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## Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems

 Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

* Corresponding author: franckemile2006@yahoo.ca

Received  April 2018 Revised  May 2018 Published  March 2019

Fund Project: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order $\alpha = 1$) and this chaos remains in the fractional case ($\alpha = 0.9$). Moreover the dynamics of the system change depending on the parameter $\alpha$, leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.

Citation: Emile Franc Doungmo Goufo. Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020034
##### References:
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Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301. doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent–Ⅱ, Geophysical Journal International, 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. [8] G. Chen and J. Lü, Dynamics of the Lorenz system family: Analysis, control and synchronization, SciencePress, Beijing. [9] Y. Chen, M. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373. doi: 10.1016/j.jocs.2012.04.008. [10] A. Coronel-Escamilla, J. Gómez-Aguilar, L. Torres and R. Escobar-Jiménez, A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. [11] A. Coronel-Escamilla, J. Gómez-Aguilar, L. Torres, R. Escobar-Jiménez and M. Valtierra-Rodríguez, Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order, Physica A: Statistical Mechanics and its Applications, 487 (2017), 1-21. doi: 10.1016/j.physa.2017.06.008. [12] A. Coronel-Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino and M. M. A. Qurashi, Bateman–feshbach tikochinsky and caldirola–kanai oscillators with new fractional differentiation, Entropy, 19 (2017), 55. doi: 10.3390/e19020055. [13] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. [14] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198. doi: 10.3846/13926292.2016.1145607. [15] E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. [16] E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator–mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187. doi: 10.1016/j.jtbi.2016.05.007. [17] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. [18] E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269. [19] G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. [20] J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13. [21] A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006. [22] Ü. Lepik and H. Hein, Haar Wavelets: With Applications, Springer Science & Business Media, 2014. doi: 10.1007/978-3-319-04295-4. [23] J. Lu, G. Chen, X. Yu and H. Leung, Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE Transactions on Circuits and Systems I: Regular Papers, 51 (2004), 2476-2490. doi: 10.1109/TCSI.2004.838151. [24] J. Lü, F. Han, X. Yu and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 40 (2004), 1677-1687. doi: 10.1016/j.automatica.2004.06.001. [25] K. S. Miller and B. Ross, An Introdution to the Fractional Calculus and Fractional Differential Equations, 1st edition, John Wiley & Sons, Inc, 1993, 1993. [26] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, 1st edition, Academic Press, Inc, 1974. [27] I. Podlubny, The laplace transform method for linear differential equations of the fractional order, Website arXiv.org/pdf/func-an/9710005.pdf, 1997. [28] S. Pooseh, H. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742. doi: 10.1063/1.3636838. [29] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87, Birkhäuser, 2013. [30] J. A. Suykens and J. Vandewalle, Generation of n-double scrolls (n = 1, 2, 3, 4, ...), IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 861-867. doi: 10.1109/81.251829. [31] W. K. Tang, G. Zhong, G. Chen and K. Man, Generation of n-scroll attractors via sine function, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48 (2001), 1369-1372. doi: 10.1109/81.964432. [32] L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. [33] M. ur Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4163-4173. doi: 10.1016/j.cnsns.2011.01.014. [34] M. E. YALÇIN, J. A. Suykens, J. Vandewalle and S. Özoğuz, Families of scroll grid attractors, International Journal of Bifurcation and Chaos, 12 (2002), 23-41. doi: 10.1142/S0218127402004164. [35] C. Zuñiga-Aguilar, J. Gómez-Aguilar, R. Escobar-Jiménez and H. Romero-Ugalde, Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13. [36] C. Zúñiga-Aguilar, H. Romero-Ugalde, J. Gómez-Aguilar, R. Escobar-Jiménez and M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons & Fractals, 103 (2017), 382-403. doi: 10.1016/j.chaos.2017.06.030.

show all references

##### References:
 [1] P. Arena, S. Baglio, L. Fortuna and G. Manganaro, Generation of n-double scrolls via cellular neural networks, International Journal of Circuit Theory and Applications, 24 (1996), 241-252. doi: 10.1002/(SICI)1097-007X(199605/06)24:3<241::AID-CTA912>3.0.CO;2-J. [2] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5. [3] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, Journal of Computational and Applied Mathematics, 225 (2009), 87-95. doi: 10.1016/j.cam.2008.07.003. [4] E. G. Bazhlekova, Subordination principle for fractional evolution equations, Fractional Calculus and Applied Analysis, 3 (2000), 213-230. [5] D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301. doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent–Ⅱ, Geophysical Journal International, 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. [8] G. Chen and J. Lü, Dynamics of the Lorenz system family: Analysis, control and synchronization, SciencePress, Beijing. [9] Y. Chen, M. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373. doi: 10.1016/j.jocs.2012.04.008. [10] A. Coronel-Escamilla, J. Gómez-Aguilar, L. Torres and R. Escobar-Jiménez, A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. [11] A. Coronel-Escamilla, J. Gómez-Aguilar, L. Torres, R. Escobar-Jiménez and M. Valtierra-Rodríguez, Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order, Physica A: Statistical Mechanics and its Applications, 487 (2017), 1-21. doi: 10.1016/j.physa.2017.06.008. [12] A. Coronel-Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino and M. M. A. Qurashi, Bateman–feshbach tikochinsky and caldirola–kanai oscillators with new fractional differentiation, Entropy, 19 (2017), 55. doi: 10.3390/e19020055. [13] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. [14] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198. doi: 10.3846/13926292.2016.1145607. [15] E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. [16] E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator–mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187. doi: 10.1016/j.jtbi.2016.05.007. [17] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. [18] E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269. [19] G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. [20] J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13. [21] A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006. [22] Ü. Lepik and H. Hein, Haar Wavelets: With Applications, Springer Science & Business Media, 2014. doi: 10.1007/978-3-319-04295-4. [23] J. Lu, G. Chen, X. Yu and H. Leung, Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE Transactions on Circuits and Systems I: Regular Papers, 51 (2004), 2476-2490. doi: 10.1109/TCSI.2004.838151. [24] J. Lü, F. Han, X. Yu and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 40 (2004), 1677-1687. doi: 10.1016/j.automatica.2004.06.001. [25] K. S. Miller and B. Ross, An Introdution to the Fractional Calculus and Fractional Differential Equations, 1st edition, John Wiley & Sons, Inc, 1993, 1993. [26] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, 1st edition, Academic Press, Inc, 1974. [27] I. Podlubny, The laplace transform method for linear differential equations of the fractional order, Website arXiv.org/pdf/func-an/9710005.pdf, 1997. [28] S. Pooseh, H. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742. doi: 10.1063/1.3636838. [29] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87, Birkhäuser, 2013. [30] J. A. Suykens and J. Vandewalle, Generation of n-double scrolls (n = 1, 2, 3, 4, ...), IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 861-867. doi: 10.1109/81.251829. [31] W. K. Tang, G. Zhong, G. Chen and K. Man, Generation of n-scroll attractors via sine function, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48 (2001), 1369-1372. doi: 10.1109/81.964432. [32] L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. [33] M. ur Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4163-4173. doi: 10.1016/j.cnsns.2011.01.014. [34] M. E. YALÇIN, J. A. Suykens, J. Vandewalle and S. Özoğuz, Families of scroll grid attractors, International Journal of Bifurcation and Chaos, 12 (2002), 23-41. doi: 10.1142/S0218127402004164. [35] C. Zuñiga-Aguilar, J. Gómez-Aguilar, R. Escobar-Jiménez and H. Romero-Ugalde, Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13. [36] C. Zúñiga-Aguilar, H. Romero-Ugalde, J. Gómez-Aguilar, R. Escobar-Jiménez and M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons & Fractals, 103 (2017), 382-403. doi: 10.1016/j.chaos.2017.06.030.
Representation of the phase portrait of saturated functions. In (a), the one-variable saturated function (4). In (b), the phase portrait of saturated function series (5) with $k = 1, \ \ h = 4$
Plot representing multi-dimensional simulations of the saturated system (1) with $\alpha = 1$ (conventional case). In (a), projection of a one-dimensional saturated chaotic attractor with six scrolls. In (b), a projection of a two-dimensional saturated chaotic attractor with a grid of $6\times6$ scrolls. In (c), a projection of a three-dimensional saturated chaotic attractor with a grid of $6\times6\times6$ scrolls
Plot representing multi-dimensional simulations of the saturated system (1) with $\alpha = 0.9$ (fractional case). The same dynamics as Fig. 2} are shown with a one-dimensional chaotic attractor with six scrolls in (a), two-dimensional saturated chaotic attractor with a grid of $6\times6$ scrolls in (b) and a three-dimensional saturated chaotic attractor with a grid of $6\times6\times6$ scrolls in (c)
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