Meriem Mansouria BELHAMITI, Zoubir DAHMANİ, Mehmet Zeki SARIKAYA

Two fractional order Langevin equation with new chaotic dynamics

In the present paper, we introduce a two-order nonlinear fractional sequential Langevin equation using the derivatives of Atangana-Baleanu and Caputo-Fabrizio. The existence of solutions is proven using a fixed point theorem under a weak topology, and an illustrative example is then given. Furthermore, we present new fractional versions of the Adams-Bashforth three-step approach for the Atangana-Baleanu and Caputo derivatives. New nonlinear chaotic dynamics are performed by numerical simulations.

Keywords:

Caputo, Caputo-Fabrizio derivative Atangana–Baleanu derivative, fixed-point theory, three-step Adams-Bashforth scheme,

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Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl., 13(2) (2012), 599-606. https://doi.org/10.1016/j.nonrwa.2011.07.052.
Almeida, R., Bastos, R.O., Teresa, M., Monteiro, T., Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39(16) (2015). https://doi.org/10.1002/mma.3818.
Atangana, A., Baleanu, D., Caputo - Fabrizio derivative applied to groundwater flow within a confined aquifer, J. Eng. Mech., (2016). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001091.
Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel, Thermal Science, 20 (2016), 763-769. https://doi.org/10.2298/TSCI160111018A.
Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction diffusion equation, Appl. Math. Comput., 273 (2015), 948-56. https://doi.org/10.1016/j.amc.2015.10.021.
Atangana, A., Alqahtani, R.T., Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Diff. Equ., 2016(1) (2016), 1-13. https://doi.org/10.1186/s13662-016-0871-x.
Atangana, A., Koca, B.I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454. https://doi.org/10.1016/j.chaos.2016.02.012
Belhamiti, M.M., Dahmani, Z., Agarwal, P., Chaotic Jerk Circuit: existence and stability of solutions for a fractional model, Progr. Fract. Differ. Appl., Accepted (2022).
Ben Amar, A., O’Regan, D., Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications, Springer International Publishing Switzerland, 2016. https://doi.org/10.1007/978-3-319-31948-3.
Bartuccelli, M.V., Gentile, G., Georgiou, K.V., On the dynamics of a vertically driven damped planar pendulum, The Royal Society, Physical and Engineering Sciences, 457 (2001), 3007-3022. https://doi.org/10.1098/rspa.2001.0841.
Bezziou, M., Dahmani, Z., Jebril, I., Belhamiti, M.M., Solvability for a differential system of Duffing type via Caputo-Hadamard approach, Appl. Math. Inf. Sci., 16(2) (2022), 341-352.
Caputo, M., Fabrizio, M., Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2(1) (2016), 1-11. https://doi.org/10.18576/pfda/020101.
Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2) (2015), 73-85.
Cao, H., Seoane, J.M., Sanju´an, M.A.F., Symmetry-breaking analysis for the general Helmholtz Duffing oscillator, Chaos, Solitons and Fractals, 34 (2007), 197-212.
Chen, A., Chen, Y., Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions, Boundary Value Problems, 3 (2011). https://doi.org/10.1155/2011/516481.
Chen, X., Fu, X., Chaos control in a special pendulum system for ultra-subharmonic resonance, American Institute of Mathematical Sciences, February, 26(2) (2021), 847-860. https://doi.org/10.3934/dcdsb.2020144.
Dahmani, Z., Belhamiti, M.M., Sarıkaya, M.Z., A three fractional order jerk equation with anti periodic conditions, Submitted paper, (2020).
Gouari, Y., Dahmani, Z. , Belhamiti, M.M., Sarıkaya, M.Z., Uniqueness of solutions, stability and simulations for a differential problem involving convergent series and time variable singularities, Rocky Mountain Journal of Mathematics, (2021). https://doi.org/10.22541/au.163673427.78470853/v1.
Hirsch, M.W., Smale, S., Devaney, R.L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier, USA, 2004.
Jeribi, A., Krichen, B., Nonlinear Functional Analysis in Banach Spaces and Banach Algebras Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications, Taylor & Francis Group, LLC., 2016.
Jeribi, A., Hammami, M.A., Masmoudi, A., Applied mathematics in Tunisia, International Conference on Advances in Applied Mathematics (ICAAM), Hammamet, Tunisia, (2013).
Kumar, S., Rashidi, M.M., New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun., 185(7) (2014), 1947-54. https://doi.org/10.1016/j.cpc.2014.03.025.
Losada, J., Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 2 (2015), 87-92.
Kpomahou, Y.J.F., Hinvi, L.A., Ad´echinan, J.A., Miwadinou, C.H., The mixed Rayleigh Lienard oscillator driven by parametric periodic pamping and external excitation, Hindawi Complexity, (2021). https://doi.org/10.1155/2021/6631094.
Owolabi, K.M., Analysis and Simulation of Herd Behaviour Dynamics Based on Derivative with Nonlocal and Nonsingular Kernel, Elsevier, 2021. https://doi.org/10.1016/j.rinp.2021.103941.
Mainardi, F., Why the Mittag-Leffler function can be considered the queen function of the fractional calculus?, Entropy, 22(12) (2020), 1359. https://doi.org/10.3390/e22121359.
Owolabi, K.M., Atangana, A., Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111–119.
Owolabi, K.M., Atangana, A., On the formulation of fractional Adams-Bashforth method with Atangana-Baleanu-Caputo derivative to model chaotic problems, (2021). https://doi.org/10.1063/1.5085490.
Peters, R.D., Chaotic pendulum based on torsion and gravity in opposition, American Journal of Physics, 63 (1995), 1128. https://doi.org/10.1119/1.18019.
Rahayu, S.U., Tamba, T., Tarigan, K., Investigation of chaos behaviour on damped and driven nonlinear simple pendulum motion simulated by mathematica, Journal of Physics Conference Series, 1811(1) 012014 (2021). https://doi.org/10.1088/1742-6596/1811/1/012014.
Salema, A., Alzahrania, F., Almaghamsia, L., Langevin equation involving one fractional order with three point boundary conditions, Nonlinear Sci. Appl., 12 (2019), 791-798. https://doi.org/10.22436/jnsa.012.12.02.
Singh, H., Kumar, D., Baleanu, D., Methods of Mathematical Modelling, Mathematics and Its Applications: Modelling, Engineering, and Social Sciences, Taylor & Francis Group, 2019. https://doi.org/10.1201/9780429274114.
Atanackovic , M.T., Pilipovic , S., Stankovic , B., Zorica, D., Fractional Calculus with Applications in Mechanics, John Wiley & Sons, 2014. https://doi.org/10.1002/9781118577530.
Tablennehas, K., Dahmani, Z., Belhamiti, M.M., Abdelnebi, A., Sarıkaya, M.Z., On a fractional problem of Lane-Emden type: Ulam type stabilities and numerical behaviors, Advances in Difference Equations, (2021).
Tarasov, V., No nonlocality no fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 62 (2018). https://doi.org/10.1016/j.cnsns.2018.02.019.
Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005.