Existence Canard Solutions For Four Dimensional Hindmarsh-Rose Model with Respect to Infinitesimal Parameter

Chiman Qadir; Ibrahim Hamad; Waleed Aziz

doi:10.21271/ZJPAS.36.2.9

Authors

Chiman Qadir Department of Mathematics, College of Science, Salahaddin University–Erbil, Kurdistan region–Iraq
Ibrahim Hamad Department of Mathematics, College of Science, Salahaddin University–Erbil, Kurdistan region–Iraq
Waleed Aziz Department of Mathematics, College of Science, Salahaddin University–Erbil, Kurdistan region–Iraq

DOI:

https://doi.org/10.21271/ZJPAS.36.2.9

Keywords:

Singularly perturbed dynamical systems, Hindmarsh-Rose model, Canard solutions, Nonstandard analysis.

Abstract

This research endeavor seeks to investigate the presence and viability of canard solutions within the context of the generalized Hindmarsh-Rose model, specifically when extended to four dimensions. To achieve this, nonstandard analysis is employed as a powerful tool for identifying and characterizing canard solutions within the four-dimensional singularly perturbed system, wherein two fast variables are considered in the folded saddle case. By undertaking this rigorous approach, we aim to contribute valuable insights to the understanding of canard phenomena in complex dynamical systems.

References

BENOÎT, É. 1983. Systemes lents-rapides dans R3 et leurs canards. Astérisque, 109-110.

BENOÎT, É. & LOBRY, C. 1982. Les canards de R3. CR Acad. Sc. Paris, 294, 483-488.

CORSON, N. & AZIZ-ALAOUI, M. 2009. Asymptotic dynamics of Hindmarsh-Rose neuronal system. Dynamics of Continuous, Discrete and Impulsive Systemes, Series B: Applications and Algorithms, p. 535.

DESROCHES, M., KAPER, T. J. & KRUPA, M. 2013. Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23.

DUMORTIER, F. & ROUSSARIE, R. H. 1996. Canard cycles and center manifolds, American Mathematical Soc.

GINOUX, J.-M. & LLIBRE, J. 2016. Canards existence in memristor’s circuits. Qualitative theory of dynamical systems, 15, 383-431.

GINOUX, J.-M., LLIBRE, J. & TCHIZAWA, K. 2019. Canards existence in the Hindmarsh–Rose model. Mathematical Modelling of Natural Phenomena, 14, 409.

HAMAD, I., QADIR, C. & AZIZ, W. 2023. Nonstandard canard solutions in a three-dimensional system and Hindmarsh-Rose neuron model. Submitted.

SZMOLYAN, P. & WECHSELBERGER, M. 2001. Canards in R3. Journal of Differential Equations, 177, 419-453.

TCHIZAWA, K. 2007. Generic conditions for duck solutions in R4 Kyoto Univ RIMS Kokyuroku, 1547, 107-113.

TCHIZAWA, K. 2010. On 4-dim duck solutions with relative stability (Dynamical Systems: with Hyperbolicity and with Large Freedom). 数理解析研究所講究録, 1688, 157-163.

TCHIZAWA, K. 2012. On relative stability in 4-dimensional duck solution. Journal of Mathematics and System Science, 2, 558.

TCHIZAWA, K. 2013. On the two methods for finding 4-dimensional duck solutions. Applied Mathematics, 2014.

TCHIZAWA, K., MIKI, H. & NISHINO, H. 2005. On the existence of a duck solution in Goodwin's nonlinear business cycle model. Nonlinear Analysis: Theory, Methods & Applications, 63, e2553-e2558.

VAN DER POL, B. 1926. On “relaxation-oscillations”. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2, 978-992.

VAN DER POL, B. 1960. A theory of the amplitude of free and forced triode vibrations, Radio Rev. 1 (1920) 701-710, 754-762. Selected scientific papers, 1.

WECHSELBERGER, M. 2012. A propos de canards (apropos canards). Transactions of the American Mathematical Society, 364, 3289-3309.