arrow
Volume 8, Issue 3
Delay Induced Hopf Bifurcation in a Nonlinear Innovation Diffusion Model: External Influences Effect

Rakesh Kumar, Anuj K. Sharma & Kulbhushan Agnihotri

East Asian J. Appl. Math., 8 (2018), pp. 422-446.

Published online: 2018-08

Export citation
  • Abstract

A nonlinear innovation diffusion model which incorporates the evaluation stage (time delay) is proposed to describe the dynamics of three population classes for non-adopter and adopter densities. The local stability of the various equilibrium points is investigated. It is observed that the system is locally asymptotically stable for a delay limit and produces periodic orbits via a Hopf bifurcation when evaluation period crosses a critical value. Applying normal form theory and center manifold theorem, we study the properties of the bifurcating periodic solutions. The model shows that the adopter population density achieves its maturity stage faster if the cumulative density of external influences increases. Several numerical examples confirm our theoretical results.

  • Keywords

Evaluation period, stability analysis, Hopf bifurcation, normal form theory, center manifold theorem.

  • AMS Subject Headings

34C23, 34D20, 37L10, 92D25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{EAJAM-8-422, author = {}, title = {Delay Induced Hopf Bifurcation in a Nonlinear Innovation Diffusion Model: External Influences Effect}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {8}, number = {3}, pages = {422--446}, abstract = {

A nonlinear innovation diffusion model which incorporates the evaluation stage (time delay) is proposed to describe the dynamics of three population classes for non-adopter and adopter densities. The local stability of the various equilibrium points is investigated. It is observed that the system is locally asymptotically stable for a delay limit and produces periodic orbits via a Hopf bifurcation when evaluation period crosses a critical value. Applying normal form theory and center manifold theorem, we study the properties of the bifurcating periodic solutions. The model shows that the adopter population density achieves its maturity stage faster if the cumulative density of external influences increases. Several numerical examples confirm our theoretical results.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.010417.200118}, url = {http://global-sci.org/intro/article_detail/eajam/12617.html} }
TY - JOUR T1 - Delay Induced Hopf Bifurcation in a Nonlinear Innovation Diffusion Model: External Influences Effect JO - East Asian Journal on Applied Mathematics VL - 3 SP - 422 EP - 446 PY - 2018 DA - 2018/08 SN - 8 DO - http://doi.org/10.4208/eajam.010417.200118 UR - https://global-sci.org/intro/article_detail/eajam/12617.html KW - Evaluation period, stability analysis, Hopf bifurcation, normal form theory, center manifold theorem. AB -

A nonlinear innovation diffusion model which incorporates the evaluation stage (time delay) is proposed to describe the dynamics of three population classes for non-adopter and adopter densities. The local stability of the various equilibrium points is investigated. It is observed that the system is locally asymptotically stable for a delay limit and produces periodic orbits via a Hopf bifurcation when evaluation period crosses a critical value. Applying normal form theory and center manifold theorem, we study the properties of the bifurcating periodic solutions. The model shows that the adopter population density achieves its maturity stage faster if the cumulative density of external influences increases. Several numerical examples confirm our theoretical results.

Rakesh Kumar, Anuj K. Sharma & Kulbhushan Agnihotri. (2020). Delay Induced Hopf Bifurcation in a Nonlinear Innovation Diffusion Model: External Influences Effect. East Asian Journal on Applied Mathematics. 8 (3). 422-446. doi:10.4208/eajam.010417.200118
Copy to clipboard
The citation has been copied to your clipboard