Numerical solutions and synchronization of a variable-order fractional chaotic system




Variable-order fractional derivative, chaotic system, Lyapunov exponent, synchronization


In the present paper, we implement a novel numerical method for solving differential equations with fractional variable-order in the Caputo sense to research the dynamics of a circulant Halvorsen system. Control laws are derived analytically to make synchronization of two identical commensurate Halvorsen systems with fractional variable-order time derivatives. The chaotic dynamics of the Halvorsen system with variable-order fractional derivatives are investigated and the identical synchronization between two systems is achieved. Moreover, graph simulations are provided to validate the theoretical analysis.


Download data is not yet available.


Lorenz, E.N. Deterministic nonperiodic flow. Journal of atmospheric sciences, 20(2), 130-141, (1963).

Azar, A.T., Sundarapandian, V. Chaos modeling and control systems design (Vol. 581). Germany: Springer, (2015).

Azar, A.T., Sundarapandian, V., Ouannas, A. Fractional order control and synchronization of chaotic systems (Vol. 688). Springer, (2017).

Effati, S., Saberi-Nadjafi, J., Saberi Nik, H. Optimal and adaptive control for a kind of 3D chaotic and 4D hyper-chaotic systems. Applied Mathematical Modelling, 38(2), 759-774, (2014).

Yang, J., Dong-Lian, Q. The feedback control of fractional order unified chaotic system. Chinese Physics B, 19(2), 020508, (2010).

Li, C., Tong, Y. Adaptive control and synchronization of a fractional-order chaotic system. Pramana, 80(4), 583-592, (2013).

Li, C.G., Chen, G.R. Chaos in the fractional order Chen system and its control. Chaos, Solitons and Fractals, 22(3), 549-554, (2004).

Li, Y., Chen, Y., Podlubny, I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag- Leffler stability. Computers and Mathematics with Applications, 59(5), 1810-1821, (2010).

Matignon, D. Stability results for fractional differential equations with applications to control processing. In Computational Engineering in Systems Applications, 963-968, (1996).

Mekkaoui, T., Hammouch, Z., Belgacem, F., Abbassi, A.E. Fractional-order nonlinear systems: Chaotic dynamics, numerical simulation and circuits design. In Fractional Dynamics, (pp.343-356). Sciendo Migraiton, (2016).

Auerbach, D., Grebogi, C., Ott, E., Yorke, J.A. Controlling chaos in high dimensional systems. Physical Review Letters, 69(24), 3479, (1992).

Pyragas, V., Pyragas, K. Continuous pole placement method for time-delayed feedback controlled systems. The European Physical Journal B, 87(11), 1-10, (2014).

Razminia, A., Baleanu, D. Fractional synchronization of chaotic systems with different orders. Proceedings of the Romanian Academy, 13, 14-321, (2012).

Bai, E.W., Lonngren, K.E. Synchronization of two Lorenz systems using active control. Chaos, Solitons Fractals, 8(1), 51-58, (1997).

Blokh, A., Cleveland, C., Misiurewicz, M. Expanding polymodials Modern Dynamical Systems and Applications ed M Brin, B Hasselblatt and Ya Pesin, 253-70, (2004).

Chamgoué, A., Yamapi, R., Woafo, P. Bifurcations in a birhythmic biological system with time-delayed noise. Nonlinear Dynamics, 73(4), 2157-2173, (2013).

Diethelm, K., Ford, N. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2), 229-248, (2002).

Diethelm, K., Ford, N., Freed, A., Luchko, Y. Algorithms for the fractional calculus: a selection of numerical method. Computer Methods in Applied Mechanics and Engineering, 194(6-8), 743-773, (2005).

Frohlich, H. Long Range Coherence and energy storage in a Biological systems. International Journal of Quantum Chemistry, 2(5), 641-649, (1968).

Frohlich, H. Quantum Mechanical Concepts in Biology. Theoretical Physics and Biology, 1, (1969).

He, G., Luo, M. Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control. Applied Mathematics and Mechanics, 33(5), 567-582, (2012).

Kadji, H.G., Orou, J.B., Yamapi, R., Woafo, P. Nonlinear Dynamics and Strange Attractors in the Biological System. Chaos Solitons and Fractals, 32(2), 862-882, (2007).

Kaiser, F. Coherent Oscillations in Biological Systems I. Bifurcations Phenomena and Phase transitions in enzyme-substrate reaction with Ferroelectric behaviour, 294, 304-333, (1978).

Kaiser, F. Coherent Oscillations in Biological Systems II. Lecture Notes in Mathematics 2007; 1907; Springer: Berlin.

Miller, K.S., Rosso, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, (1993).

Pham, V.T., Frasca, M., Caponetto, R., Hoang, T.M., Fortuna, L. Control and synchronization of fractional-order differential equations of phase-locked-loop. Chaotic Modeling and Simulation, 4, 623-631, (2012).

Pecora, L.M., Carroll, T.L. Synchronization in chaotic systems. Phys Rev Lett, 64, 821-824, (1990).

Pikovsky, A. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, (2011).

Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books Pub, (1994).

Ucar, A., Lonngren, K.E., Bai, E.W. Synchronization of the unified chaotic systems via active control. Chaos, Solitons and Fractals, 27,1292-97, (2006).

Zaslavsky, G. Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, (2008).

Katsikadelis, J.T. Numerical solution of variable order fractional differential equations. ArXiv preprint arXiv, 1802.00519, (2018).

Podlubny, I. Fractional Differential Equations. Academic Press: San Diego; Calif, USA, (1999).

Erturk, V.S, Momani, S., Odibat, Z. Application of generalized differential transform method to multi-order fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 13(8), 1642-1654, (2008).

Freihat, A., Momani, S. Application of Multistep Generalized Differential Transform Method for the Solutions of the Fractional-Order Chua’s System. Discrete Dynamics in Nature and Society, 1-12, (2012).

Hammouch, Z., Mekkaoui, T. Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex and Intelligent Systems, 4(4), 251-260, (2018).

Hongwu, W., Junhai, M. Chaos Control and Synchronization of a Fractional-order Autonomous System. WSEAS Trans. on Mathematics, 11,700-711, (2012).

Caponetto, R., Dongola, G., Fortuna, L. Fractional order systems: Modeling and control application. Singapore: World Scientific, (2010).

Escalante-Martínez, J.E., Gómez-Aguilar, J.F., Calderón-Ramón, C., Aguilar-Meléndez, A. & Padilla-Longoria, P. A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators. International Journal of Biomathematics, 11(01), 1850014, (2018).

Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F. & Valtierra-Rodríguez, M. Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order. Physica A. Statistical Mechanics and its Applications, 487, 1-21, (2017).

Yavuz, M. European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels. Numerical Methods for Partial Differential Equations, 10.1002/num.22645, (2021).

Zúniga-Aguilar, C.J., Gómez-Aguilar, J.F., Escobar-Jiménez, R.F. & Romero-Ugalde, H.M. Robust control for fractional variable-order chaotic systems with non-singular kernel. The European Physical Journal Plus, 133(1), 1-13, (2018).

Yavuz, M. Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018).

Zúniga-Aguilar, C.J., Romero-Ugalde, H.M., Gómez-Aguilar, J.F., Escobar-Jiménez, R.F. & Valtierra-Rodríguez, M. Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks. Chaos, Solitons & Fractals, 103, 382-403, (2017).

Petras, I. A note on the fractional-order Chua’s system. Chaos Soliton and Fractals, 38(1), 140-147, (2008).

Petras, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, (2011).

Caputo, M. Linear models of dissipation whose Q is almost frequency independent. Geophysical Journal International, 13(5), 529-539, (1967).

Ucar, S., Ucar, E., Ozdemir, N., Hammouch, Z. Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative. Chaos, Solitons and Fractals, 118, 300-306, (2019).

Chand, M., Hammouch, Z., Asamoa, J.K.K., Baleanu, D. Certain Fractional Integrals and Solutions of Fractional Kinetic Equations Involving the Product of S-Function. In Mathematical Methods in Engineering (pp. 213-244), Springer, Cham, (2019).

Owolabi, K.M., Hammouch, Z. Mathematical modeling and analysis of two-variable system with noninteger-order derivative. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(1), 013145, (2019).

Yavuz, M., Ozdemir, N. European vanilla option pricing model of fractional order without singular kernel. Fractal and Fractional, 2(1), 3, (2018).

Yavuz, M., Ozdemir, N. On the solutions of fractional Cauchy problem featuring conformable derivative.In ITM Web of Conferences, 22, EDP Sciences, (2018).

Yavuz, M., Ozdemir, N. Numerical inverse Laplace homotopy technique for fractional heat equations. Thermal Science, 22(1), 185-194, (2018).

Toufik, M., Atangana, A. New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. The European Physical Journal Plus, 132(10), 1-16, (2017).

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).

Mirzazadeh, M., Akinyemi, L., Şenol, M., Hosseini, K. A variety of solitons to the sixth-order dispersive (3+1)-dimensional nonlinear time-fractional Schrödinger equation with cubic-quintic-septic nonlinearities. Optik, 241, 166318, (2021).

Yavuz, M., Yokus, A. Analytical and numerical approaches to nerve impulse model of fractional-order. Numerical Methods for Partial Differential Equations, 36(6), 1348-1368, (2020).

Matar, M.M., Abbas, M.I., Alzabut, J., Kaabar, M.K.A., Etemad, S., Rezapour, S. Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Advances in Difference Equations, 2021(1), 1-18, (2021).

Kaabar, M.K., Martínez, F., Gómez-Aguilar, J.F., Ghanbari, B., Kaplan, M., Günerhan, H. New approximate analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method. Mathematical Methods in the Applied Sciences, 44(14), 11138-11156, (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).

Tariq, K.U., Younis, M., Rizvi, S.T.R., Bulut, H. M-truncated fractional optical solitons and other periodic wave structures with Schrödinger–Hirota equation. Modern Physics Letters B, 34(supp01), 2050427, (2020).

Yaghoobi, S., Moghaddam, B.P., Ivaz, K. An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dynamics, 87(2), 815-826, (2017).

Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Valtierra-Rodriguez, M. & Escobar-Jiménez, R.F. Design of a state observer to approximate signals by using the concept of fractional variable-order derivative. Digital Signal Processing, 69, 127-139, (2017).

Gómez-Aguilar, J.F. Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Physica A: Statistical Mechanics and its Applications, 494, 52-75, (2018).

Gómez-Aguilar J.F. Chaos in a nonlinear Bloch system with Atangana–Baleanu fractional derivatives. Numerical Methods for Partial Differential Equations, 34(5), 1716-1738, (2018).

Yufeng, X., Zhimin, H. Synchronization of variable-order fractional financial system via active control method. Open Physics, 11(6), 824-835, (2013).

Razminia, A., Dizaji, A.F., Majd, V.J. Solution existence for non-autonomous variable-order fractional differential equations. Mathematical and Computer Modelling, 55(3-4), 1106-1117, (2012).

Solís-Pérez, J.E., Gómez-Aguilar, J.F., Atangana, A. Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos, Solitons and Fractals, 114, 175-185, (2018).

Atangana, A., Owolabi, K.M. New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena, 13(1), 3, (2018).

Hammouch, Z., Mekkaoui, T. Chaos synchronization of a fractional nonautonomous system. Nonautononmous Dynamical Systems, 1, 61-71, (2014).

Lu, L., Zhang, C., Guo, Z.A. Synchronization between two different chaotic systems with nonlinear feedback control. Chinese Physics, 16, 1603-1607, (2007).

Olusola, O., Vincent, E., Njah, N., Ali, E. Control and Synchronization of Chaos in Biological Systems Via Backsteping Design. International Journal of Nonlinear Science and Numerical Simulation, 11(1), 121-128, (2011).

Haeri, M., Emadzadeh, A. Synchronizing different chaotic systems using active sliding mode control. Chaos, Solitons and Fractals, 31(1), 119-129, (2007).

Wang, Y., Guan, Z.H., Wang, H.O. Feedback an adaptive control for the synchronization of Chen system via a single variable. Physics Letters A, 312(1-2), 34-40, (2003).




How to Cite

Hammouch, Z., Yavuz, M., & Özdemir, N. (2021). Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation With Applications (MMNSA), 1(1), 11–23.



Research Articles