Abstract
Non-integer (fractional)-order systems are more realistic than integer-order systems, which mimic the real-time dynamics of various physical and biological systems. As a result, in this paper, we investigate the dynamical behavior of the Hartley oscillator with fractional-order JFETs (HFJ). To begin understanding the dynamical behavior of the HFJ, we perform stability of the equilibrium point analysis, and bifurcation analysis by finding local maxima of the state variables, and Lyapunov exponents. We discover that as a function of fractional order and voltage, considered HFJ exhibits the transition from periodic to chaotic dynamics via the period-doubling route. The multistability characteristics are investigated further using forward and backward bifurcation continuations, and we exemplify the presence of distinct attractors. Finally, we demonstrate the network behavior of an HFJ system, specifically, the transition to a coherent state as increasing the magnitude of fractional order or coupling strength.
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References
C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu-Batlle, Fractional-Order Systems and Controls: Fundamentals and Applications (Springer Science & Business Media, Berlin, 2010)
M.A. Jun, Chaos theory and applications: the physical evidence, mechanism are important in chaotic systems. Chaos Theory Appl. 4(1), 1–3 (2022)
M.M. Asheghan, S.S. Delshad, M.T.H. Beheshti, M.S. Tavazoei, Non-fragile control and synchronization of a new fractional order chaotic system. Appl. Math. Comput. 222, 712–721 (2013)
T.T. Hartley, C.F. Lorenzo, H.K. Qammer, Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 42(8), 485–490 (1995)
J.A. Machado, Fractional-Order Derivative Approximations in Discrete-Time Control Systems (SAMS, Bhubaneswar, 1998), pp.1–16
J.M. Muñoz-Pacheco, Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor. Eur. Phys. Journal . Spec. Top. 228(10), 2185–2196 (2019)
R.G. Li, H.N. Wu, Secure communication on fractional-order chaotic systems via adaptive sliding mode control with teaching-learning-feedback-based optimization. Nonlinear Dyn. 95(2), 1221–1243 (2019)
M. Amabili, P. Balasubramanian, I. Breslavsky, Anisotropic fractional viscoelastic constitutive models for human descending thoracic aortas. J. Mech. Behav. Biomed. Mater. 99, 186–197 (2019)
M.S. Tavazoei, Fractional order chaotic systems: history, achievements, applications, and future challenges. Eur. Phys. J. Spec. Top. 229(6), 887–904 (2020)
Z. Njitacke Tabekoueng, S. Shankar Muni, T. Fonzin Fozin, G. Dolvis Leutcho, J. Awrejcewicz, Coexistence of infinitely many patterns and their control in heterogeneous coupled neurons through a multistable memristive synapse. Chaos Interdiscip. J. Nonlinear Sci. 32(5), 053114 (2022)
S.S. Muni, H.O. Fatoyinbo, I. Ghosh, Dynamical effects of electromagnetic flux on Chialvo neuron map: nodal and network behaviors. Int. J. Bifurc. Chaos 32(09), 2230020 (2022)
D. Cafagna, G. Grassi, On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185–1197 (2012)
S.H. Weinberg, Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model. PLoS One 10(5), e0126629 (2015)
H.T. Yau, C.C. Wang, C.T. Hsieh, S.Y. Wu, Fractional order Sprott chaos synchronisation-based real-time extension power quality detection method. IET Gener. Transm. Distrib. 9(16), 2775–2781 (2015)
M.H. Wang, H.T. Yau, New power quality analysis method based on chaos synchronization and extension neural network. Energies 7(10), 6340–6357 (2014)
C.K. Chen, Y.C. Li, Machine chattering identification based on the fractional-order chaotic synchronization dynamic error. Int. J. Adv. Manuf. Technol. 100(1), 907–915 (2019)
K. Sathiyadevi, V.K. Chandrasekar, D.V. Senthilkumar, Inhomogeneous to homogeneous dynamical states through symmetry breaking dynamics. Nonlinear Dyn. 98(1), 327–340 (2019)
K. Sathiyadevi, V.K. Chandrasekar, M. Lakshmanan, Emerging chimera states under nonidentical counter-rotating oscillators. Phys. Rev. E 105(3), 034211 (2022)
K. Ponrasu, U. Singh, K. Sathiyadevi, D.V. Senthilkumar, V.K. Chandrasekar, Symmetry breaking dynamics induced by mean-field density and low-pass filter. Chaos Interdiscip. J. Nonlinear Sci. 30(5), 053120 (2020)
D. Premraj, K. Suresh, T. Banerjee, K. Thamilmaran, Bifurcation delay in a network of locally coupled slow-fast systems. Phys. Rev. E 98(2), 022206 (2018)
D. Premraj, K. Suresh, K. Thamilmaran, Effect of processing delay on bifurcation delay in a network of slow-fast oscillators. Chaos Interdiscip. J. Nonlinear Sci. 29(12), 123127 (2019)
A.T. Azar, S. Vaidyanathan, A. Ouannas (eds.), Fractional Order Control and Synchronization of Chaotic Systems, vol. 688 (Springer, Berlin, 2017)
S.K. Damarla, M. Kundu, Fractional Order Processes: Simulation, Identification, and Control (CRC Press, Boca Raton, 2018)
M.M. Al-sawalha, Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control. Adv. Differ. Equ. 2020(1), 1–17 (2020)
Y. Xu, H. Wang, Synchronization of fractional-order chaotic systems with Gaussian fluctuation by sliding mode control, in Abstract and Applied Analysis, vol. 2013 (Hindawi, London, 2013)
S.K. Agrawal, M. Srivastava, S. Das, Synchronization of fractional order chaotic systems using active control method. Chaos Solitons Fract. 45(6), 737–752 (2012)
P. Selvaraj, O.M. Kwon, S.H. Lee, R. Sakthivel, Cluster synchronization of fractional-order complex networks via uncertainty and disturbance estimator-based modified repetitive control. J. Frankl. Inst. 358(18), 9951–9974 (2021)
J. Zhou, Y. Zhao, Z. Wu, Cluster synchronization of fractional-order directed networks via intermittent pinning control. Phys. A Stat. Mech. Appl. 519, 22–33 (2019)
Y. Wang, Z. Wu, Cluster synchronization in fractional-order network with nondelay and delay coupling. Int. J. Mod. Phys. C 33(01), 2250006 (2022)
H. Fan, Y. Zhao, Cluster synchronization of fractional-order nonlinearly-coupling community networks with time-varying disturbances and multiple delays. IEEE Access 9, 60934–60945 (2021)
Z. Yaghoubi, Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller. Math. Comput. Simul. 172, 15–32 (2020)
P. Liu, Z. Zeng, J. Wang, Asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 4956–4967 (2020)
S.N. Mbouna, T. Banerjee, R. Yamapi, P. Woafo, Diverse chimera and symmetry-breaking patterns induced by fractional derivation effect in a network of Stuart–Landau oscillators. Chaos Solitons Fract. 157, 111945 (2022)
K. Rajagopal, A. Karthikeyan, S. Jafari, F. Parastesh, C. Volos, I. Hussain, Wave propagation and spiral wave formation in a Hindmarsh-Rose neuron model with fractional-order threshold memristor synaps. Int. J. Mod. Phys. B 34(17), 2050157 (2020)
K. Rajagopal, S. Panahi, M. Chen, S. Jafari, B. Bao, Suppressing spiral wave turbulence in a simple fractional-order discrete neuron map using impulse triggering. Fractals 29(08), 2140030 (2021)
K. Rajagopal, S. Jafari, C. Li, A. Karthikeyan, P. Duraisamy, Suppressing spiral waves in a lattice array of coupled neurons using delayed asymmetric synapse coupling. Chaos Solitons Fract. 146, 110855 (2021)
N. Korkmaz, İE. SAÇU, An efficient design procedure to implement the fractional-order chaotic jerk systems with the programmable analog platform. Chaos Theory Appl. 3(2), 59–66 (2021)
R. Tchitnga, P. Louodop, H. Fotsin, P. Woafo, A. Fomethe, Synchronization of simplest two-component Hartley’s chaotic circuits: influence of channel. Nonlinear Dyn. 74(4), 1065–1075 (2013)
R. Tchitnga, H.B. Fotsin, B. Nana, P.H.L. Fotso, P. Woafo, Hartley’s oscillator: the simplest chaotic two-component circuit. Chaos Solitons Fract. 45(3), 306–313 (2012)
M. Varan, A. Akgül, E. Güleryüz, K. Serbest, Synchronisation and circuit realisation of chaotic Hartley system. Zeitschrift für Naturforschung A 73(6), 521–531 (2018)
R. Kengne, R. Tchitnga, A.K.S. Tewa, G. Litak, A. Fomethe, C. Li, Fractional-order two-component oscillator: stability and network synchronization using a reduced number of control signals. Eur. Phys. J. B 91(12), 1–19 (2018)
K. Rajagopal, A. Karthikeyan, A. Srinivasan, Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components. Nonlinear Dyn. 91(3), 1491–1512 (2018)
K. Rajagopal, A. Karthikeyan, A. Srinivasan, Bifurcation and chaos in time delayed fractional order chaotic memfractor oscillator and its sliding mode synchronization with uncertainties. Chaos Solitons Fract. 103, 347–356 (2017)
K. Rajagopal, S.T. Kingni, A.J.M. Khalaf, Y. Shekofteh, F. Nazarimehr, Coexistence of attractors in a simple chaotic oscillator with fractional-order-memristor component: analysis, FPGA implementation, chaos control and synchronization. Eur. Phys. J. Spec. Top. 228(10), 2035–2051 (2019)
K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5(1), 1–6 (1997)
K. Diethelm, N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)
H.H. Sun, A. Abdelwahab, B. Onaral, Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Autom. Control 29(5), 441–444 (1984)
K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und wissenschaftliches Rechnen 1999, 57–71 (1998)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 16(3), 285–317 (1985)
K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)
M.F. Danca, Lyapunov exponents of a class of piecewise continuous systems of fractional order. Nonlinear Dyn. 81(1), 227–237 (2015)
S.S. Muni, A. Provata, Chimera states in ring-star network of Chua circuits. Nonlinear Dyn. 101(4), 2509–2521 (2020)
S.S. Muni, K. Rajagopal, A. Karthikeyan, S. Arun, Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling. Chaos Solitons Fract. 155, 111759 (2022)
K. Rajagopal, A. Karthikeyan, S. Jafari, F. Parastesh, C. Volos, I. Hussain, Wave propagation and spiral wave formation in a Hindmarsh-Rose neuron model with fractional-order threshold memristor synaps. Int. J. Mod. Phys. B 34(17), 2050157 (2020)
Y. Xu, J. Liu, W. Li, Quasi-synchronization of fractional-order multi-layer networks with mismatched parameters via delay-dependent impulsive feedback control. Neural Netw. 150, 43–57 (2022)
Acknowledgements
We gratefully acknowledge that this work is funded by the Center for Nonlinear Systems, Chennai Institute of Technology (CIT), India, vide funding number CIT/CNS/2023/RP-005.
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Kanagaraj, S., Muni, S.S., Karthikeyan, A. et al. A chaotic Hartley oscillator with fractional-order JFET and its network behaviors. Eur. Phys. J. Spec. Top. 232, 2539–2548 (2023). https://doi.org/10.1140/epjs/s11734-023-00940-3
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DOI: https://doi.org/10.1140/epjs/s11734-023-00940-3