A Novel Fractional-Order System: Chaos, Hyperchaos and ‎Applications to Linear Control

Document Type : Research Paper


1 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, 11952, Saudi Arabia

2 College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia

3 Mansoura Higher Institute for Engineering and Technology, Damietta High Way, Mansourah, Egypt


Chaos and hyperchaos are generated from a new fractional-order system. Local stability of the system’s three equilibria is analyzed when the fractional parameter belongs to (0,2]. According to Hopf bifurcation theory in fractional-order systems, approximations to the periodic solutions around the system’s three equilibria are explored. Lyapunov exponents, Lyapunov spectrum and bifurcation diagrams are computed and chaotic (hyperchaotic) attractors are depicted. Furthermore, a linear control technique (LFGC) based on Lyapunov stability theory is implemented to derive the hyperchaotic states of the proposed system to its three equilibrium points. Numerical results are used to validate the theoretical results.


Main Subjects

[1] Hegazi, A.S., Ahmed, E., Matouk, A.E., The effect of Fractional Order on Synchronization of Two Fractional Order Chaotic and Hyperchaotic Systems, J Fract Calc Appl, 1(3), 2011, 1-15.
[2] El-Sayed, A.M.A., Elsonbaty, A., Elsadany, A.A., Matouk,A.E., Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization, Int J Bifurcat Chaos, 26, 2016, Article ID 1650222, 35 pages.                 
[3] Song, P., Zhao, H., Zhang, X., Dynamic Analysis of a Fractional Order Delayed Predator-Prey System with Harvesting, Theory Biosci, 135(1-2), 2016, 59-72.
[4] Matouk, A.E., Chaos Synchronization of a Fractional-Order Modified Van der Pol-Duffing System via New Linear Control, Backstepping Control and Takagi-Sugeno Fuzzy Approaches, Complexity, 21, 2016, 116-124.
[5] Sayed, W.S., Fahmy, H.A.H., Rezk, A.A., Radwan, A.G., Generalized Smooth Transition Map Between Tent and Logistic Maps, Int J Bifurc Chaos, 27(1), 2017, Article ID 1730004.
[6] Al-khedhairi, A., Matouk, A.E., Khan, I., Chaotic Dynamics and Chaos Control for the Fractional-Order Geomagnetic Field Model, Chaos, Solitons Fractals 128, 2019, 390-401.
[7] Mondal, A., Sharma, S.K., Upadhyay, R.K., Mondal, A., Firing Activities of a Fractional-Order FitzHugh-Rinzel Bursting Neuron Model and its Coupled Dynamics, Scientific Reports, 9, 2019, Article number 15721. 11 pages.
[8] Chen, Y., Fiorentino, F., Negro, L.D., A Fractional Diffusion Random Laser, Scientific Reports, 9, 2019, Article number 8686. 14 pages.
[9] Kyriakis, P., Pequito, S., Bogdan, P., On the Effects of Memory and Topology on the Controllability of Complex Dynamical Networks, Scientific Reports, 10, 2020, Article number 17346. 13 pages.
[10] Chu, Y-M., Ali, R., Asjad, M.I., Ahmadian, A., Senu, N., Heat Transfer Flow of Maxwell Hybrid Nanofluids Due to Pressure Gradient Into Rectangular Region, Scientific Reports, 10, 2020, Article number 16643. 18 pages.
[11] Caputo, M., Linear Models of Dissipation Whose Q is Almost Frequency Independent-II, Geophys J R Astron Soc, 13, 1967, 529-539.
[12] Matouk, A.E., Chaos, Feedback Control and Synchronization of a Fractional-Order Modified Autonomous Van der Pol-Duffing Circuit, Commun Nonlinear Sci Numer Simul, 16, 2011, 975-986.
[13] Hegazi, A.S., Ahmed, E., Matouk, A.E., On Chaos Control and Synchronization of the Commensurate Fractional Order Liu System, Commun Nonlinear Sci Numer Simul, 18, 2013, 1193-1202.
[14] Ahmed, E., Matouk, A.E., Complex Dynamics of Some Models of Antimicrobial Resistance on Complex Networks, Math Meth Appl Sci, Accepted 2020.
[15] Matouk, A.E., Stability Conditions, Hyperchaos and Control in a Novel Fractional Order Hyperchaotic System, Phys Lett A, 373, 2009, 2166-2173.
[16] Hegazi, A.S., Matouk, A.E., Dynamical Behaviors and Synchronization in the Fractional Order Hyperchaotic Chen System, Appl Math Lett, 24, 2011, 1938-1944.
[17] Matignon, D., Stability Results for Fractional Differential Equations with Applications to Control Processing, Proccedings of IMACS, IEEE-SMC,Lille, 1996, 2, 963.
[18] Ahmed, E., Elgazzar, A.S., On Fractional Order Differential Equations Model for Nonlocal Epidemics, Physica A, 379, 2007, 607-614.
[19] Radwan, A.G., Moaddy, K., Salama, K.N., Momani, S., Hashim, I., Control and Switching Synchronization of Fractional Order Chaotic Systems Using Active Control Technique, J Adv Res, 5, 2014, 125.
[20] Mahmoud, G.M., Ahmed, M.E., Abed-Elhameed, T.M., Active Control Technique of Fractional-Order Chaotic Complex Systems, Eur Phys J Plus, 131, 2016, 200. DOI: 10.1140/epjp/i2016-16200-x.
[21] Matouk, A.E., Dynamics and Control in a Novel Hyperchaotic System, Int J Dyn Control, 7, 2019, 241.
[22] Al-Khedhairi, A., Matouk, A.E., Askar, S.S., Computations of Synchronisation Conditions in Some Fractional-Order Chaotic and Hyperchaotic Systems, Pramana–J Phys, 92, 2019, 72, 11 pages.
[23] Matouk, A.E.M., On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System, In Advanced Applications of Fractional Differential Operators to Science and Technology, pp. 115-132, IGI Global, 2020.
[24] Matouk, A.E., Complex Dynamics in Susceptible-Infected Models for COVID-19 with Multi-Drug Resistance, Chaos Solit Fract, 140, 2020, 110257.
[25] Matouk, A.E., Khan, I., Complex Dynamics and Control of a Novel Physical Model Using Nonlocal Fractional Differential Operator With Singular Kernel, J Adv Res, 24, 2020, 463-474.
[26] He, J.H., Approximate Solution of Non Linear Differential Equations With Convolution Product Nonlinearities, Computer Meth Appl Mech Eng, 167, 1998, 69-73.
[27] Hartley, T.T., Lorenzo, C.F., Qammer, H.K., Chaos in a Fractional-Order Chua’s System, IEEE Trans CAS-I, 42, 1995, 485-490.
[28] El-Sayed, A.M.A., Behiry, S.H., Raslan, W.E., Adomian’s Decomposition Method for Solving an Intermediate Fractional Advection-Dispersion Equation, Computers & Mathematics with Applications, 59, 2010, 1759-1765.
[29] Diethelm, K., Ford, N.J., Freed, A.D., A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations, Nonlinear Dynam, 29, 2002, 3-22.
[30] Tavazoei, M.S., Haeri, M., A Proof for Non Existence of Periodic Solutions in Time Invariant Fractional Order Systems, Automatica, 45(8), 2009, 1886-1890.
[31] Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A. Determining Lyapunov Exponents From a Time Series, Physica D: Nonlinear Phenomena, 16(3), 1985, 285-317.