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Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (UR-MACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

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Balamurali Ramakrishnan

Center for Nonlinear Systems, Chennai Institute of Technology, India

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Adélaïde Nicole Kengnou Telem

Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon

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Karthikeyan Rajagopal

Center for Nonlinear Systems, Chennai Institute of Technology, India

Department of Electronics and Communications Engineering and University Center for Research and Development, Charligarh University, Mohali, 10413, Punjab, India

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, and

Jacques Kengne

Laboratoire d’Automatique et Informatique Appliquée (LAIA), IUT-FV Bandjoun, University of Dschang, Cameroon

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https://doi.org/10.1142/S021812742250211XCited by:9 (Source: Crossref)

Abstract

It is known that the unidirectional coupling between a double-scroll system (drive) in the oscillation regime with another system (response) in the steady-state, gives rise to more complex dynamics such as multiscroll attractors [Dana et al., 2008]. This paper shows the way to move from n-scroll to 2n-scroll in a bidirectional coupling scheme between two jerk oscillators. The coupling considered can be interpreted as a perturbation to each oscillator’s amplitude through a signal proportional to the amplitude of the other. Indeed, starting from a situation of two oscillators which can generate double-scroll dynamics individually, we perform a bidirectional coupling of both oscillators which results in more complex dynamics such as multistability and 4-scroll chaotic attractor in the coupled system (not observed in an isolated oscillator). The underlined coupling scheme provides an efficient method for generating higher-order multiscroll attractors. The route to the 4-scroll chaotic attractor is explained with the help of bifurcation diagrams, plots of several phase portraits, and basins of attractions. Indirect validation of the theoretical analysis is carried out by using the PSpice simulation software.

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References

Ahmad, W. M. [2005] “ Generation and control of multi-scroll chaotic attractors in fractional order systems,” Chaos Solit. Fract. 25, 727–735. Crossref, Web of Science, Google Scholar
Ahmadi, A. et al. [2019] “ Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation,” Adv. Mech. Engin. 11, 1687814019888046. Crossref, Web of Science, Google Scholar
Argyris, J. H., Faust, G. & Haase, M. [1994] An Exploration of Chaos: An Introduction for Natural Scientists and Engineers (North Holland). Google Scholar
Bao, B. et al. [2015] “ Self-excited and hidden attractors found simultaneously in a modified Chua’s circuit,” Int. J. Bifurcation and Chaos 25, 1550075-1–10. Link, Web of Science, Google Scholar
Bao, B. et al. [2016] “ Coexisting infinitely many attractors in active band-pass filter-based memristive circuit,” Nonlin. Dyn. 86, 1711–1723. Crossref, Web of Science, Google Scholar
Campos-Cantón, E. et al. [2010] “ Multiscroll attractors by switching systems,” Chaos 20, 013116. Crossref, Web of Science, Google Scholar
Campos-Cantón, E. [2016] “ Chaotic attractors based on unstable dissipative systems via third-order differential equation,” Int. J. Mod. Phys. C 27, 1650008. Link, Web of Science, Google Scholar
Carroll, T. L. & Pecora, L. M. [1995] “ Synchronizing chaotic circuits,” Nonlinear Dynamics in Circuits (World Scientific), pp. 215–248. Link, Google Scholar
Chen, M. et al. [2015] “ Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit,” Nonlin. Dyn. 81, 215–226. Crossref, Web of Science, Google Scholar
Chen, L. et al. [2020] “ Chaos in fractional-order discrete neural networks with application to image encryption,” Neural Networks 125, 174–184. Crossref, Web of Science, Google Scholar
Chua, L., Komuro, M. & Matsumoto, T. [1986] “ The double scroll family,” IEEE Trans. Circuits Syst. 33, 1072–1118. Crossref, Google Scholar
Chunxia, L. et al. [2012] “ Research on the multi-scroll chaos generation based on Jerk mode,” Procedia Engin. 29, 957–961. Crossref, Google Scholar
Dana, S. K. et al. [2008] “ Multiscroll in coupled double scroll type oscillators,” Int. J. Bifurcation and Chaos, 18, 2965–2980. Link, Web of Science, Google Scholar
Deng, Q. & Wang, C. [2019] “ Multi-scroll hidden attractors with two stable equilibrium points,” Chaos 29, 093112. Crossref, Web of Science, Google Scholar
Escalante-González, R. D. J. & Campos-Cantón, E. [2018] “ A class of piecewise linear systems without equilibria with 3-D grid multiscroll chaotic attractors,” IEEE Trans. Circuits Syst.-II: Express Briefs 66, 1456–1460. Crossref, Web of Science, Google Scholar
Escalante-González, R. & Campos, E. [2021a] “ Multistable systems with nested hidden and self-excited double scroll attractors,” European Phys. J. Special Topics 231, 1–7. Web of Science, Google Scholar
Escalante-González, R. D. J. & Campos, E. [2021b] “ Emergence of hidden attractors through the rupture of heteroclinic-like orbits of switched systems with self-excited attractors,” Complexity 2021, 1–24. Crossref, Web of Science, Google Scholar
Faghani, Z. et al. [2019] “ Simple chaotic systems with specific analytical solutions,” Int. J. Bifurcation and Chaos 29, 1950116-1–11. Link, Web of Science, Google Scholar
García-Martínez, M. et al. [2015] “ Hyperchaotic encryption based on multi-scroll piecewise linear systems,” Appl. Math. Comput. 270, 413–424. Crossref, Web of Science, Google Scholar
Guan, Z.-H., Huang, F. & Guan, W. [2005] “ Chaos-based image encryption algorithm,” Phys. Lett. A 346, 153–157. Crossref, Web of Science, Google Scholar
Güunay, E. & Kiliç, R. [2011] “ A new way of generating N-scroll attractors via trigonometric function,” Int. J. Bifurcation and Chaos 21, 897–901. Link, Web of Science, Google Scholar
Han, F. et al. [2007] “ Fingerprint images encryption via multi-scroll chaotic attractors,” Appl. Math. Comput. 185, 931–939. Crossref, Web of Science, Google Scholar
Huang, X., Ye, G. & Wong, K. -W. [2013] “ Chaotic image encryption algorithm based on circulant operation,” Abstract and Applied Analysis (Hindawi). Google Scholar
Kengne, J., Njitacke, Z. & Fotsin, H. [2016] “ Dynamical analysis of a simple autonomous jerk system with multiple attractors,” Nonlin. Dyn. 83, 751–765. Crossref, Web of Science, Google Scholar
Kengne, J. & Mogue, R. L. T. [2019] “ Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: Chaos, multi-scroll, and multiple coexisting attractors,” Int. J. Dyn. Contr. 7, 112–133. Crossref, Google Scholar
Kengne, J. et al. [2019] “ Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting,” Chaos Solit. Fract. 121, 63–84. Crossref, Web of Science, Google Scholar
Kengne, L. K., Kengne, J. & Pone, J. R. M. [2020] “ Coexisting bubbles, multiple attractors, and control of multistability in a simple jerk system under the influence of a constant excitation force,” Pramana 94, 1–25. Crossref, Web of Science, Google Scholar
Kengne, J. et al. [2021] “ A simple anti-parallel diodes based chaotic jerk circuit with arcsinh function: Theoretical analysis and experimental verification,” Anal. Integr. Circuits Sign. Process. 108, 597–623. Crossref, Web of Science, Google Scholar
Kuznetsov, Y. A. [1998] “ Elements of applied bifurcation theory,” Appl. Math. Sci. 112, 591. Google Scholar
Leonov, G. A. & Kuznetsov, N. V. [2013] “ Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits,” Int. J. Bifurcation and Chaos 23, 1330002-1–69. Link, Web of Science, Google Scholar
Lü, J. et al. [2004a] “ Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method,” Automatica 40, 1677–1687. Crossref, Web of Science, Google Scholar
Lü, J. et al. [2004b] “ Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Trans. Circuits Syst.-I: Regular Papers 51, 2476–2490. Google Scholar
Lü, J. et al. [2008] “ Generating multi-scroll chaotic attractors by thresholding,” Phys. Lett. A 372, 3234–3239. Crossref, Web of Science, Google Scholar
Nayfeh, A. H. & Balachandran, B. [2008] Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (John Wiley & Sons). Google Scholar
Nguenjou, L. N. et al. [2019] “ A window of multistability in Genesio–Tesi chaotic system, synchronization and application for securing information,” AEU — Int. J. Electron. Commun. 99, 201–214. Crossref, Web of Science, Google Scholar
Nguimdo, R. M. et al. [2012] “ Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Expr. 20, 28603–28613. Crossref, Web of Science, Google Scholar
Nicolis, J. S. [1991] Chaos and Information Processing (World Scientific). Link, Google Scholar
Pano-Azucena, A. D. et al. [2017] “ Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators,” Nonlin. Dyn. 87, 2203–2217. Crossref, Web of Science, Google Scholar
Pastor, I. et al. [1993] “ Ordered and chaotic behavior of two coupled van der Pol oscillators,” Phys. Rev. E 48, 171. Crossref, Web of Science, Google Scholar
Prakash, P. et al. [2018] “ Megastability, multistability in a periodically forced conservative and dissipative system with signum nonlinearity,” Int. J. Bifurcation and Chaos 28, 1830030-1–16. Link, Web of Science, Google Scholar
Pulido-Luna, J. R. et al. [2021] “ A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator,” Integration 81, 34–42. Crossref, Web of Science, Google Scholar
Rajagopal, K. et al. [2018] “ A novel parametrically controlled multi-scroll chaotic attractor along with electronic circuit design,” Europ. Phys. J. Plus 133, 1–8. Crossref, Web of Science, Google Scholar
Ren, X. et al. [2021] “ Parameter and initial offset boosting dynamics in two-memristor-based Colpitts system,” Europ. Phys. J. Special Topics 230, 1709–1721. Crossref, Web of Science, Google Scholar
Sánchez-López, C. et al. [2010] “ N-scroll chaotic attrac- tors from saturated function series employing CCII+s,” Nonlin. Dyn. 61, 331–341. Crossref, Web of Science, Google Scholar
Sprott, J. C. et al. [2017] “ Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping,” Europ. Phys. J. Special Topics 226, 1979–1985. Crossref, Web of Science, Google Scholar
Strogatz, S. H. [1994] Nonlinear Dynamics and Chaos: Lab Demonstrations. Google Scholar
Strogatz, S. H. [2018] Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press). Crossref, Google Scholar
Tamba, V. K. et al. [2021] “ Hidden extreme multistability generated from a novel memristive two-scroll chaotic system,” Mem-Elements for Neuromorphic Circuits with Artificial Intelligence Applications (Elsevier), pp. 147–164. Crossref, Google Scholar
Tolba, M. F. et al. [2017] “ FPGA implementation of two fractional order chaotic systems,” AEU — Int. J. Electron. Commun. 78, 162–172. Crossref, Web of Science, Google Scholar
Tuna, M. et al. [2019] “ Hyperjerk multiscroll oscillators with megastability: Analysis, FPGA implementation and a novel ANN-ring-based true random number generator,” AEU — Int. J. Electron. Commun. 112, 152941. Crossref, Web of Science, Google Scholar
Wang, S. et al. [2002] “ Chaos-based secure communications in a large community,” Phys. Rev. E 66, 065202. Crossref, Web of Science, Google Scholar
Wang, F.-Q. & Liu, C.-X. [2008] “ Generation of multi-scroll chaotic attractors via the saw-tooth function,” Int. J. Mod. Phys. B 22, 2399–2405. Link, Web of Science, Google Scholar
Wang, C., Liu, X. & Xia, H. [2017] “ Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system,” Chaos 27, 033114. Crossref, Web of Science, Google Scholar
Wang, N. et al. [2019] “Generating multi-scroll Chua’s attractors via simplified piecewise-linear Chua’s diode,” IEEE Trans. Circuits Syst.-I: Regular Papers 66, 4767–4779. Google Scholar
Wang, N., Zhang, G. & Li, H. [2020] “ Parametric control for multi-scroll attractor generation via nested sine-PWL function,” IEEE Trans. Circuits Syst.-II: Express Briefs 68, 1033–1037. Crossref, Web of Science, Google Scholar
Wolf, A. et al. [1985] “ Determining Lyapunov exponents from a time series,” Physica D 16, 285–317. Crossref, Web of Science, Google Scholar
Xie, Q., Chen, G. & Bollt, E. M. [2002] “ Hybrid chaos synchronization and its application in information processing,” Math. Comput. Model. 35, 145–163. Crossref, Web of Science, Google Scholar
Xu, F., Yu, P. & Liao, X. [2013] “ Synchronization and stabilization of multi-scroll integer and fractional order chaotic attractors generated using trigonometric functions,” Int. J. Bifurcation and Chaos 23, 1350145-1–33. Link, Web of Science, Google Scholar
Yu, S. et al. [2006] “ A general multiscroll Lorenz system family and its realization via digital signal processors,” Chaos 16, 033126. Crossref, Web of Science, Google Scholar
Yu, S., Tang, W. K. & Chen, G. [2007] “ Generation of $n \times m$ $n \times m$ -scroll attractors under a Chua-circuit framework,” Int. J. Bifurcation and Chaos 17, 3951–3964. Link, Web of Science, Google Scholar
Yu, S. et al. [2008] “ Multi-wing butterfly attractors from the modified Lorenz systems,” 2008 IEEE Int. Symp. Circuits and Systems (IEEE), pp. 768–771. Google Scholar
Yu, S. et al. [2010] “ Design and implementation of multi-wing butterfly chaotic attractors via Lorenz-type systems,” Int. J. Bifurcation and Chaos 20, 29–41. Link, Web of Science, Google Scholar
Yu, F. et al. [2021] “ A new multi-scroll Chua’s circuit with composite hyperbolic tangent-cubic nonlinearity: Complex dynamics, hardware implementation and image encryption application,” Integration 81, 71–83. Crossref, Web of Science, Google Scholar
Zhang, K. et al. [2021] “ A novel megastable oscillator with a strange structure of coexisting attractors: Design, analysis, and FPGA implementation,” Complexity 2021. Web of Science, Google Scholar

Vol. 32, No. 14

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Received 11 June 2022

Revised 15 August 2022

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Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

Abstract

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Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

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