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Dynamics of an Innovation Diffusion Model with Time Delay

Part of: Qualitative theory Manifolds Stability theory Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  07 September 2017

Rakesh Kumar ,
Anuj K. Sharma  and
Kulbhushan Agnihotri
Rakesh Kumar*
Affiliation:
Department of Applied Sciences, SBS State Technical Campus, Ferozepur-152004, Punjab, India
Anuj K. Sharma*
Affiliation:
Department of Mathematics, LRDAV College, Jagraon, Ludhiana-142026, Punjab, India
Kulbhushan Agnihotri*
Affiliation:
Department of Applied Sciences, SBS State Technical Campus, Ferozepur-152004, Punjab, India
*
*Corresponding author. Email addresses:keshav20070@gmail.com (R. Kumar), anujsumati@gmail.com (A. K. Sharma), agnihotri69@gmail.com (K. Agnihotri)
*Corresponding author. Email addresses:keshav20070@gmail.com (R. Kumar), anujsumati@gmail.com (A. K. Sharma), agnihotri69@gmail.com (K. Agnihotri)
*Corresponding author. Email addresses:keshav20070@gmail.com (R. Kumar), anujsumati@gmail.com (A. K. Sharma), agnihotri69@gmail.com (K. Agnihotri)
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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 modelevaluation periodstability analysisHopf bifurcationnormal form theorycentre manifold theorem

MSC classification

Secondary: 34C23: Bifurcation 34D20: Stability 37L10: Normal forms, center manifold theory, bifurcation theory 49Q12: Sensitivity analysis 92D25: Population dynamics (general)
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 455 - 481
Copyright
Copyright © Global-Science Press 2017 

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