0
Research Papers

Chaotic and Hyperchaotic Dynamics of a Modified Murali–Lakshmanan–Chua Circuit

[+] Author and Article Information
I. Manimehan

Department of Physics,
M. R. Government Arts College,
Mannargudi 614001, Tamil Nadu, India
e-mail: manimehan@gmail.com

M. Paul Asir

Department of Physics,
A.V.V.M Sri Pushpam College,
Poondi,
Thanjavur 613503, Tamil Nadu, India
e-mail: paulphy.asir@gmail.com

P. Philominathan

Department of Physics,
A.V.V.M Sri Pushpam College,
Poondi,
Thanjavur 613503, Tamil Nadu, India
e-mail: philominathan@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 6, 2018; final manuscript received January 9, 2019; published online February 15, 2019. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 14(5), 051001 (Feb 15, 2019) (9 pages) Paper No: CND-18-1394; doi: 10.1115/1.4042692 History: Received September 06, 2018; Revised January 09, 2019

The present study uncovers the hyperchaotic dynamical behavior of the famous Murali-Lakshmanan-Chua (MLC) circuit, when suitably modified. In the conventional MLC oscillator, an inductor is introduced in parallel between the nonlinear element and the capacitor. Many novel and interesting dynamical behaviors such as reverse period-3 doubling, torus breakdown to chaos and hyperchaos, etc., were observed. Characterization techniques includes spectrum of Lyapunov exponents, one parameter bifurcation diagram, recurrence quantification analysis, correlation dimension, etc., were employed to analyze the different dynamical regimes. Explicit analytical solution of the model is derived and the results are corroborated with the numerical outcomes.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Feki, M. , 2003, “An Adaptive Chaos Synchronization Scheme Applied to Secure Communication,” Chaos Solitons Fractals, 18(1), pp. 141–148. [CrossRef]
Cenys, A. , Tamasevicius, A. , Baziliauskas, A. , Krivickas, R. , and Lindberg, E. , 2003, “Hyperchaos in Coupled Colpitts Oscillators,” Chaos Solitons Fractals, 17, pp. 349–353. [CrossRef]
Li, C. , Liao, X. , and Wong, K. W. , 2005, “Lag Synchronization of Hyperchaos With Application to Secure Communications,” Chaos Solitons Fractals, 23(1), pp. 183–193. [CrossRef]
Vicente, R. , Mirasso, C. R. , and Fischer, I. , 2007, “Simultaneous Bidirectional Message Transmission in a Chaos-Based Communication Scheme,” Opt. Lett., 32(4), pp. 403–405. [CrossRef] [PubMed]
Rossler, O. E. , 1979, “An Equation for Hyperchaos,” Phys. Lett. A, 71(2–3), pp. 155–157. [CrossRef]
Matsumoto, T. , Chua, L. , and Kobayashi, K. , 1986, “Hyperchaos: Laboratory Experiment and Numerical Confirmation,” IEEE Trans. Circuits Syst., 33(11), pp. 1143–1147. [CrossRef]
Cafagna, D. , and Grassi, G. , 2003, “Hyperchaotic Coupled Chua's Circuits: An Approach for Generating New n × m-Scroll Attractors,” Int. J. Bifurcation Chaos, 13(9), pp. 2537–2550. [CrossRef]
Cafagna, D. , and Grassi, G. , 2003, “New 3D-Scroll Attractors in Hyperchaotic Chua's Circuits Forming a Ring,” Int. J. Bifurcation Chaos, 13(10), pp. 2889–2903. [CrossRef]
Li, Y. , Tang, W. K. S. , and Chen, G. , 2005, “Generating Hyperchaos Via State Feedback Controller,” Int. J. Bifurcation Chaos, 15(10), pp. 367–375.
Gao, T. G. , Chen, Z. Q. , Yuan, Z. , and Chen, G. , 2006, “A Hyperchaos Generated From Chen's System,” Int. J. Mod. Phys. C., 17(4), pp. 471–478. [CrossRef]
Chen, A. M. , Lu, J. , Lu, J. , and Yu, S. , 2006, “Generating Hyperchaotic Lu Attractor Via State Feedback Control,” Physica A, 364, pp. 103–110. [CrossRef]
Barboza, R. , 2007, “Dynamics of Hyperchaotic Lorenz System,” Int. J. Bifurcation Chaos, 17(12), pp. 4285–4294. [CrossRef]
Li, Y. , Chen, G. , and Tang, W. K. S. , 2005, “Controlling a Unified Chaotic System to Hyperchaotic,” IEEE Trans. Circuits Syst.-II, 52(4), pp. 204–207. [CrossRef]
Wu, X. , Lu, J. , Iu, H. H. C. , and Wong, S. C. , 2007, “Suppression and Generation of Chaos for a 3-Dimensional Autonomous System Using Parametric Perturbations,” Chaos Solitons Fractals, 31(4), pp. 811–819. [CrossRef]
Thamilmaran, K. , Lakshmanan, M. , and Venkatesan, A. , 2004, “Hyperchaos in Modified Canonical Chua's Circuit,” Int. J. Bifurcation Chaos, 14(1), pp. 221–243. [CrossRef]
Kennedy, M. P. , 1992, “Robust op Amp Realization of Chua's Circuit,” Frequenz, 46(3–4), pp. 66–80.
Golmankhaneh, A. K. , Arefi, R. , and Baleanu, D. , 2015, “Synchronization in a Nonidentical Fractional Order of a Proposed Modified System,” J. Vib. Control, 21(6), pp. 1154–1161. [CrossRef]
Golmankhaneh, A. K. , Arefi, R. , and Baleanu, D. , 2013, “The Proposed Modified Liu System With Fractional Order,” Adv. Math. Phys., 2013, p. 186037. [CrossRef]
Murali, K. , Lakshmanan, M. , and Chua, L. O. , 1994, “Bifurcation and Chaos in Simplest Dissipative Non-Autonomous Circuit,” Int. J. Bifurcation Chaos, 4(6), pp. 1511–1524. [CrossRef]
Inaba, N. , and Mori, S. , 1991, “Chaos Via Torus Breakdown in a Piecewise-Linear Forced Van-der-Pol Oscillator With a Diode,” IEEE Trans. Circuits Syst., 38(4), pp. 398–409. [CrossRef]
Thamilmaran, K. , and Lakshmanan, M. , 2002, “Classification of Bifurcations and Routes to Chaos in a Variant of Murali-Lakshmanan-Chua Circuit,” Int. J. Bifurcation Chaos, 12(4), pp. 783–813. [CrossRef]
Eckmann, J. P. , Kamphorst, S. O. , and Ruelle, D. , 1987, “Recurrence Plots of Dynamical Systems,” Europhys. Lett., 4(9), p. 973. [CrossRef]
Souza, E. G. , Viana, R. L. , and Lopes, S. R. , 2008, “Using Recurrences to Characterize the Hyperchaos-Chaos Transition,” Phys. Rev. E., 78(6 Pt. 2), p. 066206. [CrossRef]
Marwan, N. , Romano, M. C. , Thiel, M. , and Kurths, J. , 2007, “Recurrence Plots for the Analysis of Complex Systems,” Phys. Rep., 438(5–6), pp. 237–329. [CrossRef]
Grassberger, P. , and Procaccia, I. , 1983, “Characterization of Strange Attractors,” Phys. Rev. Lett., 50(5), p. 346. [CrossRef]
Kapitaniak, T. , Maistrenko, Y. , and Popovych, S. , 2000, “Chaos-Hyperchaos Transition,” Phys. Rev. E, 62(2 Pt. A), p. 1972. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Circuit realization of modified MLC circuit. Here NR is the nonlinear resistor (Chua's diode).

Grahic Jump Location
Fig. 2

Numerical phase portraits in the (xy2) plane: (a) period-1 limit cycle, f  = 0.6; (b) torus, f  = 0.52; (c) period-16 limit cycle, f  =0.47; (d) chaotic attractor, f  =0.45, and (e) hyperchaotic attractor, f  =0.3

Grahic Jump Location
Fig. 3

Numerical phase portraits of hyperchaotic attractor for f  =0.3 in different projection: (a) (xy1) plane, (b) (xy2) plane, (c) (x−sin(ωt)) plane, (d) (y1y2) plane, (e) (y1−sin(ωt)) plane, and (f) (y2−sin(ωt)) plane

Grahic Jump Location
Fig. 4

Hyperchaos: (a) one parameter bifurcation diagram in the (fx) plane and Lyapunov Exponent spectra diagram in the (b) (fλ1,2,4) plane (c) (fλ3) plane

Grahic Jump Location
Fig. 5

Expanded view of hyperchaotic region: (a) one parameter bifurcation diagram in the (fx) plane and Lyapunov exponent spectra diagram in the (b) (fλ1,2,4) plane (c) (fλ3) plane

Grahic Jump Location
Fig. 6

Poincaré maps for (a) chaotic attractor (b) hyperchaotic attractor

Grahic Jump Location
Fig. 7

Torus breakdown to chaos: (a) one parameter bifurcation diagram in the (fx) plane and (b) Lyapunov exponent spectra diagram in the (fλmax) plane

Grahic Jump Location
Fig. 8

Reverse period-3 doubling to chaos: (a) one parameter bifurcation diagram in the (fx) plane and (b) Lyapunov exponent spectra diagram in the (fλmax) plane

Grahic Jump Location
Fig. 9

(a) Recurrence plot for f  =  0.4, chaos (b) RP for f  =  0.36, hyperchaos (c) corresponding recurrence rates (These plots were computed with dimension m  =3, delay τ = 5, and threshold ε = 0.2)

Grahic Jump Location
Fig. 10

The correlation dimension (D2) of f =0.4, chaotic (filled circle) and f =0.36, hyperchaotic attractor (filled square) as a function of embedding dimension M

Grahic Jump Location
Fig. 11

(a) Phase portrait in xy1 plane of a chaotic attractor, f  =0.4 (b) hyperchaotic attractor, f = 0.36 (The lines are the nullclines of the model and open triangle represents the unstable node (repeller) fixed point)

Grahic Jump Location
Fig. 12

Analytical study—phase portraits in the (xy1) plane: (a) torus, f = 0.25; (b) period-7 limit cycle, f = 0.4; (c) chaotic attractor, f = 0.5; (d) period-4 limit cycle, f = 0.56, and (e) period-1 limit cycle, f = 0.7

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In