Geodesic
Chaos control and synchronization of a new chaotic financial system with integer and fractional order
Dousseh, P. Y. 1 ; Ainamon, C. 1 ; Miwadinou, C. H. 2 ; Monwanou, A. V. 1 ; Chabi-Orou, J. B. 1
1 Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin
2 Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin;Departement de Physique, ENS-Natitingou, Universite Nationale des Sciences, Technologies, Ingenierie et Mathematiques (UNSTIM), Abomey, Benin
Journal of nonlinear sciences and its applications, Tome 14 (2021) no. 6, p. 372-389.

Voir la notice de l'article provenant de la source International Scientific Research Publications

Synchronization of chaotic dynamical systems with fractional order is receiving great attention in recent literature because of its applications in a variety of fields including optics, secure communications of analog and digital signals, and cryptographic systems. In this paper, chaos control of a new financial system, and chaos synchronization between two identical financial systems, and non-identical financial systems with integer and fractional order are investigated. Chaos control is based on a linear feedback controller for stabilizing chaos to unstable equilibrium. In addition, chaos synchronization, not only between two identical new chaotic financial systems, but also between the new financial system and an another financial system given in the literature is realized by using active control technique. The synchronization is done for integer and fractional order in each case. It is shown that chaotic behavior can be controlled easily to any unstable equilibrium point of the new financial system. Also, it is observed that synchronization is enhanced when the fractional order increases and approximates to one. Numerical simulations are used to verify the proposed methods.
DOI : 10.22436/jnsa.014.06.01
Classification : 34H10, 65P20, 34D06, 26A33
Keywords: Chaos control, synchronization, chaotic financial systems, fractional order systems, active control, feedback controller

Dousseh, P. Y.  1 ; Ainamon, C.  1 ; Miwadinou, C. H.  2 ; Monwanou, A. V.  1 ; Chabi-Orou, J. B.  1

1 Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin
2 Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin;Departement de Physique, ENS-Natitingou, Universite Nationale des Sciences, Technologies, Ingenierie et Mathematiques (UNSTIM), Abomey, Benin 
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     title = {Chaos control and synchronization of a new chaotic financial system  with integer  and  fractional order},
     journal = {Journal of nonlinear sciences and its applications},
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Dousseh, P. Y. ; Ainamon, C. ; Miwadinou, C. H. ; Monwanou, A. V. ; Chabi-Orou, J. B. . Chaos control and synchronization of a new chaotic financial system  with integer  and  fractional order. Journal of nonlinear sciences and its applications, Tome 14 (2021) no. 6, p. 372-389. doi : 10.22436/jnsa.014.06.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.06.01/

[1] Abd-Elouahab, M. S.; Hamri, N.-E.; Wang, J. Chaos control of a fractional-order financial system, Math. Probl. Eng.,, Volume 2010 (2010), pp. 1-18 | DOI

[2] Agiza, H. N.; Yassen, M. T. Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys. Lett. A, Volume 278 (2001), pp. 191-197 | DOI

[3] Ahmad, H.; Akgul, A.; Khan, T. A.; Stanimirovic, P. S.; Chu, Y.-M. New perspective on the conventional solutions of the nonlinear time-fractional partial differential equations, Complexity, Volume 2020 (2020), pp. 1-10 | DOI

[4] Ahmad, H.; Khan, T. A.; Ahmad, I.; Stanimirovic, P. S.; Chu, Y.-M. A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations, Results Phys., Volume 19 (2020), pp. 1-8 | DOI

[5] Bai, E.-W.; Lonngren, K. E. Synchronization of two Lorenz systems using active control, Chaos Solitons Fractals, Volume 8 (1997), pp. 51-58 | DOI

[6] Bhalekar, S.; Daftardar-Gejji, V. Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul., Volume 15 (2010), pp. 3536-3546 | DOI

[7] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S. The synchronization of chaotic systems, Phys. Rep., Volume 366 (2002), pp. 1-101 | DOI

[8] Chen, W.-C. Nonlinear dynamics and chaos in a fractional-order financial system, Chaos Solitons Fractals, Volume 36 (2008), pp. 1305-1314 | DOI

[9] Chen, L.; Chai, Y.; Wu, R. Control and synchronization of fractional-order financial system based on linear control, Discrete Dyn. Nat. Soc., Volume 2011 (2011), pp. 1-21 | DOI

[10] Daftardar-Gejji, V.; Bhalekar, S. Chaos in fractional ordered Liu system, Comput. Math. Appl., Volume 59 (2010), pp. 1117-1127 | DOI

[11] David, S. A.; Machado, J. A. T.; Quintino, D. D.; Balthazar, J. M. Partial chaos suppression in a fractional order macroeconomic model, Math. Comput. Simulation, Volume 122 (2016), pp. 55-68 | DOI

[12] Diethelm, K.; Ford, N. J.; Freed, A. D. A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., Volume 29 (2002), pp. 3-22 | DOI

[13] Fradkov, A. L.; Evans, R. J. Control of chaos: method and applications in engineering, Annu. Rev. Control, Volume 29 (2005), pp. 33-56 | DOI

[14] Hajipour, A.; Tavakoli, H. Dynamic analysis and adaptive sliding mode controller for a chaotic fractional incommensurate order financial system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 27 (2017), pp. 1-14 | DOI

[15] He, R.; Vaidya, P. G. Implementation of chaotic cryptography with chaotic synchronization, Phys. Rev. E, Volume 57 (1998), pp. 1532-1535

[16] Hilfer, R. Applications of fractional calculus in physics, World Scientific, USA, 2000

[17] Hołyst, J. A.; Urbanowicz, K. Chaos control in economical model by time-delayed feedback method, Phys. A: Stat. Mech. Appl., Volume 287 (2000), pp. 587-598 | DOI

[18] Huang, L.; Feng, R.; Wang, M. Synchronization of chaotic systems via nonlinear control, Phys. Lett. A, Volume 320 (2004), pp. 271-275 | DOI

[19] Jia, Q. Chaos control and synchronization of the Newton-Leipnik chaotic system, Chaos Solitons Fractals, Volume 35 (2008), pp. 814-824 | DOI

[20] Kaplan, J. L.; Yorke, J. A. Preturbulence: a regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., Volume 67 (1979), pp. 93-108

[21] Li, C.; Peng, G. Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals, Volume 22 (2004), pp. 443-450 | DOI

[22] Liao, T.-L. Adaptive synchronization of two Lorenz systems, Chaos Solitons Fractals, Volume 9 (1998), pp. 1555-1561 | DOI

[23] Liao, Y.; Zhou, Y.; Xu, F.; Shu, X.-B. A Study on the Complexity of a New Chaotic Financial System, Complexity, Volume 2020 (2020), pp. 1-5 | DOI

[24] Lorenz, E. N. Deterministic nonperiodic flow, J. Atmospheric Sci., Volume 20 (1963), pp. 130-141 | DOI

[25] Lorenz, H.-W. Nonlinear Dynamical Economics and Chaotic Motion, Springer, Berlin, 1993 | DOI

[26] Lu, J. G. Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Solitons Fractals, Volume 26 (2005), pp. 1125-1133 | DOI

[27] Ma, J. H.; Chen, Y. S. Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. I, Appl. Math. Mech., Volume 22 (2001), pp. 1119-1128 | DOI

[28] Ma, J. H.; Chen, Y. S. Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. II, Appl. Math. Mech.,, Volume 22 (2001), pp. 1375-1382 | DOI

[29] Ma, C.; Wang, X. Hopf bifurcation and topological horseshoe of a novel finance chaotic system, Commun. Nonlinear Sci. Numer. Simul., Volume 17 (2012), p. 721-130 | DOI

[30] Matignon, D. Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl.,, Volume 2 (1996), pp. 963-968

[31] Miwadinou, C. H.; Hinvi, L. A.; Monwanou, A. V.; Orou, J. B. Chabi Nonlinear dynamics of a φ6 - modified Duffing oscillator: resonant oscillations and transition to chaos, Nonlinear Dyn.,, Volume 88 (2017), pp. 97-113 | DOI

[32] Ott, E.; Grebogi, C.; Yorke, J. A.; Controlling chaos, Phys. Rev. Lett., Volume 64 (1990), pp. 1196-1199 | DOI

[33] Park, J. H. Chaos synchronization of a chaotic system via nonlinear control, Chaos Solitons Fractals, Volume 25 (2005), pp. 579-584 | DOI

[34] Podlubny, I. Fractional differential equations, Academic Press, San Diego, 1999

[35] Saif, M.; Khan, F.; Nisar, K. S.; Araci, S. Modified Laplace transform and its properties, J. Math. Comput. Sci., Volume 21 (2020), pp. 127-135

[36] Strotz, R. H.; McAnulty, J. C.; Naines, J. B. Goodwin’s nonlinear theory of the business cycle: an electro-analog solution, Econometrica, Volume 390--411 (1953) | DOI

[37] Suthar, D. L.; Purohit, S. D.; Araci, S. Solution of fractional kinetic equations associated with the (p, q)-Mathieu-type series, Discrete Dyn. Nat. Soc., Volume 2020 (2020), pp. 1-7 | DOI

[38] Tavazoei, M. S.; Haeri, M. Chaotic attractors in incommensurate fractional order systems, Phys. D, Volume 237 (2008), pp. 2628-2637 | DOI

[39] Wang, X.-Y.; He, Y.-J.; Wang, M.-J. Chaos control of a fractional modified coupled dynamos system, Nonlinear Anal., Volume 71 (2009), pp. 6126-6134 | DOI

[40] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A. Determining Lyapunov exponents from a time series, Phys. D, Volume 16 (1985), pp. 285-317 | DOI

[41] Yassen, M. T. Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput., Volume 135 (2003), pp. 113-128 | DOI

[42] Yassen, M. T. Chaos control of chaotic dynamical system using backstepping design, Chaos Solitons Fractals, Volume 27 (2006), pp. 537-548 | DOI

[43] Yu, H.; Cai, G.; Li, Y. Dynamic analysis and control of a new hyperchaotic finance system, Nonlinear Dynam., Volume 67 (2012), pp. 2171-2182 | DOI

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