Article: Bifurcation behaviour of a nonlinear innovation diffusion model with external influences Journal: International Journal of Dynamical Systems and Differential Equations (IJDSDE) 2020 Vol.10 No.4 pp.329 - 357 Abstract: A nonlinear form of Bass model for innovation diffusion consisting of a system of two variables viz. for adopters and nonadopters population is proposed to lay stress on the evaluation period. The local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analysing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and nonadopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. The direction and the stability of bifurcating periodic solutions is determined by using the normal form theory and centre manifold theorem. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage (final adoption stage) in the system. Numerical computations are executed to confirm the correctness of theoretical investigations. Inderscience Publishers - linking academia, business and industry through research

Title: Bifurcation behaviour of a nonlinear innovation diffusion model with external influences

Authors: Rakesh Kumar; Anuj Kumar Sharma; Kulbhushan Agnihotri

Addresses: Department of Applied Sciences, Shaheed Bhagat Singh State Technical Campus, Moga Road, Ferozepur, Punjab, 152004, India ' Department of Mathematics, L.R.D.A.V. College, Jagraon, Ludhiana, Punjab, 142026, India ' Department of Applied Sciences, Shaheed Bhagat Singh State Technical Campus, Moga Road, Ferozepur, Punjab, 152004, India

Abstract: A nonlinear form of Bass model for innovation diffusion consisting of a system of two variables viz. for adopters and nonadopters population is proposed to lay stress on the evaluation period. The local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analysing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and nonadopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. The direction and the stability of bifurcating periodic solutions is determined by using the normal form theory and centre manifold theorem. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage (final adoption stage) in the system. Numerical computations are executed to confirm the correctness of theoretical investigations.

Keywords: dynamical system; innovation diffusion model; evaluation period; variable external influences; word of mouth; stability analysis; sensitivity analysis; Hopf bifurcation; centre manifold theorem; normal form theory.

DOI: 10.1504/IJDSDE.2020.109107

International Journal of Dynamical Systems and Differential Equations, 2020 Vol.10 No.4, pp.329 - 357

Received: 13 Aug 2018
Accepted: 15 Dec 2018
Published online: 20 Aug 2020 *

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Title: Bifurcation behaviour of a nonlinear innovation diffusion model with external influences

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