered. The analysis using the Liapunov functional V shows that the endemic equilibrium E₊ is feasible and also globally asymptotically stable. Results show that for small time delay the condition gives the threshold for an epidemic to occur.

关键词： SIR epidemic model distributed time delay global asymptotic stability invariance principle Liapunov functional

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Mathematical and dynamic analysis of an ecological model with an impulsive control strategy and distributed time delay

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MATHEMATICAL AND COMPUTER MODELLING 2009年第11-12期50卷 1622-1635页

作者： Yu, Hengguo Zhong, Shouming Ye, Mao Chen, Wufan Univ Elect Sci & Technol China Sch Appl Math Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Minist Educ Key Lab NeuroInformat Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Sch Comp Sci & Engn Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Sch Automat Chengdu 610054 Sichuan Peoples R China

In this paper, on the basis of the theories and methods of ecology and ordinary differential equations, an ecological model with an impulsive control strategy and distributed time delay is established. By using impulsive equation theories, small amplitude perturbation skills and the comparison technique, we get the condition which guarantees the global asymptotical stability of the prey (x) and predator(y) eradication periodic solutions. Further, the influences of impulsive perturbations on the inherent oscillations are studied numerically;these show rich dynamics features, such as period-halving bifurcation, a chaotic band, a periodic window, chaotic crises, etc. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： Impulsive control strategy The largest Lyapunov exponent Globally asymptotically stable Chaotic behavior Periodic solution distributed time delay

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Bifurcations in an economic growth model with a distributed time delay transformed to ODE

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NONLINEAR DYNAMICS 2020年第2期101卷 1263-1279页

作者： Guerrini, Luca Krawiec, Adam Szydlowski, Marek Polytech Univ Marche Dept Management Piazza Martelli 8 I-60121 Ancona AN Italy Jagiellonian Univ Inst Econ Finance & Management Lojasiewicza 4 PL-30348 Krakow Poland Jagiellonian Univ Astron Observ Orla 171 PL-30244 Krakow Poland Jagiellonian Univ Mark Kac Complex Syst Res Ctr Lojasiewicza 11 PL-30348 Krakow Poland

In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three- and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment function show the emergence of two Hopf bifurcations with respect to the rate growth parameter. In this case, we have been able to detect the existence of stable long-period cycles in the economy. According to the time-delay and adjustment speed parameters, the range of admissible values of the rate of growth parameter breaks down into three intervals. First, we have stable focus, then the limit cycle and finally again the stable solution with two Hopf bifurcations. Such behavior appears for some middle interval of the admissible range of values of the rate of growth parameter.

关键词： Kaldor-Kalecki growth model distributed time delay Bifurcation analysis Hopf bifurcation Linear chain trick

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Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay

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MATHEMATICS AND COMPUTERS IN SIMULATION 2009年第3期80卷 619-632页

作者： Yu, Hengguo Zhong, Shouming Ye, Mao Univ Elect Sci & Technol China Sch Appl Math Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Minist Educ Key Lab Neuroinformat Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Sch Engn & Comp Sci Chengdu 610054 Sichuan Peoples R China

In this paper, by using the theories and methods of ecology and ordinary differential equation, an ecological model with impulsive control strategy and distributed time delay is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest level prey and top predator eradication periodic solution. Further, influences of the impulsive perturbation and the parameter a on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises, etc. Moreover, computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. Meanwhile. we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Impulsive control strategy The largest Lyapunov exponent Locally asymptotically stable Chaotic behavior distributed time delay Periodic solution

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Species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2010年第12期59卷 3824-3835页

作者： Yu, Hengguo Zhong, Shouming Agarwal, Ravi P. Xiong, Lianglin Univ Elect Sci & Technol China Sch Math Sci Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Key Lab NeuroInformat Minist Educ Chengdu 610054 Sichuan Peoples R China Florida Inst Technol Dept Math Sci Melbourne FL 32901 USA King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Yunnan Nationalities Univ Sch Math & Comp Sci Kunming 650031 Peoples R China

In this paper, on the basis of the theories and methods of ecology and ordinary differential equations, an ecological system with impulsive harvest and distributed time delay is established. By using the theories of impulsive equations, small amplitude perturbation skills and comparison techniques, we get a condition which guarantees the global asymptotical stability of the prey-(x) eradication and predator-(z) eradication periodic solution. Further, the influences of the impulsive perturbation on the inherent oscillation are studied numerically, and shows rich dynamics, such as period-doubling bifurcation, chaotic bands, periodic windows, chaotic crises, etc. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Impulsive harvest The largest Lyapunov exponent Globally asymptotically stable Dynamical behavior Periodic solution distributed time delay

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Stabilization for Schrodinger equation with a distributed time delay in the boundary input

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IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION 2019年第4期36卷 1305-1324页

作者： Cui, Haoyue Xu, Genqi Chen, Yunlan Tianjin Univ Commerce Dept Math Tianjin 300134 Peoples R China Tianjin Univ Sch Math Tianjin 300350 Peoples R China

In this study, the stabilization problem for Schrodinger equation with distributed input time delay is considered. The main idea of solving the stabilization problem is transformation. The original time delay system is firstly transformed into the undelayed system, and then the feedback control law which can stabilize the undelayed system is found. Finally, we prove that the feedback control law can also exponentially stabilize the time delay system.

关键词： Schrodinger equation distributed time delay boundary feedback control exponential stability

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Analysis of mathematics and dynamics in a food web system with impulsive perturbations and distributed time delay

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APPLIED MATHEMATICAL MODELLING 2010年第12期34卷 3850-3863页

作者： Wang, Xiaomei Yu, Hengguo Zhong, Shouming Agarwal, Ravi P. Univ Elect Sci & Technol China Sch Math Sci Chengdu 610054 Sichuan Peoples R China Univ Elect Sci & Technol China Key Lab NeuroInformat Minist Educ Chengdu 610054 Sichuan Peoples R China Florida Inst Technol Dept Math Sci Melbourne FL 32901 USA King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, a food web system with impulsive perturbations and distributed time delay is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the prey and intermediate predator eradication periodic solution. On this basis, we get that the food web system is permanent if some parameters are satisfied with certain conditions. In order to show that these conditions are effective, the influences of impulsive perturbations on the inherent oscillation and distributed time delay are studied numerically: these show rich dynamics, such as period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Impulsive perturbations The largest Lyapunov exponent Globally asymptotically stable Fourier spectra Periodic solution distributed time delay

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Harmonic resonance and bifurcation of fractional Rayleigh oscillator with distributed time delay

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MATHEMATICS AND COMPUTERS IN SIMULATION 2024年 221卷 281-297页

作者： Zhang, Yufeng Li, Jing Zhu, Shaotao Ma, Zerui Beijing Univ Technol Interdisciplinary Res Inst Fac Sci Beijing Peoples R China Beijing Univ Technol Fac Informat Technol Beijing Peoples R China

Resonance and bifurcation are prominent and significant features observed in various nonlinear systems, often leading to catastrophic failure in practical engineering. This paper investigates, under an analytical and numerical perspective, the dynamical characteristics of a fractional Rayleigh oscillator with distributed time delay. Firstly, through the application of the multiple scales method, we derive approximated analytical solutions and amplitude-frequency equations for the regions near both primary and secondary resonances. The stability conditions of steady-state motions and the existence region of the subharmonic response are also obtained. Furthermore, to validate the accuracy of the approximated solutions, the results are compared with numerical solutions derived from the Caputo scheme, revealing a high concordance between them. Then, a comprehensive study on response curves is conducted for the system under different nonlinear damping, fractional parameters and delay strength. Finally, we identify and discuss the presence of the forked bifurcation within the system.

关键词： Fractional Rayleigh oscillator Harmonic resonance Forked bifurcation distributed time delay Multiple scales method

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Bifurcations in a predator-prey model with discrete and distributed time delay

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NONLINEAR DYNAMICS 2012年第3期67卷 2207-2223页

作者： Xu, Changjin Shao, Yuanfu Guizhou Coll Finance & Econ Guizhou Key Lab Econ Syst Simulat Guiyang 550004 Peoples R China Guilin Univ Technol Dept Math & Phys Guilin 541004 Guangxi Peoples R China

In this paper, a class of predator-prey model with discrete and distributed time delay is considered. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By using the normal form theory and center manifold theory, we derive some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.

关键词： Predator-prey model distributed time delay Stability Hopf bifurcation Periodic solution

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