Chaotic and Hopf Bifurcation Analysis of the Stretch-Twist-Fold Flow System in a Fractional Order

Sardar Rashid Fattah; Niazy Hady Hussein; Sheelan Abdulkader Osman

doi:10.21271/ZJPAS.38.2.6

Authors

Sardar Rashid Fattah Department of Mathematics, Faculty of Science, Soran University, Kawa St.-Soran, Erbil, Iraq
Niazy Hady Hussein Department of Mathematics, College of Education, Salahadin University-Erbil, Erbil ,Kurdistan, Iraq
Sheelan Abdulkader Osman Department of Mathematics, Faculty of Science, Soran University, Kawa St.-Soran, Erbil, Iraq

DOI:

https://doi.org/10.21271/ZJPAS.38.2.6

Keywords:

stretch-twist-fold flow system, Caputo fractional order derivative, Hopf bifurcation, Adams-type predictor-corrector approach, Lyapunov exponents

Abstract

This study investigates a modified fractional Stretch-Twist-Fold (STF) model with a Caputo fractional-order derivative. The local stability and Hopf bifurcation of equilibrium points are analyzed by using an Adams-type predictor-corrector method implemented in MATLAB software. We study the influence of variation of parameters on the system's behavior and verify chaotic dynamics by computing maximal Lyapunov exponents. In addition to supporting analytical findings, numerical simulations are used to reveal chaotic characteristics such as bifurcations, phase portraits, limit cycles, and attractive chaotic sets, highlighting the crucial role of the fractional-order derivative in the system's dynamic behavior.

References

Abdelouahab, M.-S., et al. (2012). "Hopf bifurcation and chaos in fractional-order modified hybrid optical system." Nonlinear Dynamics 69: 275-284.

Abualhomos, M., et al. (2023). "Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders." Mathematics 11(19): 4166.

Ahmed, E., et al. (2007). "Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models." Journal of Mathematical Analysis and Applications 325(1): 542-553.

Asgari-Targhi, M. and M. A. Berger (2009). "Writhe in the stretch-twist-fold dynamo." Geophysical and Astrophysical Fluid Dynamics 103(1): 69-87.

Azam, A., et al. (2017). "Chaotic behavior of modified stretch-twist-fold (STF) flow with fractal property." Nonlinear Dynamics 90(1): 1-12.

Bajer, K. (1989). Flow kinematics and magnetic equilibria, University of Cambridge.

Bandyopadhyay, B. and S. Kamal (2015). "Stabilization and control of fractional order systems: a sliding mode approach."

Bao, J. and Q. Yang (2010). "Complex dynamics in the stretch-twist-fold flow." Nonlinear Dynamics 61: 773-781.

Bao, J. and Q. Yang (2014). "Bifurcation analysis of the generalized stretch-twist-fold flow." Applied Mathematics and Computation 229: 16-26.

Bao, J. and Q. Yang (2014). "Darboux integrability of the stretch-twist-fold flow." Nonlinear Dynamics 76: 797-807.

Bhalekar, S. and D. Gupta (2022). "Stability and bifurcation analysis of a fractional order delay differential equation involving cubic nonlinearity." Chaos, Solitons & Fractals 162: 112483.

Cafagna, D. and G. Grassi (2008). "Bifurcation and chaos in the fractional-order Chen system via a time-domain approach." International Journal of Bifurcation and Chaos 18(07): 1845-1863.

Chettouh, B. and T. Menacer (2024). "Effect of fractional order on the stability and the localisation of the critical Hopf bifurcation value in a fractional chaotic system." International Journal of Dynamics and Control 12(6): 1707-1716.

Childress, S. and A. D. Gilbert (2008). Stretch, twist, fold: the fast dynamo, Springer Science & Business Media.

Danca, M.-F. (2023). "Symmetry-breaking and bifurcation diagrams of fractional-order maps." Communications in Nonlinear Science and Numerical Simulation 116: 106760.

Danca, M.-F. and N. Kuznetsov (2018). "Matlab code for Lyapunov exponents of fractional-order systems." International Journal of Bifurcation and Chaos 28(05): 1850067.

Das, S. (2011). Introduction to fractional calculus. Functional fractional calculus, Springer: 1-50.

Deshpande, A. S., et al. (2017). "On Hopf bifurcation in fractional dynamical systems." Chaos, Solitons & Fractals 98: 189-198.

Diethelm, K. and N. Ford (2010). "The analysis of fractional differential equations." Lecture notes in mathematics 2004.

Diethelm, K., et al. (2002). "A predictor-corrector approach for the numerical solution of fractional differential equations." Nonlinear Dynamics 29: 3-22.

Edelman, M., et al. (2023). "Bifurcations and transition to chaos in generalized fractional maps of the orders 0< α< 1." Chaos: An Interdisciplinary Journal of Nonlinear Science 33(6).

El-Saka, H., et al. (2009). "On stability, persistence, and Hopf bifurcation in fractional order dynamical systems." Nonlinear Dynamics 56: 121-126.

Ghaziani, R. K., et al. (2016). "Stability and dynamics of a fractional order Leslie–Gower prey–predator model." Applied Mathematical Modelling 40(3): 2075-2086.

Grigorenko, I. and E. Grigorenko (2003). "Chaotic dynamics of the fractional Lorenz system." Physical review letters 91(3): 034101.

Hu, W., et al. (2017). "Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system." Optik 130: 189-200.

Hussein, N. H. and A. I. Amen (2019). "Zero-Hopf bifurcation in the generalized stretch-twist-fold flow." Sultan Qaboos University Journal for Science [SQUJS] 24(2): 122-128.

Jafari, S. and J. Sprott (2013). "Simple chaotic flows with a line equilibrium." Chaos, Solitons & Fractals 57: 79-84.

Jafari, S., et al. (2016). "A simple chaotic flow with a plane of equilibria." International Journal of Bifurcation and Chaos 26(06): 1650098.

Ji, Y., et al. (2018). "Bifurcation and chaos of a new discrete fractional-order logistic map." Communications in Nonlinear Science and Numerical Simulation 57: 352-358.

Kaslik, E. and S. Sivasundaram (2012). "Nonlinear dynamics and chaos in fractional-order neural networks." Neural networks 32: 245-256.

Kiani-B, A., et al. (2009). "A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter." Communications in Nonlinear Science and Numerical Simulation 14(3): 863-879.

Kilbas, A. A., et al. (2006). Theory and applications of fractional differential equations, elsevier.

Leonov, G. A. and N. V. Kuznetsov (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits." International Journal of Bifurcation and Chaos 23(01): 1330002.

Leung, A. Y., et al. (2014). "Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay." Communications in Nonlinear Science and Numerical Simulation 19(4): 1142-1155.

Li, X. and R. Wu (2014). "Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system." Nonlinear Dynamics 78: 279-288.

Liu, X. and L. Hong (2013). "Chaos and adaptive synchronizations in fractional-order systems." International Journal of Bifurcation and Chaos 23(11): 1350175.

Liu, X., et al. (2016). "Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method." Chaos: An Interdisciplinary Journal of Nonlinear Science 26(8).

Lu, J. G. and G. Chen (2006). "A note on the fractional-order Chen system." Chaos, Solitons & Fractals 27(3): 685-688.

Lyubomudrov, O., et al. (2003). "Pseudochaotic systems and their fractional kinetics." International Journal of Modern Physics B 17(22n24): 4149-4167.

Machado, J. T. and A. M. Galhano (2018). "A fractional calculus perspective of distributed propeller design." Communications in Nonlinear Science and Numerical Simulation 55: 174-182.

Maciejewski, A. J. and M. Przybylska (2020). "Integrability analysis of the stretch–twist–fold flow." Journal of Nonlinear Science 30(4): 1607-1649.

Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. Computational engineering in systems applications, Lille, France.

Moffatt, H. (1989). "Stretch, twist and fold." Nature 341(6240): 285-286.

Osman, S. and T. Langlands (2022). "Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations." Fractional Calculus and Applied Analysis 25(6): 2166-2192.

Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, elsevier.

Poincaré, H. (1893). Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars et fils, imprimeurs-libraires.

Sheu, L. J., et al. (2010). "A two-channel secure communication using fractional chaotic systems." World Academy of Science, Engineering and Technology 65: 1057-1061.

Sprott, J. C. (2011). "A proposed standard for the publication of new chaotic systems." International Journal of Bifurcation and Chaos 21(09): 2391-2394.

Suryanto, A., et al. (2025). "Bifurcation Analysis and Chaos Control of a Discrete–Time Fractional Order Predator-Prey Model with Holling Type II Functional Response and Harvesting." CHAOS Theory and Applications 7(1): 87-98.

Tavazoei, M. S. and M. Haeri (2007). "A necessary condition for double scroll attractor existence in fractional-order systems." Physics Letters A 367(1-2): 102-113.

Tavazoei, M. S. and M. Haeri (2008). "Limitations of frequency domain approximation for detecting chaos in fractional order systems." Nonlinear Analysis: Theory, Methods & Applications 69(4): 1299-1320.

Tavazoei, M. S. and M. Haeri (2009). "Describing function based methods for predicting chaos in a class of fractional order differential equations." Nonlinear Dynamics 57(3): 363-373.

Tenreiro Machado, J., et al. (2015). "Fractional state space analysis of economic systems." Entropy 17(8): 5402-5421.

Uddin, M. J. (2022). "Analysis of bifurcation and chaos to fractional order Brusselator model." Journal of Applied Mathematics and Computation 6(3).

Vainshtein, S. I., et al. (1997). "Stretch-twist-fold and ABC nonlinear dynamos: restricted chaos." Physical Review E 56(2): 1605.

Vainshtein, S. I., et al. (1996). "Fractal properties of the stretch-twist-fold magnetic dynamo." Physical Review E 53(5): 4729.

Vainshtein, S. I. and Y. B. Zel’dovich (1972). "Origin of magnetic fields in astrophysics (turbulent “dynamo” mechanisms)." Soviet Physics Uspekhi 15(2): 159-172.

Wang, C., et al. (2023). "Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay." Fractal and Fractional 7(8): 638.

Xiao, M. and W. X. Zheng (2012). Nonlinear dynamics and limit cycle bifurcation of a fractional-order three-node recurrent neural network. 2012 IEEE International Symposium on Circuits and Systems (ISCAS), IEEE.

Yue, B. and M. Aqeel (2013). "Chaotification in the stretch-twist-fold (STF) flow." Chinese Science Bulletin 58: 1655-1662.

Zhi, Y., et al. (2022). "Computational methods for nonlinear analysis of Hopf bifurcations in power system models." Electric Power Systems Research 212: 108574.