In recent years, fractional-order dynamical systems have attracted significant attention because of their ability to describe memory effects and hereditary properties found in many physical and engineering processes. Fractional calculus extends the concept of differentiation and integration to non-integer orders, providing a more flexible mathematical framework for modeling and analyzing complex system behaviors. Within this framework, the study of chaotic dynamics in fractional-order systems has become an active research area since many well-known chaotic models show new and qualitatively different behaviors when generalized to fractional derivatives. This study revisits a classical chaotic system originally defined in the integer-order setting and analyzes its behavior using fractional-order calculus. The main objective is to examine how fractional differentiation influences the system's chaotic dynamics and sensitivity to initial conditions. To achieve this, numerical simulations are performed to demonstrate the effects of different fractional orders on the development and evolution of chaotic behavior.
Citation: Seda İĞRET ARAZ, Mehmet Akif ÇETİN. Investigation of chaotic behavior in a Lorenz-like system with fractional derivatives[J]. Communications in Analysis and Mechanics, 2026, 18(1): 208-227. doi: 10.3934/cam.2026008
In recent years, fractional-order dynamical systems have attracted significant attention because of their ability to describe memory effects and hereditary properties found in many physical and engineering processes. Fractional calculus extends the concept of differentiation and integration to non-integer orders, providing a more flexible mathematical framework for modeling and analyzing complex system behaviors. Within this framework, the study of chaotic dynamics in fractional-order systems has become an active research area since many well-known chaotic models show new and qualitatively different behaviors when generalized to fractional derivatives. This study revisits a classical chaotic system originally defined in the integer-order setting and analyzes its behavior using fractional-order calculus. The main objective is to examine how fractional differentiation influences the system's chaotic dynamics and sensitivity to initial conditions. To achieve this, numerical simulations are performed to demonstrate the effects of different fractional orders on the development and evolution of chaotic behavior.
| [1] | E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130–141. |
| [2] | C. Sparrow, The Lorenz equations: Bifurcations, chaos, and strange attractors, New York: Springer-Verlag, 1982. https://doi.org/10.1007/978-1-4612-5767-7 |
| [3] |
G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (1999), 1465–1466. https://doi.org/10.1142/S0218127499001024 doi: 10.1142/S0218127499001024
|
| [4] |
J. Lü, G. Chen, D. Cheng, A new chaotic attractor coined, Int. J. Bifurcation Chaos, 12 (2002), 659–661. https://doi.org/10.1142/S0218127402004620 doi: 10.1142/S0218127402004620
|
| [5] |
C. Li, G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals, 22 (2004), 549–554. https://doi.org/10.1016/j.chaos.2004.02.035 doi: 10.1016/j.chaos.2004.02.035
|
| [6] | I. Petráš, Fractional-order nonlinear systems: Modeling, analysis and simulation, Berlin: Springer-Verlag, 2011. https://doi.org/10.1007/978-3-642-18101-6 |
| [7] |
M. S. Tavazoei, M. Haeri, A proof for nonexistence of chaos in linear fractional-order systems, Automatica, 43 (2007), 918–920. https://doi.org/10.1016/j.automatica.2009.04.001 doi: 10.1016/j.automatica.2009.04.001
|
| [8] |
Z. M. Odibat, N. Corson, M. A. Aziz-Alaooi, C. Bertelle, Synchronization of chaotic fractional order systems via linear control, Int. J. Bifurcation Chaos, 20 (2010), 81–97. https://doi.org/10.1142/S0218127410025429 doi: 10.1142/S0218127410025429
|
| [9] | S. Boulaaras, S. Arunachalam, S. Sriramulu, Results on Ulam–Hyers stability of nonlinear Chen system with fractional-order derivative, Asian J. Control, 2025, 1–14. https://doi.org/10.1002/asjc.3656 |
| [10] |
I. Akbulut Arık, S. Igret Araz, Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos, Commun. Anal. Mech., 16 (2024), 169–192. https://doi.org/10.3934/cam.2024008 doi: 10.3934/cam.2024008
|
| [11] |
K. A. Abro, A. Atangana, J. F. Gomez-Aguilar, Chaos control and characterization of brushless DC motor via integral and differential fractal-fractional techniques, Int. J. Model Simul., 43 (2022), 416–425. https://doi.org/10.1080/02286203.2022.2086743 doi: 10.1080/02286203.2022.2086743
|
| [12] |
M. Sinan, J. Leng, K. Shah, T. Abdeljawad, Advances in numerical simulation with a clustering method based on K–means algorithm and Adams Bashforth scheme for fractional order laser chaotic system, Alex. Eng. J., 75 (2023), 165–179. https://doi.org/10.1016/j.aej.2023.05.080 doi: 10.1016/j.aej.2023.05.080
|
| [13] |
I. Haq, N. Ali, H. Ahmad, Analysis of a chaotic system using fractal-fractional derivatives with exponential decay type kernels, Math. Model. Control, 2 (2022), 185–199. https://doi.org/10.3934/mmc.2022019 doi: 10.3934/mmc.2022019
|
| [14] |
X. Zhang, B. Sang, B. Li, J. Liu, L. Fan, N. Wang, Hidden chaotic mechanisms for a family of chameleon systems, Math. Model. Control, 3 (2023), 400–415. https://doi.org/10.3934/mmc.2023032 doi: 10.3934/mmc.2023032
|
| [15] | S. Igret Araz, M. B. Riaz, Qualitative behavior and bifurcation structure of a new Lorenz-type chaotic model, preprint, hal-05175933. |
| [16] | M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. |
| [17] |
M. Caputo, Linear model of dissipation whose Q is almost frequency independent II, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
|
| [18] |
A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9
|
| [19] | J. Kaplan, J. Yorke, Chaotic behavior of multidimensional difference equations, In: Functional Differential Equations and the Approximation of Fixed Points, Berlin: Springer, 1979. https://doi.org/10.1007/BFb0064319 |
| [20] | A. Atangana, S. Igret Araz, Step forward on nonlinear differential equations with the Atangana-Baleanu derivative: Inequalities, existence, uniqueness and method, Chaos Solitons Fractals, 173 (2023). https://doi.org/10.1016/j.chaos.2023.113700 |
| [21] |
M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444. https://doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0
|
| [22] |
K. Dielthem, R. Garrappa, A. Giusti, M. Stynes, Why fractional derivatives with nonsingular kernels should not be used, Fract. Calc. Appl. Anal., 23 (2010), 610–634. https://doi.org/10.1515/fca-2020-0032 doi: 10.1515/fca-2020-0032
|
| [23] |
V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl., 316 (2006), 753–763. https://doi.org/10.1016/j.jmaa.2005.05.009 doi: 10.1016/j.jmaa.2005.05.009
|
| [24] |
S. M. Sivalingam, V. Govindaraj, J. V. C. Sousa, A. S. Hendy, L1-predictor–corrector method for $\psi $-Caputo type fractional differential equations, Comput. Appl. Math., 44 (2025), 753–763. https://doi.org/10.1007/s40314-025-03192-0 doi: 10.1007/s40314-025-03192-0
|
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Seda İĞRET ARAZ, Mehmet Akif ÇETİN. Investigation of chaotic behavior in a Lorenz-like system with fractional derivatives[J]. Communications in Analysis and Mechanics, 2026, 18(1): 208-227. doi: 10.3934/cam.2026008
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