Three-dimensional fractional system with the stability condition and chaos control

Year 2022, , 41 - 47, 31.03.2022

Molood Gholami Reza Khoshsiar Ghaziani Zohreh Eskandari

https://doi.org/10.53391/mmnsa.2022.01.004

Cited By: 15

Abstract

A three-dimensional system is introduced in this paper and its local stability is analyzed. Our study establishes the validity and uniqueness of the linear feedback control for the proposed system and proves its existence and uniqueness. The numerical simulation algorithm described by Atanackovic and Stankovic is finally applied. The analytical results are analyzed and the dynamics of the system are explored in more detail.

Keywords

Fractional-order system, stability, control, chaotic system

References

• Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A., & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).
• Özköse, F., Şenel, M.T., & Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).
• Ikram, R., Khan, A., Zahri, M., Saeed, A., Yavuz, M., & Kumam, P. Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay. Computers in Biology and Medicine, 141, 105115, (2022).
• Özköse, F., Yılmaz, S., Yavuz, M., Öztürk, İ., Şenel, M.T., Bağcı, B.Ş., ... & Önal, Ö. A Fractional Modeling of Tumor–Immune System Interaction Related to Lung Cancer with Real Data. The European Physical Journal Plus, 137(1), 1-28, (2022).
• Oud, M.A.A., Ali, A., Alrabaiah, H., Ullah, S., Khan, M.A., & Islam, S. A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. Advances in Difference Equations, 2021(1), 1-19, (2021).
• Ali, A., Alshammari, F.S., Islam, S., Khan, M. A., & Ullah, S. Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative. Results in Physics, 20, 103669, (2021).
• Ali, A., Islam, S., Khan, M.R., Rasheed, S., Allehiany, F.M., Baili, J., ... & Ahmad, H. Dynamics of a fractional order Zika virus model with mutant. Alexandria Engineering Journal, 61(6), 4821-4836, (2021).
• Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
• Daşbaşı, B. Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 44-55, (2021).
• Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).
• Naik, P.A., Owolabi, K.M., Yavuz, M., & Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).
• Naik, P.A. Global dynamics of a fractional-order SIR epidemic model with memory. International Journal of Biomathematics, 13(08), 2050071, (2020).
• Alidousti, J., & Eskandari, Z. Dynamical behavior and Poincare section of fractional-order centrifugal governor system. Mathematics and Computers in Simulation, 182, 791-806, (2021).
• Naik, P.A., Eskandari, Z., & Shahraki, H.E. Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 95-101, (2021).
• Sene, N. Study of a Fractional-Order Chaotic System Represented by the Caputo Operator. Complexity, 2021, Article ID 5534872, (2021).
• Naik, P.A., Zu, J. & Naik, M. Stability analysis of a fractional-order cancer model with chaotic dynamics. International Journal of Biomathematics, 14(6), 2150046, (2021).
• Magin, R. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 32(1), 1-104, (2004).
• Naik, P.A., Yavuz, M. & Zu, J. The role of prostitution on HIV transmission with memory: A modeling approach. Alexandria Engineering Journal, 59(4), 2513-2531, (2020).
• Naik, P.A., Zu, J. & Owolabi, K. Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos Solitons & Fractals, 138, 109826, (2020).
• Naik, P.A., Owolabi, K.M, Zu, J. & Naik, M. Modeling the transmission dynamics of COVID-19 pandemic in Caputo type fractional derivative. Journal of Multiscale Modelling, 12(3), 2150006-107, (2021).
• Yavuz, M. & Sene, N. Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model. Journal of Ocean Engineering and Science, 6(2), 196-205, (2021).
• Yavuz, M., Sulaiman, T.A., Usta, F. & Bulut, H. Analysis and numerical computations of the fractional regularized long-wave equation with damping term. Mathematical Methods in the Applied Sciences, 44(9), 7538-7555, (2021).
• Sene, N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 13-25, (2022).
• Ozkose, F. & Yavuz, M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in Biology and Medicine, 141, 105044, (2022).
• Caputo, M. Linear models of dissipation whose Q is almost frequency independent—II. Geophysical Journal International, 13(5), 529-539, (1967).
• Matignon, D. (1996, July). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications (Vol. 2, No. 1, pp. 963-968).
• Lai, Q., & Wang, L. Chaos, bifurcation, coexisting attractors and circuit design of a three-dimensional continuous autonomous system. Optik, 127(13), 5400-5406, (2016).
• Ahmed, E., El-Sayed, A.M.A., & El-Saka, H.A. On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Physics Letters A, 358(1), 1-4, (2006).
• Atanackovic, T.M., & Stankovic, B. On a numerical scheme for solving differential equations of fractional order. Mechanics Research Communications, 35(7), 429-438, (2008).