Abstract
We consider systems of ordinary differential equations, arising in applications. The focus is on attractors and their properties. Since attractors determine the evolution of systems and the behavior of their solutions, this information is necessary to monitor and manage the networks that are modeled by these systems. Particular attention is paid to the types of attractors, including chaotic ones. Three classes of systems are considered. The first is the occurrence of chaos in a certain system which is a generalization of the Duffing equation is analyzed. Then, the three- and four-dimensional systems, arising in the theory of gene regulatory networks, are studied. Examples of systems, which have attractors of different types, are given. A significant part of the chapter is devoted to four-, five-, and six-dimensional systems found in the theory of neuronal networks. Here, the hyperbolic tangent is chosen as the activating function. Several examples of chaotic behavior of solutions are given. The presentation is accompanied by multiple illustrations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Tamaševičius, S. Bumelienė, R. Kirvaitis, G. Mykolaitis, E. Tamaševičiūtė, E. Lindberg, Autonomous Duffing-Holmes type chaotic oscillator. Elektronika ir Elektrotechnika. 5(93), 43–46 (2009). https://www.eejournal.ktu.lt/index.php/elt/article/download/10178/5036
Y. Koizumi, T. Miyamura, S. Arakawa, E. Oki, K. Shiomoto, and M. Murata, Application of attractor selection to adaptive virtual network topology control, in Proceedings of BIONETICS (2008), pp. 1–8
L.-Z. Wang, R.-Q. Su, Z.-G. Huang, X. Wang, W.-X. Wang, C. Grebogi, Y.-C. Lai, A geometrical approach to control and controllability of nonlinear dynamical networks. Nat. Commun. 7, 11323 (2016). https://doi.org/10.1038/ncomms11323
O.E. Rössler, Chaotic oscillations: an example of hyperchaos, in Nonlinear Oscillations in Biology. Lectures in Applied Mathematics, vol. 17 (AMS, Providence, 1979)
P. Holmes, A nonlinear oscillator with a strange attractor. Philos. Trans. R. Soc. Lond. A: Math. Phys. Sci. 292, 419–448 (1979). https://royalsocietypublishing.org/doi/10.1098/rsta.1979.0068
D.K. Arrowsmith, C.M. Place, Dynamical Systems. Differential Equations, Maps and Haotic Behavior (Chapman and Hall/CRC, London, 1992)
V.W. Noonburg, Differential Equations: From Calculus to Dynamical Systems, 2nd edn. (MAA Press, Providence, 2019)
F. Sadyrbaev F., V. Sengileyev, Remarks on inhibition. Int. J. Math. Comput. Methods 7, 11–17 (2022)
D. Ogorelova, F. Sadyrbaev F., V. Sengileyev, Control in inhibitory genetic regulatory network models. Contemp. Math. 1(5), 421–428
A. Krogh, What are artificial neural networks? Nat. Biotechnol. 26(2), 195–197 (2008). https://doi.org/10.1038/nbt1386
S. Haykin, Neural networks expand SP’s horizons. IEEE Signal Process. Mag. 13(2), 24–49 (1996). https://doi.org/10.1109/79.487040
F. Gafarov, A. Galimyanov, Artificial Neural Networks and Their Applications (Kazan, 2018).
K. Suzuki, Artificial Neural Networks. Methodological Advances and Biomedical Applications (InTech, Rijeka, 2011)
S. Walczak, N. Cerpa, Artificial Neural Networks, 3rd edn. Encyclopedia of Physical Science and Technology (2003)
J.C. Sprott, Elegant Chaos Algebraically Simple Chaotic Flows (World Scientific Publishing Company, Singapore, 2010), 302 p. https://doi.org/10.1142/7183
J.C. Sprott, Chaotic dynamics on large networks. Chaos 18(2), 023135 (2008). https://doi.org/10.1063/1.2945229
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Sadyrbaev, F., Samuilik, I. (2023). From Duffing Equation to Bio-oscillations. In: Pinto, C.M., Ionescu, C.M. (eds) Computational and Mathematical Models in Biology. Nonlinear Systems and Complexity, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-031-42689-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-42689-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-42688-9
Online ISBN: 978-3-031-42689-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)