Araştırma Makalesi
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Yıl 2021, Cilt: 1 Sayı: 1, 11 - 23, 30.09.2021
https://doi.org/10.53391/mmnsa.2021.01.002

Öz

Kaynakça

  • 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).

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

Yıl 2021, Cilt: 1 Sayı: 1, 11 - 23, 30.09.2021
https://doi.org/10.53391/mmnsa.2021.01.002

Öz

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.

Kaynakça

  • 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).
Toplam 74 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Araştırma Makalesi
Yazarlar

Zakia Hammouch Bu kişi benim 0000-0001-7349-6922

Mehmet Yavuz 0000-0002-3966-6518

Necati Özdemir Bu kişi benim 0000-0002-6339-1868

Yayımlanma Tarihi 30 Eylül 2021
Gönderilme Tarihi 13 Temmuz 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 1 Sayı: 1

Kaynak Göster

APA 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, 1(1), 11-23. https://doi.org/10.53391/mmnsa.2021.01.002

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