Mathematical Modelling of Bifurcation Analysis on the Effect of Random Perturbation Value on a Dynamical System: Alternative Numerical Approach

I. C. Eli; O. T. Daniel

doi:https://doi.org/10.14445/22315373/IJMTT-V69I2P502

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 2 | Year 2023 | Article Id. IJMTT-V69I2P502 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I2P502

Mathematical Modelling of Bifurcation Analysis on the Effect of Random Perturbation Value on a Dynamical System: Alternative Numerical Approach

I. C. Eli, O. T. Daniel

Received	Revised	Accepted	Published
16 Dec 2022	17 Jan 2023	30 Jan 2023	10 Feb 2023

Citation :

I. C. Eli, O. T. Daniel, "Mathematical Modelling of Bifurcation Analysis on the Effect of Random Perturbation Value on a Dynamical System: Alternative Numerical Approach," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 2, pp. 15-23, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I2P502

Abstract

Mathematical modelling of a Bifurcation analysis on the effect of random perturbation value of 0.64 on a dynamical system was investigated with the help of numerical approach of ordinary differential equation of order 45 (ODE45) and it was observed that the proposed dynamical system was purely unstable when the length of the growing season ranges from 19 days to 44 days. But when the length of the growing season increases to 49 days, Bifurcation was noticed when the length of the growing season is 54 days up to the harvesting season (99 days) and beyond. The randomization equally affects the steady-state since their values are fluctuating.

Keywords

Bifurcation, Dynamical system, Numerical approach, Steady-state, Random perturbation value and length of growing season.

References

[1] H Poincare, “Memory on the Curves Defined by the Differential Equations I – IV,” Journal of Pure and Applied Mathematics, vol. 7, pp. 375-422, 1881.
[2] Douglas Adams, The Hitchiker’s Guide to the Galaxy, Pocket Books, New York, 1979.
[3] I. C. Eli, and K. W. Bunonyo, “Mathematical Modelling of the Effect of HIV/AIDS on Sickle Cell Genotype,” International Journal of Scientific Engineering and Applied Science, vol. 6, no. 6, pp. 92-106, 2020.
[4] Hiroki Sayama, Basics of Dynamical Systems, Opensung publisher, 2020.
[5] M. J. Crawley, Plant Ecology, Blackwell Scientific Publications, 1986.
[6] Paul Glendinning, Stability, Instability and Chaos, An Introduction to the Theory of Nonlinear Differential Equation, Cambridge University Press, 1994.
[7] J. D. Murray, Mathematical Biology 1: An Introduction, Springer-Verlag, Third Edition, New York, 2002. Crossref, https://doi.org/10.1007/b98868
[8] J. W. Silvertown, and Lovett Doust J, Introduction to Plant Biology, Third Edition, Blackwell Science, 1995.
[9] Ali H. Nayfeh, and Balakumar Balachandran, Applied Nonlinear Dynamics: Analytical Computation and Experimental Methods, A Wiley-Inter-Science Publications, 1995.
[10] Richard L. Burden, and J. Douglas Faires, Numerical Analysis, Ninth Edition, Canada, 2011.
[11] I. C. Eli, and E. N. Ekaka-a, “Effect of Discrete Time Delays on the Stability of a Dynamical System,” International Journal of Mathematics Trends and Technology, vol. 67, no. 8, pp. 45-49, 2021. Crossref, http://doi.org/10.14445/22315373/IJMTT-V67I8P505
[12] Isobeye George et al., “Deterministic Stabilization of a Dynamical System using a Computational Approach,” International Journal of Advanced Engineering, Management and Science, vol. 4, no. 1, pp. 24-28, 2018. Crossref, https://dx.doi.org/10.22161/ijaems.4.1.6
[13] Chyi Hwang, and Yi-Cheng Cheng, “A Note on the Use of the Lambert W Function in the Stability Analysis of Time-Delay System,” Automatic, vol. 41, no. 11, pp. 1979-1985, 2005. Crossref, https://doi.org/10.1016/j.automatica.2005.05.020
[14] Z. H. Wang, “Numerical Stability test of Neutral Delay Differential Equations,” Mathematical Problems in Engineering, 2008. Crossref, https://doi.org/10.1155/2008/698043
[15] R. E. Akpodee, and E. N. Ekaka-a, “Deterministic Stability Analysis using a Numerical Simulation Approach,” 300k of Proceedings – Academic Conference Publications and Research International on Sub-Sahara African Potentials in the New Millennium, vol. 3, no. 1, 2015.
[16] Jang bahadur Shukla et al., “Existence and Survival of Two Competing Species in a Polluted Environment: A Mathematical Model,” Journal of Biological System, vol. 9, no. 2, pp. 89-103, 2001. Crossref, https://doi.org/10.1142/S0218339001000359
[17] J.B. Shukla et al., “Modelling the Survival of a Resource Dependent Population: Effects of Toxicants (Pollutants) Emitted from External Sources as Well as Formed by its Precursors,” Nonlinear Analysis; Real World Application, vol. 10, no. 1, pp. 54-70, 2009. Crossref, https://doi.org/10.1016/j.nonrwa.2007.08.014
[18] E. N. Ekaka-a, and V. M. Nafo, “Stability a Mathematical Model of Stock Market Population System,” Scientia Africa, vol. 11, no. 1, pp. 92-97, 2012.
[19] Yubin Yan, and Enu-Obari N. Ekaka-a, “Stability of a Mathematical Model of Population System,” Journal of the Franklin Institute, vol. 348, no. 10, pp. 2744-2758, 2011. Crossref, https://doi.org/10.1016/j.jfranklin.2011.08.014
[20] Peter A. Abrams, and William G. Wilson, “Co-Existence and Competitors in Meta Communities Due to Spatial Variation in Resource Growth Rates Does R* Predict the Outcome of Completion?,” American Naturalist, vol. 7, pp. 929-940, 2004. Crossref, https://doi.org/10.1111/j.1461-0248.2004.00644.x
[21] O. A. Sultanov, “Stochastic Stability of a Dynamical System Perturbed by White Nose,” Springer, vol. 101, pp. 149-156, 2017. Crossref, https://doi.org/10.1134/S0001434617010151
[22] M. Agarwal, and S. Devi, “The Effect of Environmental Tax on the Survival of Biological Species in a Polluted Environment: A Mathematical Model,” Nonlinear Analysis Modelling and Control, vol. 15, no. 3, pp. 271-286.
[23] Innocent C. Eli, and Godspower C. Abanum, “Comparism between Analytical and Numerical Result of Stability Analysis of a Dynamical System,” Communication in Physical Sciences, vol. 5, no. 4, pp. 437-440, 2020.
[24] Leonid Shaikhet, “Stability of Equilibrium States of a Nonlinear Delay Differential Equation with Stochastic Perturbation,” International Journal of Robust and Nonlinear Control, vol. 27, no. 6, pp. 915-924, 2021. Crossref, https://doi.org/10.1002/rnc.360