数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (2): 709-732.doi: 10.1016/S0252-9602(18)30776-8

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STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

Rakesh KUMAR1,2, Anuj Kumar SHARMA3, Kulbhushan AGNIHOTRI1   

  1. 1. Department of Applied Sciences, S. B. S. State Technical Campus, Ferozepur, Punjab 152004, India;
    2. Research Scholar with I. K. G. Punjab Technical University, Kapurthala, Punjab 144603, India;
    3. Department of Mathematics, L. R. D. A. V. College, Jagraon, Ludhiana, Punjab 142026, India
  • 收稿日期:2016-11-17 修回日期:2017-12-18 出版日期:2018-04-25 发布日期:2018-04-25
  • 通讯作者: Rakesh KUMAR E-mail:keshav20070@gmail.com
  • 作者简介:Anuj Kumar SHARMA,E-mail:anujsumati@gmail.com;Kulbhushan AGNIHOTRI,E-mail:agnihotri69@gmail.com
  • 基金资助:

    The Authors Gratefully Acknowledge the Support Provided by the I.K.G. Punjab Technical University, Kapurthala, Punjab, India, where one of us (RK) is a Research Scholar.

STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

Rakesh KUMAR1,2, Anuj Kumar SHARMA3, Kulbhushan AGNIHOTRI1   

  1. 1. Department of Applied Sciences, S. B. S. State Technical Campus, Ferozepur, Punjab 152004, India;
    2. Research Scholar with I. K. G. Punjab Technical University, Kapurthala, Punjab 144603, India;
    3. Department of Mathematics, L. R. D. A. V. College, Jagraon, Ludhiana, Punjab 142026, India
  • Received:2016-11-17 Revised:2017-12-18 Online:2018-04-25 Published:2018-04-25
  • Contact: Rakesh KUMAR E-mail:keshav20070@gmail.com
  • Supported by:

    The Authors Gratefully Acknowledge the Support Provided by the I.K.G. Punjab Technical University, Kapurthala, Punjab, India, where one of us (RK) is a Research Scholar.

摘要:

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

关键词: Innovation diffusion model, stability analysis, Hopf-bifurcation, normal form theory, center manifold theorem

Abstract:

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Key words: Innovation diffusion model, stability analysis, Hopf-bifurcation, normal form theory, center manifold theorem