Chaos control and synchronization of a new chaotic financial system with integer and fractional order
Volume 14, Issue 6, pp 372--389
http://dx.doi.org/10.22436/jnsa.014.06.01
Publication Date: April 25, 2021
Submission Date: November 17, 2020
Revision Date: January 03, 2021
Accteptance Date: March 18, 2021
-
1291
Downloads
-
2788
Views
Authors
P. Y. Dousseh
- 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.
C. Ainamon
- 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.
C. H. Miwadinou
- 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.
A. V. Monwanou
- 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.
J. B. Chabi-Orou
- 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.
Abstract
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.
Share and Cite
ISRP Style
P. Y. Dousseh, C. Ainamon, C. H. Miwadinou, A. V. Monwanou, J. B. Chabi-Orou, Chaos control and synchronization of a new chaotic financial system with integer and fractional order, Journal of Nonlinear Sciences and Applications, 14 (2021), no. 6, 372--389
AMA Style
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. J. Nonlinear Sci. Appl. (2021); 14(6):372--389
Chicago/Turabian Style
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 Applications, 14, no. 6 (2021): 372--389
Keywords
- Chaos control
- synchronization
- chaotic financial systems
- fractional order systems
- active control
- feedback controller
MSC
References
-
[1]
M. S. Abd-Elouahab, N.-E. Hamri, J. Wang, Chaos control of a fractional-order financial system, Math. Probl. Eng.,, 2010 (2010), 18 pages
-
[2]
H. N. Agiza, M. T. Yassen, Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys. Lett. A, 278 (2001), 191--197
-
[3]
H. Ahmad, A. Akgul, T. A. Khan, P. S. Stanimirovic, Y.-M. Chu, New perspective on the conventional solutions of the nonlinear time-fractional partial differential equations, Complexity, 2020 (2020), 10 pages
-
[4]
H. Ahmad, T. A. Khan, I. Ahmad, P. S. Stanimirovic, Y.-M. Chu, A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations, Results Phys., 19 (2020), 1--8
-
[5]
E.-W. Bai, K. E. Lonngren, Synchronization of two Lorenz systems using active control, Chaos Solitons Fractals, 8 (1997), 51--58
-
[6]
S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3536--3546
-
[7]
S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, C. S. Zhou, The synchronization of chaotic systems, Phys. Rep., 366 (2002), 1--101
-
[8]
W.-C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos Solitons Fractals, 36 (2008), 1305--1314
-
[9]
L. Chen, Y. Chai, R. Wu, Control and synchronization of fractional-order financial system based on linear control, Discrete Dyn. Nat. Soc., 2011 (2011), 21 pages
-
[10]
V. Daftardar-Gejji, S. Bhalekar, Chaos in fractional ordered Liu system, Comput. Math. Appl., 59 (2010), 1117--1127
-
[11]
S. A. David, J. A. T. Machado, D. D. Quintino, J. M. Balthazar, Partial chaos suppression in a fractional order macroeconomic model, Math. Comput. Simulation, 122 (2016), 55--68
-
[12]
K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3--22
-
[13]
A. L. Fradkov, R. J. Evans, Control of chaos: method and applications in engineering, Annu. Rev. Control, 29 (2005), 33--56
-
[14]
A. Hajipour, H. Tavakoli, Dynamic analysis and adaptive sliding mode controller for a chaotic fractional incommensurate order financial system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 14 pages
-
[15]
R. He, P. G. Vaidya, Implementation of chaotic cryptography with chaotic synchronization, Phys. Rev. E, 57 (1998), 1532--1535
-
[16]
R. Hilfer, Applications of fractional calculus in physics, World Scientific, USA (2000)
-
[17]
J. A. HoĆyst, K. Urbanowicz, Chaos control in economical model by time-delayed feedback method, Phys. A: Stat. Mech. Appl., 287 (2000), 587--598
-
[18]
L. Huang, R. Feng, M. Wang, Synchronization of chaotic systems via nonlinear control, Phys. Lett. A, 320 (2004), 271--275
-
[19]
Q. Jia, Chaos control and synchronization of the Newton-Leipnik chaotic system, Chaos Solitons Fractals, 35 (2008), 814--824
-
[20]
J. L. Kaplan, J. A. Yorke, Preturbulence: a regime observed in a fluid flow model of Lorenz, Comm. Math. Phys., 67 (1979), 93--108
-
[21]
C. Li, G. Peng, Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals, 22 (2004), 443--450
-
[22]
T.-L. Liao, Adaptive synchronization of two Lorenz systems, Chaos Solitons Fractals, 9 (1998), 1555--1561
-
[23]
Y. Liao, Y. Zhou, F. Xu, X.-B. Shu, A Study on the Complexity of a New Chaotic Financial System, Complexity, 2020 (2020), 5 pages
-
[24]
E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130--141
-
[25]
H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, Springer, Berlin (1993)
-
[26]
J. G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Solitons Fractals, 26 (2005), 1125--1133
-
[27]
J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. I, Appl. Math. Mech., 22 (2001), 1119--1128
-
[28]
J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. II, Appl. Math. Mech.,, 22 (2001), 1375--1382
-
[29]
C. Ma, X. Wang, Hopf bifurcation and topological horseshoe of a novel finance chaotic system, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 721--130
-
[30]
D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl.,, 2 (1996), 963--968
-
[31]
C. H. Miwadinou, L. A. Hinvi, A. V. Monwanou, J. B. Chabi Orou, Nonlinear dynamics of a φ6 - modified Duffing oscillator: resonant oscillations and transition to chaos, Nonlinear Dyn.,, 88 (2017), 97--113
-
[32]
E. Ott, C. Grebogi, J. A. Yorke, , Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196--1199
-
[33]
J. H. Park, Chaos synchronization of a chaotic system via nonlinear control, Chaos Solitons Fractals, 25 (2005), 579--584
-
[34]
I. Podlubny, Fractional differential equations, Academic Press, San Diego (1999)
-
[35]
M. Saif, F. Khan, K. S. Nisar, S. Araci, Modified Laplace transform and its properties, J. Math. Comput. Sci., 21 (2020), 127--135
-
[36]
R. H. Strotz, J. C. McAnulty, J. B. Naines, Goodwin’s nonlinear theory of the business cycle: an electro-analog solution, Econometrica, 390--411 (1953),
-
[37]
D. L. Suthar, S. D. Purohit, S. Araci, Solution of fractional kinetic equations associated with the (p, q)-Mathieu-type series, Discrete Dyn. Nat. Soc., 2020 (2020), 7 pages
-
[38]
M. S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems, Phys. D, 237 (2008), 2628--2637
-
[39]
X.-Y. Wang, Y.-J. He, M.-J. Wang, Chaos control of a fractional modified coupled dynamos system, Nonlinear Anal., 71 (2009), 6126--6134
-
[40]
A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285--317
-
[41]
M. T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput., 135 (2003), 113--128
-
[42]
M. T. Yassen, Chaos control of chaotic dynamical system using backstepping design, Chaos Solitons Fractals, 27 (2006), 537--548
-
[43]
H. Yu, G. Cai, Y. Li, Dynamic analysis and control of a new hyperchaotic finance system, Nonlinear Dynam., 67 (2012), 2171--2182