Volume 7, Issue 3
Dynamics of an Innovation Diffusion Model with Time Delay

Rakesh Kumar, Anuj K. Sharma & Kulbhushan Agnihotri

East Asian J. Appl. Math., 7 (2017), pp. 455-481.

Published online: 2018-02

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  • Abstract

A nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

  • Keywords

Innovation diffusion model, evaluation period, stability analysis, Hopf bifurcation, normal form theory, centre manifold theorem.

  • AMS Subject Headings

34C23, 34D20, 37L10, 49Q12, 92D25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-455, author = {}, title = {Dynamics of an Innovation Diffusion Model with Time Delay}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {3}, pages = {455--481}, abstract = {

A nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.201216.230317a}, url = {http://global-sci.org/intro/article_detail/eajam/10759.html} }
TY - JOUR T1 - Dynamics of an Innovation Diffusion Model with Time Delay JO - East Asian Journal on Applied Mathematics VL - 3 SP - 455 EP - 481 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.201216.230317a UR - https://global-sci.org/intro/article_detail/eajam/10759.html KW - Innovation diffusion model, evaluation period, stability analysis, Hopf bifurcation, normal form theory, centre manifold theorem. AB -

A nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

Rakesh Kumar, Anuj K. Sharma & Kulbhushan Agnihotri. (2020). Dynamics of an Innovation Diffusion Model with Time Delay. East Asian Journal on Applied Mathematics. 7 (3). 455-481. doi:10.4208/eajam.201216.230317a
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