Studying simple chaotic systems with fractional-order derivatives improves modeling accuracy, increases complexity, and enhances control capabilities and robustness against noise. This paper investigates the dynamics of the simple Sprott-B chaotic system using fractional-order derivatives. This study involves a comprehensive dynamical analysis conducted through bifurcation diagrams, revealing the presence of coexisting attractors. Additionally, the synchronization behavior of the system is examined for various derivative orders. Finally, the integer-order and fractional-order electronic circuits are implemented to validate the theoretical findings. This research contributes to a deeper understanding of the Sprott-B system and its fractional-order dynamics, with potential applications in diverse fields such as chaos-based secure communications and nonlinear control systems.

Centre for Nonlinear Systems Chennai Institute of Technology Chennai 600069 India

Department of Biomedical Engineering Amirkabir University of Technology Tehran 15916 34311 Iran

Department of Computer Technology Engineering College of Information Technology Imam Ja'afar Al Sadiq University Baghdad 10001 Iraq

Department of Electronics and Communication Engineering Vemu Institute of Technology Chithoor 517112 India

Department of Electronics and Communications Engineering University Centre for Research and Development Chandigarh University Mohali 140413 India

Department of Radio Electronics Brno University of Technology 616 00 Brno Czech Republic

Health Technology Research Institute Amirkabir University of Technology Tehran 15916 34311 Iran

School of Electronic Engineering Changzhou College of Information Technology Changzhou 213164 China

See more in PubMed

Sprott J.C. Elegant Chaos: Algebraically Simple Chaotic Flows. World Scientific; Singapore: 2010.

Shukla J. Predictability in the midst of chaos: A scientific basis for climate forecasting. Science. 1998;282:728–731. doi: 10.1126/science.282.5389.728. PubMed DOI

Dumitrescu C. Contributions to modeling the behavior of chaotic systems with applicability in economic systems. Intern. Audit. Risk Manag. 2019;56:98–107.

Wilder J. Effect of initial condition sensitivity and chaotic transients on predicting future outbreaks of gypsy moths. Ecol. Modell. 2001;136:49–66. doi: 10.1016/S0304-3800(00)00385-9. DOI

Hsieh D.A. Chaos and nonlinear dynamics: Application to financial markets. J. Financ. 1991;46:1839–1877. doi: 10.1111/j.1540-6261.1991.tb04646.x. DOI

Buizza R. Chaos and weather prediction-A review of recent advances in Numerical Weather Prediction: Ensemble forecasting and adaptive observation targeting. Il Nuovo C. C. 2001;24:273–302.

Amigo J., Kocarev L., Szczepanski J. Theory and practice of chaotic cryptography. Phys. Lett. A. 2007;366:211–216. doi: 10.1016/j.physleta.2007.02.021. DOI

Volos C., Akgul A., Pham V.-T., Stouboulos I., Kyprianidis I. A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn. 2017;89:1047–1061. doi: 10.1007/s11071-017-3499-9. DOI

Wu H., Zhang Y., Bao H., Zhang Z., Chen M., Xu Q. Initial-offset boosted dynamics in memristor-sine-modulation-based system and its image encryption application. AEU-Int. J. Electron. Commun. 2022;157:154440. doi: 10.1016/j.aeue.2022.154440. DOI

Ma X., Wang C. Hyper-chaotic image encryption system based on N+ 2 ring Joseph algorithm and reversible cellular automata. Multimed. Tools Appl. 2023:1–26. doi: 10.1007/s11042-023-15119-0. DOI

Ma X., Wang C., Qiu W., Yu F. A fast hyperchaotic image encryption scheme. Int. J. Bifurc. Chaos. 2023;33:2350061. doi: 10.1142/S021812742350061X. DOI

Sprott J.C. Simple chaotic systems and circuits. Am. J. Phys. 2000;68:758–763. doi: 10.1119/1.19538. DOI

Lin H., Wang C., Sun Y. A Universal Variable Extension Method for Designing Multiscroll/Wing Chaotic Systems. IEEE Trans. Indust. Electron. 2023;68:12708–12719. doi: 10.1109/TIE.2020.3047012. DOI

Yu F., Zhang W., Xiao X., Yao W., Cai S., Zhang J., Wang C., Li Y. Dynamic analysis and FPGA implementation of a new, simple 5D memristive hyperchaotic Sprott-C system. Mathematics. 2023;11:701. doi: 10.3390/math11030701. DOI

Lin H., Wang C., Du S., Yao W., Sun Y. A family of memristive multibutterfly chaotic systems with multidirectional initial-based offset boosting. Chaos Solitons Fractals. 2023;172:113518. doi: 10.1016/j.chaos.2023.113518. DOI

Fiori S., Di Filippo R. An improved chaotic optimization algorithm applied to a DC electrical motor modeling. Entropy. 2017;19:665. doi: 10.3390/e19120665. DOI

Sabatier J., Agrawal O.P., Machado J.T. Advances in Fractional Calculus. Volume 4 Springer; Berlin/Heidelberg, Germany: 2007.

Khennaoui A.-A., Ouannas A., Bendoukha S., Grassi G., Lozi R.P., Pham V.-T. On fractional–order discrete–time systems: Chaos, stabilization and synchronization. Chaos Solitons Fractals. 2019;119:150–162. doi: 10.1016/j.chaos.2018.12.019. DOI

Kumar D., Baleanu D. Fractional Calculus and Its Applications in Physics. Volume 7. Frontiers Media SA; Lausanne, Switzerland: 2019. p. 81.

Gutierrez R.E., Rosário J.M., Tenreiro Machado J. Fractional order calculus: Basic concepts and engineering applications. Math. Prob. Engin. 2010;2010:375858. doi: 10.1155/2010/375858. DOI

Zhang L., Sun K., He S., Wang H., Xu Y. Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings. Euro. Phys. J. Plus. 2017;132:31. doi: 10.1140/epjp/i2017-11310-7. DOI

Gu S., He S., Wang H., Du B. Analysis of three types of initial offset-boosting behavior for a new fractional-order dynamical system. Chaos Solitons Fractals. 2021;143:110613. doi: 10.1016/j.chaos.2020.110613. DOI

Birs I., Muresan C., Nascu I., Ionescu C. A survey of recent advances in fractional order control for time delay systems. IEEE Access. 2019;7:30951–30965. doi: 10.1109/ACCESS.2019.2902567. DOI

Azar A.T., Vaidyanathan S., Ouannas A. Fractional Order Control and Synchronization of Chaotic Systems. Volume 688 Springer; Berlin/Heidelberg, Germany: 2017.

Yan B., Parastesh F., He S., Rajagopal K., Jafari S., Perc M. Interlayer and intralayer synchronization in multiplex fractional-order neuronal networks. Fractals. 2022;30:2240194. doi: 10.1142/S0218348X22401946. DOI

Zhao J., Wang S., Chang Y., Li X. A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dyn. 2015;80:1721–1729. doi: 10.1007/s11071-015-1911-x. DOI

Yao Z., Sun K., He S. Firing patterns in a fractional-order FithzHugh–Nagumo neuron model. Nonlinear Dyn. 2022;110:1807–1822. doi: 10.1007/s11071-022-07690-2. DOI

Rajagopal K., Karthikeyan A., Jafari S., Parastesh F., Volos C., Hussain I. Wave propagation and spiral wave formation in a Hindmarsh–Rose neuron model with fractional-order threshold memristor synaps. Int. J. Mod. Phys. B. 2020;34:2050157. doi: 10.1142/S021797922050157X. DOI

Nosrati K., Shafiee M. Fractional-order singular logistic map: Stability, bifurcation and chaos analysis. Chaos Solitons Fractals. 2018;115:224–238. doi: 10.1016/j.chaos.2018.08.023. DOI

Nosrati K., Belikov J., Tepljakov A., Petlenkov E. Image Encryption Using Fractional Singular Chaotic Systems: An Extended Kalman Filtering Approach; Proceedings of the 2022 International Conference on Electrical, Computer and Energy Technologies (ICECET); Prague, Czech Republic. 20–22 July 2022; pp. 1–6.

Nosrati K., Belikov J., Tepljakov A., Petlenkov E. Extended fractional singular kalman filter. Appl. Math. Comput. 2023;448:127950. doi: 10.1016/j.amc.2023.127950. DOI

Wei Y.-Q., Liu D.-Y., Boutat D., Chen Y.-M. An improved pseudo-state estimator for a class of commensurate fractional order linear systems based on fractional order modulating functions. Syst. Control Lett. 2018;118:29–34. doi: 10.1016/j.sysconle.2018.05.011. DOI

Sprott J.C., Thio W.J.-C. Elegant Circuits: Simple Chaotic Oscillators. World Scientific; Singapore: 2022.

Petrzela J. Chaos in analog electronic circuits: Comprehensive review, solved problems, open topics and small example. Mathematics. 2022;10:4108. doi: 10.3390/math10214108. DOI

Bao B., Xu L., Wang N., Bao H., Xu Q., Chen M. Third-order RLCM-four-elements-based chaotic circuit and its coexisting bubbles. AEU-Int. J. Electron. Commun. 2018;94:26–35. doi: 10.1016/j.aeue.2018.06.042. DOI

Bao B., Wang N., Chen M., Xu Q., Wang J. Inductor-free simplified Chua’s circuit only using two-op-amp-based realization. Nonlinear Dyn. 2016;84:511–525. doi: 10.1007/s11071-015-2503-5. DOI

Ogorzalek M.J. Chaos and Complexity in Nonlinear Electronic Circuits. Volume 22 World Scientific; Singapore: 1997.

Gokyildirim A. Circuit Realization of the Fractional-Order Sprott K Chaotic System with Standard Components. Fractal Fract. 2023;7:470. doi: 10.3390/fractalfract7060470. DOI

Ahmad W.M., Sprott J.C. Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals. 2003;16:339–351. doi: 10.1016/S0960-0779(02)00438-1. DOI

Sprott J.C. Some simple chaotic flows. Phys. Rev. E. 1994;50:R647. doi: 10.1103/PhysRevE.50.R647. PubMed DOI

Diethelm K., Freed A.D. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forsch. Und Wiss. Rechn. 1998;1999:57–71.

Petrzela J. Fractional-order chaotic memory with wideband constant phase elements. Entropy. 2020;22:422. doi: 10.3390/e22040422. PubMed DOI PMC

Sene N. Study of a fractional-order chaotic system represented by the Caputo operator. Complexity. 2021;2021:5534872. doi: 10.1155/2021/5534872. DOI