Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviors have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise differentiation and integration, this approach helps to capture processes with crossover effects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for different cases.
Citation: Abdon ATANGANA, Seda İĞRET ARAZ. Deterministic-Stochastic modeling: A new direction in modeling real world problems with crossover effect[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3526-3563. doi: 10.3934/mbe.2022163
Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviors have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise differentiation and integration, this approach helps to capture processes with crossover effects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for different cases.
| [1] |
A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.98/TSCI160111018A doi: 10.98/TSCI160111018A
|
| [2] |
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
|
| [3] |
A. Atangana, Extension of rate of change concept: From local to nonlocal operators with applications, Results Phys., 19 (2021), 103515. https://doi.org/10.1016/j.rinp.2020.103515 doi: 10.1016/j.rinp.2020.103515
|
| [4] |
A. Atangana, S. Igret Araz, New concept in calculus: Piecewise differential and integral operators, Chaos Solit. Fract., 145 (2021), 110638. https://doi.org/10.1016/j.chaos.2020.110638 doi: 10.1016/j.chaos.2020.110638
|
| [5] |
M. Ababneh A new four-dimensional chaotic attractor Ain Shams Eng. J., 9 (2018), 1849–1854. https://doi.org/10.1016/j.asej.2016.08.020 doi: 10.1016/j.asej.2016.08.020
|
| [6] |
A. Atangana, J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Solit. Fract., 114 (2018), 516–535. https://doi.org/10.1016/j.chaos.2018.07.033 doi: 10.1016/j.chaos.2018.07.033
|
| [7] |
D. Faranda, Y. Sato, B. Saint-Michel, C. Wiertel, V. Padilla, B. Dubrulle, et.al., Stochastic chaos in a turbulent swirling flow, Phys. Rev. Lett., 119 (2017), 014502. https://doi.org/10.1103/PhysRevLett.119.014502 doi: 10.1103/PhysRevLett.119.014502
|
| [8] |
E. K. Kosmidis, K. Pakdaman, Stochastic Chaos in a neuronal model, Int. J. Bifurcat. Chaos, 16 (2006), 395–410. https://doi.org/10.1142/S0218127406014873 doi: 10.1142/S0218127406014873
|
| [9] |
C. C. Canavier, P. D. Shepard, Chaotic versus stochastic dynamics: A critical look at the evidence for nonlinear sequence dependent structure in dopamine neurons, J. Neural Transm. Suppl., 73 (2009), 121–128. https://doi.org/10.1007/978-3-211-92660-4 doi: 10.1007/978-3-211-92660-4
|
| [10] | D. Toker, F. T. Sommer, M. D'Esposito, A simple method for detecting chaos in nature, Commun. Biol., 3 (2020). https://doi.org/10.1038/s42003-019-0715-9 |
| [11] |
W. Cunli, L. Youming, F. Tong, Stochastic chaos in a Duffing oscillator and its control, Chaos Solit. Fract., 27 (2006), 459–469. https://doi.org/10.1016/j.chaos.2005.04.035 doi: 10.1016/j.chaos.2005.04.035
|
| [12] | A. Atangana, S. Igret Araz, New numerical scheme with Newton polynomial: Theory, Methods and Applications, Academic Press, (2021). Available from: https://www.sciencedirect.com/book/9780323854481/new-numerical-scheme-with-newton-polynomial |
| [13] |
J. L. Hindmarsh, R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Royal Soc. B. P. Roy. Soc. B. Biol. Sci. 221, (1984), 87–102. https://doi.org/10.1098/rspb.1984.0024 doi: 10.1098/rspb.1984.0024
|
| [14] |
Han Chunyan, Yu Simin, Wang Guangyi, A Sinusoidally Driven Lorenz System and Circuit Implementation, Math. Probl. Eng., 2015 (2015), Article ID 706902. https://doi.org/10.1155/2015/706902 doi: 10.1155/2015/706902
|
| [15] | S. Bouali, A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited, 24th ABCM International Congress of Mechanical Engineering, (2013). |
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Abdon ATANGANA, Seda İĞRET ARAZ. Deterministic-Stochastic modeling: A new direction in modeling real world problems with crossover effect[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3526-3563. doi: 10.3934/mbe.2022163
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