Exploring vibrational resonance in biophysical systems with fractional-order damping

Authors

  • T. O. Roy-Layinde
    Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Ogun State, Nigeria
    https://orcid.org/0000-0001-5285-7231
  • K. O. Olonade
    Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Ogun State, Nigeria | Department of Science Laboratory Technology, Moshood Abiola Polytechnic, Ojere, Abeokuta, Nigeria.
  • K. A. Omoteso
    School of Computing and Engineering, University of Derby, Markeaton Street, Derby, United Kingdom
    https://orcid.org/0000-0002-3753-1425
  • H. T. Oladunjoye
    Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Ogun State, Nigeria
    https://orcid.org/0000-0003-2897-5849
  • B. A. Oyero
    Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Ogun State, Nigeria
  • J. A. Laoye
    Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Ogun State, Nigeria
    https://orcid.org/0000-0003-0633-4995

Keywords:

Vibrational Resonance, Biological System, Enzyme, Brain Wave, Differential Equation

Abstract

Vibrational Resonance (VR), which is characterised by the enhancement of the maximum response of a weakly driven system’s output signal induced by a high-frequency (HF) periodic signal, was numerically examined in a bi-harmonically driven dimensionless model of an enzymesubstrate reaction with fractional-order damping, which models coherent oscillations in brain waves. The model incorporates a damping force that depends on a non-integer (fractional) order derivative rather than the typical first-order derivative in classical damping models. The output response was obtained by solving the model numerically using Grunwald-Letnikov’s fractional derivatives definition. The response amplitude,¨ computed from the Fourier spectrum of the output signal, was used to characterise VR. The effect of the fractional-order damping coefficient on the observed VR was considered for different damping strength coefficients. Single-peak resonances were observed. The fractional-order damping modulated the observed VR in a manner similar to the damping strength in an integer-order system, by reducing the high-frequency signal amplitude at which VR occurs. Increased brain wave activity from enzyme-substrate reaction may be due to inherent energy transfers from changes in the rate of decay, hence significant behavioural changes in brain wave activity could be linked to inherent changes in the decay rate of the excited enzymes, even when there is no change in the number of enzyme-substrate carriers. This study reveals the potential of fractional-order damping for enhancing biophysical system modelling, with implications for understanding brain wave activities.

Dimensions

[1] H. Frohlich, “Bose condensation of strongly excited longitudinal elec-¨ tric modes”, Physics Letters A 26 (1968) 402. https://doi.org/10.1016/0375-9601(68)90242-9.

[2] H. Frohlich,¨ The extraordinary dielectric properties of biological materials and the action of enzymes, Proceedings of the National Academy of Sciences, 1975, pp. 4211–4215. https://doi.org/10.1073/pnas.72.11.4211.

[3] H. Frohlich, “Theoretical physics and biology” in¨ Biological coherence and response to external stimuli, H. Frohlich (Ed.), Springer, Berlin, Hei-¨ delberg, 1988, pp. 1–24. https://doi.org/10.1007/978-3-642-73309-3 1.

[4] U. E. Vincent, & O. Kolebaje, “Introduction to the dynamics of driven nonlinear systems”, Contemporary Physics 61 (2020) 169. https://doi.org/10.1080/00107514.2020.1850003.

[5] S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, CRC Press, Boca Raton, United States, 2018. https://doi.org/10.1201/9780429492563.

[6] E. Ott, Chaos in dynamical systems, Cambridge University Press, Cambridge, United Kingdom, 2002. https://doi.org/10.1017/CBO9780511803260.

[7] T. O. Roy-Layinde, K. A. Omoteso, D. O. Ogooluwa, H. T. Oladunjoye & J. A. Laoye, “Chaotic and periodic behavior in a fractionalorder biological system”, Acta Physica Polonica B 51 51 (2020) 1885. https://doi.org/10.5506/APhysPolB.51.1885.

[8] I. Petra´s,? Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer-Verlag, Berlin, Germany, 2011. https://doi.org/10.1007/978-3-642-18101-6.

[9] J. T. Machado, V. Kiryakova & F. Mainardi, “Recent history of fractional calculu”, Communications in nonlinear science and numerical simulation 16 (2011) 1140. https://doi.org/10.1016/j.cnsns.2010.05.027.

[10] S. K. Guin, A. S. Ambolikar, S. Das & A. K. Poswal, “Advantage of fractional calculus based Hybrid-Theoretical–Computational–Experimental approach for alternating current voltammetry”, Electroanalysis 32 (2020) 1629. https://doi.org/10.1002/elan.201900552.

[11] A. P. Singh & K. Bingi, “Applications of fractional-order calculus in robotics”, Fractal and Fractional 8 (2024) 403. https://doi.org/10.3390/fractalfract8070403.

[12] L. C. Vieira, R. S. Costa & D. Vlerio, “An overview of mathematical mod-´ elling in cancer research: fractional calculus as modelling tool”, Fractal and Fractional 7 (2023) 595. https://doi.org/10.3390/fractalfract7080595.

[13] E. H. Robles, F. J. Torres, A. J. Balvant´?n-Garc´?a, I. Mart´?nez-Ram´?rez, G. Capilla, & J. P. Ram´?rez-Paredes, “Fractional calculus applied to the generalized model and control of an electrohydraulic system”, Fractal and Fractional 8 (2024) 679. https://doi.org/10.3390/fractalfract8120679.

[14] A. O. Yunus & M. O. Olayiwola, “The analysis of a novel COVID19 model with the fractional-order incorporating the impact of the vaccination campaign in Nigeria via the Laplace-Adomian Decomposition Method”, Journal of the Nigerian Society of Physical Sciences 6 (2024) 1830. https://doi.org/10.46481/jnsps.2024.1830.

[15] P. L. Butzer & U. Westphal, “An introduction to fractional calculus”, in Applications of fractional calculus in physics, World Scientific, Danvers, United States, 2000, pp. 1–85. https://doi.org/10.1142/97898128177470001.

[16] H. Khan, R. Shah, J. F. Gomez-Aguilar, D. Baleanu & P. Kumam, “Trav-´ elling waves solution for fractional-order biological population model”, Mathematical Modelling of Natural Phenomena 16 (2021) 32. https: //doi.org/10.1051/mmnp/2021016.

[17] K. S. Cole, “Electric conductance of biological systems”, Cold Spring Harbor symposia on quantitative biology 1 (1933) 107. https://symposium.cshlp.org/content/1/107.short.

[18] X. Liu & H. Fang, “Periodic pulse control of hopf bifurcation in a fractional-order delay predator-prey model incorporating a prey refuge”, Advances in Difference Equations 2019 (2019) 479. https://doi.org/10.1186/s13662-019-2413-9.

[19] K. S. Nisar, M. Farman, M. Abdel-Aty & C. Ravichandran, “A review of fractional order epidemic models for life sciences problems: Past, present and future”, Alexandria Engineering Journal 95 (2024) 283. https://doi.org/10.1016/j.aej.2024.03.059.

[20] F. Mansal & N. Sene, “Analysis of fractional fishery model with reserve area in the context of time-fractional order derivative”, Chaos, Solitons & Fractals 140 (2020) 110200. https://doi.org/10.1016/j.chaos.2020.110200.

[21] M. A. Khan, Z. Hammouch & D. Baleanu, “Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative”, Mathematical Modelling of Natural Phenomena 14 (2019) 311. https://doi.org/10.1051/mmnp/2018074.

[22] S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch & D. Baleanu, “A fractional model for the dynamics of tuberculosis infection using CaputoFabrizio derivative”, Discrete and Continuous Dynamical Systems - Series S 13 (2020) 975. http://hdl.handle.net/20.500.12416/3489.

[23] M. S. Tavazoei & M. Haeri, “Chaotic attractors in incommensurate fractional order systems”, Physica D: Nonlinear Phenomena 237 (2008) 2628. https://doi.org/10.1016/j.physd.2008.03.037.

[24] A. E. Matouk, “Chaotic attractors that exist only in fractional-order case”, Journal of Advanced Research 45 (2023) 183. https://doi.org/10.1016/j.jare.2022.03.008.

[25] M. F. Danca, “Hidden chaotic attractors in fractional-order systems”, Nonlinear Dynamics 89 (2017) 577. https://doi.org/10.1007/s11071-017-3472-7.

[26] T. Liu, H. Yan, S. Banerjee & J. Mou, “A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation”, Chaos, Solitons & Fractals 145 (2021) 110791. https://doi.org/10.1016/j.chaos.2021.110791.

[27] J. Yang, S. Rajasekar & M. A. Sanjuan, “Vibrational resonance: A review”, Physics Reports 1067 (2024) 1. https://doi.org/10.1016/j.physrep. 2024.03.001.

[28] P. Landa & P. V. McClintock, “Vibrational resonance”, Journal of Physics A: Mathematical and general 33 (2000) L433. https://doi.org/10.1088/0305-4470/33/45/103.

[29] S. Rajasekar & M. A. Sanjuan, Nonlinear resonances, Springer International Publishing, Switzerland, 2016. https://doi.org/10.1007/978-3-319-24886-8.

[30] U. E. Vincent, P. V. E. McClintock, I. A. Khovanov & S. Rajasekar, “Vibrational and stochastic resonances in driven nonlinear systems”, Philosophical Transactions of the Royal Society A 379 (2021) 20200226. https://doi.org/10.1098/rsta.2020.0226.

[31] T. O. Roy-Layinde, U. E. Vincent, S. A. Abolade, O. O. Popoola, J. A. Laoye & P. V. E. McClintock, “Vibrational resonances in driven oscillators with position-dependent mass”, Philosophical Transactions of the Royal Society A 379 (2021) 20200227. https://doi.org/10.1098/rsta.2020.0227.

[32] T. O. Roy-Layinde, J. A. Laoye, O. O. Popoola & U. E. Vincent, “Analysis of vibrational resonance in bi-harmonically driven plasma”, Chaos: An Interdisciplinary Journal of Nonlinear Science 26 (2016) 093117. https://doi.org/10.1063/1.4962403.

[33] T. O. Roy-Layinde, K. A. Omoteso, B. A. Oyero, J. A. Laoye & U. E. Vincent, “Vibrational resonance of ammonia molecule with doubly singular position-dependent mass”, The European Physical Journal B 95 (2022) 80. https://doi.org/10.1140/epjb/s10051-022-00342-9.

[34] T. O. Roy-Layinde, K. A. Omoteso, U. H. Diala, J. A. Runsewe & J. A. Laoye, “Analysis of vibrational resonance in an oscillator with exponential mass variation”, Chaos, Solitons & Fractals 178 (2024) 114310. https://doi.org/10.1016/j.chaos.2023.114310.

[35] K. A. Omoteso, T. O. Roy-Layinde, J. A. Laoye, U. E. Vincent & P. V. McClintock, “Delay-induced vibrational resonance in the Rayleigh–Plesset bubble oscillator”, Journal of Physics A: Mathematical and Theoretical 55 (2022) 495701. https://doi.org/10.1088/1751-8121/aca7e3.

[36] K. A. Omoteso, O. Ozioko, O. Bagdasar, T. O. Roy-Layinde & U. H. Diala, “Numerical analyses of acoustic vibrational resonance in a Helmholtz resonator”, Nonlinear Dynamics (2024) 1. https://doi.org/10.1007/s11071-024-10534-w.

[37] K. A. Omoteso, T. O. Roy-Layinde & U. H. Diala, “Performance boost of an electromagnetic energy harvester using vibrational resonance”, International Journal of Non-Linear Mechanics 170 (2025) 104989. https: //doi.org/10.1016/j.ijnonlinmec.2024.104989.

[38] P. O. Adesina, U. E. Vincent, T. O. Roy-Layinde, O. T. Kolebaje & P. V. E. McClintock, “Crisis-induced vibrational resonance in a phase-modulated periodic structure”, Physical Review E 110 (2024) 034215. https://doi.org/10.1103/PhysRevE.110.034215.

[39] J. Yang & H. Zhu, “Vibrational resonance in duffing systems with fractional-order damping”, Chaos: An Interdisciplinary Journal of Nonlinear Science 22 (2012) 013112. https://doi.org/10.1063/1.3678788.

[40] J. Yang, M. A. Sanjuan & H. Liu, “Enhancing the weak signal with ar-´ bitrary high-frequency by vibrational resonance in fractional-order duffing oscillators”, Journal of Computational and Nonlinear Dynamics 12 (2017) 051011. https://doi.org/10.1115/1.4036479.

[41] T. L. M. Djomo Mbong, M. S. Siewe & C. Tchawoua, “The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator”, Mechanics Research Communications 78 (2016) 13. https://doi.org/10.1016/j.mechrescom.2016.10.004.

[42] W. Guo & L. Ning, “Vibrational resonance in a fractional order quintic oscillator system with time delay feedback”, International Journal of Bifurcation and Chaos 30 (2020) 2050025. https://doi.org/10.1142/S021812742050025X.

[43] T. Qin, T. Xie, M. Luo & K. Deng, “Vibrational resonance in fractionalorder overdamped multistable systems”, Chinese Journal of Physics 55 (2017) 546. https://doi.org/10.1016/j.cjph.2016.11.005.

[44] J. Wang, R. Zhang & J. Liu, “Vibrational resonance analysis in a fractional order Toda oscillator model with asymmetric potential”, International Journal of Non-Linear Mechanics 148 (2023) 104258. https://doi.org/10.1016/j.ijnonlinmec.2022.104258.

[45] J. W. Mao & D. L. Hu, “Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System”, Mathematics 10 (2021) 87. https://doi.org/10.3390/math10010087.

[46] P. Fu, C. J. Wang, K. L. Yang, X. B. Li & B. Yu, “Reentrance-like vibrational resonance in a fractional-order birhythmic biological system”, Chaos, Solitons & Fractals 155 (2022) 111649. https://doi.org/10.1016/j.chaos.2021.111649.

[47] F. Kaiser, “Specific effects in externally driven self-sustained oscillating biophysical model systems”, in Coherent excitations in biological systems, Springer, Berlin Heidelberg, 1983, pp. 128–133. https://doi.org/10.1007/978-3-642-69186-713.

[48] C. Meyers, “Quantum coherent-like state observed in a biological protein for the first time”, American Institute of Physics, 2015. https://www.aip.org/publishing/journal-highlights/ quantum-coherent-state-observed-biological-protein-first-time.

[49] H. E. Kadji, J. C. Orou, R. Yamapi & P. Woafo, “Nonlinear dynamics and strange attractors in the biological system ”, Chaos, Solitons And Fractals 32 (2007) 862. https://doi.org/10.1016/j.chaos.2005.11.063.

[50] I. Petras,´ “Fractional derivatives, fractional integrals, and fractional differential equations in Matlab”, in A.H. Assi (Ed.) Engineering Education and Research Using MATLAB, IntechOpen, London, UK, 2011. https://dx.doi.org/10.5772/19412.

[51] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Elsevier, 1999. https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9.

[52] K. O. Olonade, T. O. Roy-Layinde, B. A. Oyero, K. A. Omoteso, R. K. Odunaike & J. A. Laoye, “Analysis of vibrational resonance in a Biophysical System”, Nigerian Journal of Physics 33 (2024) 75. https://doi.org/10.62292/njp.v33i3.2024.268.

Published

2025-08-01

How to Cite

Exploring vibrational resonance in biophysical systems with fractional-order damping. (2025). Journal of the Nigerian Society of Physical Sciences, 7(3), 2594. https://doi.org/10.46481/jnsps.2025.2594

Issue

Volume 7, Issue 3, August 2025

Section

Physics & Astronomy
Copyright & Licensing

Copyright (c) 2025 Kehinde A. Omoteso, Taiwo O. Roy-Layinde, Kayode O. Olonade, Hamid T. Oladunjoye, Babatunde A. Oyero, John A. Laoye

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Exploring vibrational resonance in biophysical systems with fractional-order damping. (2025). Journal of the Nigerian Society of Physical Sciences, 7(3), 2594. https://doi.org/10.46481/jnsps.2025.2594

Similar Articles

11-20 of 185

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)