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,
Thanjavur 613503, Tamil Nadu, India
e-mail: paulphy.asir@gmail.com

P. Philominathan

Department of Physics,
A.V.V.M Sri Pushpam College,
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.

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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

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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

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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



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