Abstract
This work introduces a chaotic satellite system including the Caputo–Prabhakar fractional derivative and investigates the characteristics and complex dynamics of the system. The asymptotic stability of the nonlinear Caputo–Prabhakar fractional system is analyzed by assessing the eigenvalues of the Jacobian matrix of system in the complex eigenvalues plane. To solve the Caputo–Prabhakar fractional chaotic satellite system numerically and exhibit the dynamic characteristics of the system, we state a numerical algorithm. Next, we prove the existence and uniqueness of the solution to the system and analyze the dynamical behaviors of the system around the equilibria. Then, we control the chaotic vibration of the system by means of the feedback control procedure and the Lyapunov second method. Further, chaos synchronization is achieved between two identical Caputo–Prabhakar fractional chaotic satellite systems by designing control laws. Furthermore, we illustrate the effect of the parameters of Caputo–Prabhakar fractional derivative on the dynamic behaviors of the system. Choosing the suitable values of the mentioned derivative parameters is able to successfully increase the stability region and achieve chaos control without any controllers. By numerical simulations, we show that the appropriate values of the derivative parameters in the Caputo–Prabhakar fractional satellite system are able to return back the satellite’s attitude to its equilibrium point, when the satellite attitude is tilted of this point. This is despite the fact that the integer-order form of the system and even the fractional system with the Caputo fractional derivative remain chaotic.
Similar content being viewed by others
References
Alban MT, Antonia JJ (2000) The control of higher dimensional chaos: comparative results for the chaotic satellite attitude control problem. Phys D 135:41–62
Arqub OA, Hayat T, Alhodaly M (2022) Analysis of Lie symmetry, explicit series solutions, and conservation laws for the nonlinear time-fractional Phi-four equation in two-dimensional space. Int J Appl Comput Math 8(145):1–17. https://doi.org/10.1007/s40819-022-01334-0
Beghami W, Maayah B, Bushnaq S, Arqub OA (2022) The Laplace optimized decomposition method for solving systems of partial differential equations of fractional order. Int J Appl Comput Math 8(52):1–18. https://doi.org/10.1007/s40819-022-01256-x
Chu YM, Inc M, Hashemi MS, Eshaghi S (2022) Analytical treatment of regularized Prabhakar fractional differential equations by invariant subspaces. Comput Appl Math 41(6):1–17
Diethelm K (2010) The analysis of fractional differential equations. Springer, New York
D’Ovidio M, Polito F (2013) Fractional diffusion-telegraph equations and their associated stochastic solutions. Theory Probab Appl 62(4):552–574
Dracopoulos DC, Jones AJ (1997) Adaptive neuro-genetic control of chaos applied to the attitude control problem. Neural Comput Appl 6:102–115
Eshaghi S, Ansari A, Khoshsiar Ghaziani R, Ahmadi Darani M (2017) Fractional Black-Scholes model with regularized Prabhakar derivative. Publ Inst Math 102(116):121–132
Eshaghi S, Khoshsiar Ghaziani R, Ansari A (2019) Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay. Math Methods Appl Sci 42(7):2302–2323
Eshaghi S, Khoshsiar Ghaziani R, Ansari A (2020) Stability and dynamics of neutral and integro–differential regularized Prabhakar fractional differential systems. Comput Appl Math 39(4):1–21
Eshaghi S, Ansari A, Khoshsiar Ghaziani R (2021) Generalized Mittag-Leffler stability of nonlinear fractional regularized Prabhakar differential systems. Int J Nonlinear Anal Appl 12(2):665–678
Garra R, Gorenflo R, Polito F, Tomovski Z (2014) Hilfer-Prabhakar derivatives and some applications. Appl Math Comput 242:576–589
Garrappa R (2016) Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models. Commun Nonlinear Sci Numer Simul 38:178–191
Garrappa R, Kaslik E (2020) Stability of fractional-order systems with Prabhakar derivatives. Nonlinear Dyn 102:567–578
Giusti A, Colombaro I (2018) Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simulat 56:138–143
Havriliak S, Negami S (1966) A complex plane analysis of \(\alpha\)-dispersions in some polymer systems. J Polym Sci C 14:99–117
Kilbas AA, Saigo M, Saxena RK (2004) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform Spec Funct 15:31–49
Kumar S, Matouk AE, Chaudhary H, Kant S (2021) Control and synchronization of fractional-order chaotic satellite systems using feedback and adaptive control techniques. Int J Adapt Control Signal Process 35(4):484–497
Mainardi F, Garrappa R (2015) On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics. J Comput Phys 293:70–80
Momani S, Arqub OA, Maayah B (2020) Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu-Caputo model of the Lienard’s equation. Fractals 28(8):2040007
Momani S, Maayah B, Arqub OA (2020) The reproducing kernel algorithm for numerical solution of Van der Pol damping model in view of the Atangana-Baleanu fractional approach. Fractals 28(08):2040010
Niknam MR, Kheiri H, Sobouhi NA (2022) Optimal control of satellite attitude and its stability based on quaternion parameters. Comput Methods Differ Equ 10(1):168–178
Prabhakar TR (1971) A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 19:7–15
Shafiq M, Ahmad I, Almatroud OA, Al-Sawalha MM (2022) Robust attitude control of the three-dimensional unknown chaotic satellite system. Trans Inst Meas Control 44(7):1484–1504
Sidi MJ (1997) Spacecraft dynamics and control- a practical engineering approach. Cambridge University Press, Cambridge, UK
Tarasov VE (2020) Fractional nonlinear dynamics of learning with memory. Nonlinear Dyn 100(2):1231–1242
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interest.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Eshaghi, S., Ordokhani, Y. Dynamical Behaviors of the Caputo–Prabhakar Fractional Chaotic Satellite System. Iran J Sci Technol Trans Sci 46, 1445–1459 (2022). https://doi.org/10.1007/s40995-022-01358-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-022-01358-7