Analysis and electronic circuit implementation of an integer-and fractional-order Shimizu-Morioka system

Authors

  • François Kapche Tagne University of Dschang
  • Guillaume Honoré KOM University of Dschang
  • Marceline Motchongom Tingue The University of Bamenda
  • Pierre Kisito Talla University of Dschang
  • V. Kamdoum Tamba University of Dschang

DOI:

https://doi.org/10.31349/RevMexFis.67.061401

Keywords:

Shimizu-Morioka system, fractional-order derivative, bistable attractors, metastable chaos, spiking oscillations, electronic circuit implementation

Abstract

The dynamics of an integer-order and fractional-order Lorenz like system called Shimizu-Morioka system is investigated in this paper. It is shown thatinteger-order Shimizu-Morioka system displays bistable chaotic attractors, monostable chaotic attractors and coexistence between bistable and monostable chaotic attractors. For suitable choose of parameters, the fractional-order Shimizu-Morioka system exhibits bistable chaotic attractors, monostable chaotic attractors, metastable chaos (i.e. transient chaos) and spiking oscillations. The bifurcation structures reveal that the fractional-order derivative affects considerably the dynamics of Shimizu-Morioka system. The chain fractance circuit is used to designand implement the integer- and fractional-order Shimizu-Morioka system in Pspice. A close agreement is observed between PSpice based circuit simulations and numerical simulations analysis. The results obtained in this work were not reported previously in the interger as well as in fractional-order Shimizu-Morioka system and thus represent an important contribution which may help us in better understanding of the dynamical behavior of this class of systems.

References

H. Li, X. Liao, and M. Luo, A novel non-equilibrium fractionalorder chaotic system and its complete synchronization by circuit implementation, Nonlinear Dyn. 68 (2012) 137, https://doi.org/10.1007/s11071-011-0210-4.

A. K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectrics Press, London, 1983).

O. Heaviside, Electromagnetic Theory (Academic Press, New York, 1971)

R. L. Bagley and R. A. Calico, Fractional Order State Equations for the Control of Viscoelastically Damped Structures, J. Guid. Control Dyn. 14 (1991) 304, https://doi.org/10.2514/3.20641.

A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys. 35 (1996) 311, https://doi.org/10.1007/BF02083817.

D. Kusnezov, A. Bulgac, and G. D. Dang, Quantum Levy Processes and Fractional Kinetics, ´ Phys. Rev. Lett. 82 (1999) 1136, https://doi.org/10.1103/PhysRevLett.82.1136.

N. Laskin, Fractional market dynamics, Phys. A 287 (2000) 482, https://doi.org/10.1016/S0378-4371(00)00387-3.

M. Altaf Khan, S. Ullah, and S. Kumar, A robust study on 2019-nCOV outbreaks through non-singular derivative, Eur. Phys.

J. Plus 136 (2021) 168, https://doi.org/10.1140/epjp/s13360-021-01159-8.

Fatmawati, M. Altaf Khan, and H. Putra Odinsyah, Fractional model of HIV transmission with awareness effect, Chaos Solitons Fractals 138 (2020) 109967, https://doi.org/10.1016/j.chaos.2020.109967.

R. Jan, M. Altaf Khan, and J. F. Gomez-Aguilar, Asymptomatic carriers in transmission dynamics of dengue with control interventions, Optim. Control Appl. Methods 41 (2019) 430, https://doi.org/10.1002/oca.2551.

M. Altaf Khan, M. Ismail, S. Ullah, and M. Farhan, Fractional order SIR model with generalized incidence rate, AIMS Math. 5 (2020) 1856, https://doi.org/10.3934/math.2020124.

M. Altaf Khan and A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alex. Eng. J. 59 (2020) 2379, https://doi.org/10.1016/j.aej.2020.02.033.

S. Ullah, M. Altaf Khan, and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Solitons Fractals 116 (2018) 63, https://doi.org/10.1016/j.chaos.2018.09.001.

P. A. Naik, M. Yavuz, S. Qureshi, J. Zu, and S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivativs using real data from Pakistan, Eur. Phys. J. Plus 135 (2020) 795, https://doi.org/10.1140/epjp/s13360-020-00819-5.

X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing’s oscillators, Chaos Solitons Fractals 24 (2005) 1097, https://doi.org/10.1016/j.chaos.2004.09.090.

T. T. Hartley, C. F. Lorenzo, and H. Killory Qammer, Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst. I 42

(1995) 485, https://doi.org/10.1109/81.404062.

C. Li and G. Peng, Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals 22 (2004) 443, https://doi.org/10.1016/j.chaos.2004.02.013.

J. G. Lu, Chaotic dynamics of the fractional-order Lu system and its synchronization, Phys. Lett. A 354 (2006) 305, https://doi.org/10.1016/j.physleta.2006.01.068.

J. Wang and Y. Zhang, Designing synchronization schemes for chaotic fractional-order unified systems, Chaos Solitons

Fractals 30 (2006) 1265, https://doi.org/10.1016/j.chaos.2005.09.027.

I. Grigorenko and E. Grigorenko, Chaotic Dynamics of the Fractional Lorenz System, Phys. Rev. Lett. 91 (2003) 034101, https://doi.org/10.1103/PhysRevLett.91.034101.

C. Li and G. Chen, Chaos and hyperchaos in the fractionalorder Rossler equations, ¨ Phys. A 341 (2004) 55, https://doi.org/10.1016/j.physa.2004.04.113.

T. Shimizu and N. Morioka, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A 76 (1980) 201, https://doi.org/10.1016/0375-9601(80)90466-1.

M. M. El-Dessoky, M. T. Yassen, and E. S. Aly, Bifurcation analysis and chaos control in Shimizu-Morioka chaotic system with delayed feedback, Appl. Math. Comput. 243 (2014) 283, https://doi.org/10.1016/j.amc.2014.05.072.

M. Islam, B. Islam, and N. Islam, Chaos control in Shimizu Morioka system by Lie algebraic exact linearization, Int. J. Dyn. Control 2 (2014) 386, https://doi.org/10.1007/s40435-013-0051-8.

I. Podlubny, Fractional Differential Equation (Academic Press, San Diego, 1999).

R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, New Jersey, 2000), https://doi.org/10.1142/3779.

I. Petras, Fractional-Order Nonlinear Systems (SpringerVerlag, BerlinHeidelberg, 2011), https://doi.org/10.1007/978-3-642-18101-6.

I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).

C. F. Lorenzo and T. T. Hartley, Initialized fractional calculus, Int. J. Appl. Math. 3 (2000) 249.

C. F. Lorenzo and T. T. Hartley, Variable Order and Distributed Order Fractional Operators, Nonlinear Dyn. 29 (2002) 57, https://doi.org/10.1023/A:1016586905654.

I. Podlubny and A. M. A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science Preprint No. UEF 03-96, 1996.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D 16 (1985) 285, https://doi.org/10.1016/0167-2789(85)90011-9.

K. Diethelm, N. J. Ford, and A. D. Freed, A PredictorCorrector Approach for the Numerical Solution of Fractional

Differential Equations, Nonlinear Dyn. 29 (2002) 3, urlhttps://doi.org/10.1023/A:1016592219341.

M. S. Tavazoei and M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems, Phys.

Lett. A 367 (2007) 102, https://doi.org/10.1016/j.physleta.2007.05.081.

J. A. Yorke and E. D. Yorke, Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys. 21 (1979) 263, https://doi.org/10.1007/BF01011469.

A. Souza de Paula, M. Amorim Savi, and F. H. Iunes PereiraPinto, Chaos and transient chaos in an experimental nonlinear pendulum, J. Sound Vib. 294 (2006) 585, https://doi.org/10.1016/j.jsv.2005.11.015.

F. M. Izrailev, B. Timmermann, and W. Timmermann, Transient chaos in a generalized HA˜ °c non map on the torus, Phys.

Lett. A 126 (1988) 405, https://doi.org/10.1016/0375-9601(88)90801-8.

X.-S. Yang and Q. Yuan, Chaos and transient chaos in simple Hopfield neural networks, Neurocomputing 69 (2005) 232, https://doi.org/10.1016/j.neucom.2005.06.005.

V. Kamdoum Tamba, H. B. Fotsin, J. Kengne, E. B. Megam Ngouonkadi, and P. K. Talla, Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos, Int. J. Dyn. Control 5 (2017) 395, https://doi.org/10.1007/s40435-016-0223-4.

X. Han, B. Jiang, and Q. Bi, 3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/foldcycle bursting and their relation, Nonlinear Dyn. 61 (2010) 667, https://doi.org/10.1007/s11071-010-9678-6.

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT Press, Cambridge, 2007).

J. Rinzel, Bursting oscillations in an excitable membrane model, in Ordinary and Partial Differential Equations, edited by B. D. Sleeman and R. J. Jarvis (Springer, Berlin-Heidelberg, 1985), https://doi.org/10.1007/BFb0074739.

J. Rinzel, A Formal Classification of Bursting Mechanisms in Excitable Systems, in Mathematical Topics in Population Biology, edited by E. Teramoto and M. Yumaguti (Springer, Berlin-Heidelberg, 1987), https://doi.org/10.1007/978-3-642-93360-8 26.

E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, Int. J. Bifurcation Chaos 10 (2000) 1171, https://doi.org/10.1142/S0218127400000840.

J. Keener and J. Sneyd, Mathematical Physiology (Springer, New York, 1998), https://doi.org/10.1007/978-0-387-75847-3.

P. Gray, G. Nicolis, F. Baras, P. Borckmans, and S. K. Scott, Spatial Inhomogeneities and Transient Behaviour in Chemical Kinetics (Wiley, New York, 1992).

V. K. Vanag, L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, Oscillatory cluster patterns in a homogeneous chemical system with global feedback, Nature 406 (2000) 389, https://doi.org/10.1038/35019038.

P. Zhou and K. Huang, A new 4-D non-equilibrium fractional order chaotic system and its circuit implementation, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 2005, https://doi.org/10.1016/j.cnsns.2013.10.024.

Downloads

Published

2021-11-01

How to Cite

[1]
F. Kapche Tagne, G. H. KOM, M. Motchongom Tingue, . P. Kisito Talla, and V. Kamdoum Tamba, “Analysis and electronic circuit implementation of an integer-and fractional-order Shimizu-Morioka system”, Rev. Mex. Fís., vol. 67, no. 6 Nov-Dec, pp. 061401 1–, Nov. 2021.

Issue

Vol. 67 No. 6 Nov-Dec (2021): Revista Mexicana de Física

Section

14 Other areas in Physics

License

Copyright (c) 2021 François Kapche Tagne, Guillaume Honoré KOM, Marceline Motchongom Tingue, Pierre Kisito Talla, V. Kamdoum Tamba

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.