Abstract

In this study, we report an explicit analytical solution of state-controlled cellular neural network (SC-CNN) based second-order nonautonomous system. The proposed system is modeled with an aid of a generalized two-state-controlled cellular neural network (CNN) equations and experimentally realized by imposing a suitable connection of simple two-state-controlled generalized CNN cells following the report of Swathi et al. [2014]. The chaotic and quasi-periodic dynamics observed from this system have been investigated through an analytical approach for the first time. The intriguing dynamics observed from the system where further substantiated by phase portraits, Poincaré map, power spectra, and “01 test.” We trace the transition of the system from periodic to chaos through analytical solutions, which are in good agreement with hardware experiments. Additionally, we show PSpice circuit simulation results for validating our analytical and experimental studies.

References

References
1.
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
.10.1142/S0218127402004681
2.
Kingston
,
S. L.
, and
Thamilmaran
,
K.
,
2017
, “
Bursting Oscillations and Mixed-Mode Oscillations in Driven Liénard System
,”
Int. J. Bifurcation Chaos
,
27
(
7
), p.
1730025
.10.1142/S0218127417300257
3.
Li
,
Y.
,
Qian
,
H.
, and
Yi
,
Y.
,
2010
, “
Nonlinear Oscillations and Multiscale Dynamics in a Closed Chemical Reaction System
,”
J. Dyn. Differ. Equation
,
22
(
3
), pp.
491
507
.10.1007/s10884-010-9156-3
4.
Manimehan
,
I.
,
Thamilmaran
,
K.
, and
Philominathan
,
P.
,
2011
, “
Torus Breakdown to Chaos Via Periods Doubling Route in a Modified Canonical Chua's Circuit
,”
Int. J. Bifurcation Chaos
,
21
(
7
), p.
1987
.10.1142/S0218127411029586
5.
Thamilmaran
,
K.
,
Lakshmanan
,
M.
, and
Murali
,
K.
,
2000
, “
Rich Variety of Bifurcations and Chaos in a Variant of Murali-Lakshmanan-Chua Circuit
,”
Int. J. Bifurcation Chaos
,
10
(
7
), pp.
1781
1785
.10.1142/S0218127400001109
6.
Thamilmaran
,
K.
,
Senthilkumar
,
D. V.
,
Venkatesan
,
A.
, and
Lakshmanan
,
M.
,
2006
, “
Experimental Realization of Strange Nonchaotic Attractors in a Quasiperiodically Forced Electronic Circuit
,”
Phys. Rev. E
,
74
(
3
), p.
036205
.10.1103/PhysRevE.74.036205
7.
Kennedy
,
M. P.
,
1993
, “
Three Steps to Chaos—Part I: Evolution
,”
IEEE Trans. Circuits Syst.-I
,
40
(
10
), pp.
640
656
.10.1109/81.246140
8.
Kennedy
,
M. P.
,
1993
, “
Three Steps to Chaos—Part II: A Chua's Circuit Primer
,”
IEEE Trans. Circuits Syst.-I
,
40
(
10
), pp.
657
674
.10.1109/81.246141
9.
Lacy
,
J. G.
,
1996
, “
A Simple Piecewise-Linear Non-Autonomous Circuit With Chaotic Behavior
,”
Int. J. Bifurcation Chaos
,
6
(
11
), pp.
2097
2100
.10.1142/S0218127496001375
10.
Murali
,
K.
, and
Lakshmanan
,
M.
,
1996
,
Chaos in Nonlinear Oscillators: Controlling and Synchronization
,
World Scientific
,
Singapore
.
11.
Venkatesan
,
A.
, and
Lakshmanan
,
M.
,
1997
, “
Bifurcation and Chaos in the Double Well Duffing Van-Der-Pol Oscillator Numerical and Analytical Studies
,”
Phys. Rev. E
,
56
(
6
), p.
6321
.10.1103/PhysRevE.56.6321
12.
Venkatesh
,
P. R.
,
Venkatesan
,
A.
, and
Lakshmanan
,
M.
,
2016
, “
Analytical Treatment for Synchronizing Chaos Through Unidirectional Coupling and Implementation of Logic Gates
,”
Pramana J. Phys.
,
86
(
6
), pp.
1195
1207
.10.1007/s12043-016-1199-5
13.
Bhatnagar
,
G.
, and
Wu
,
Q. M. J.
,
2012
, “
Chaos-Based Security Solution for Fingerprint Data During Communication and Transmission
,”
IEEE Trans. Instrum. Meas.
,
61
(
4
), pp.
876
887
.10.1109/TIM.2011.2179330
14.
Shameem Banu
,
H.
, and
Sheik Uduman
,
P. S.
,
2018
, “
An Analytical Study of SC-CNN Based Quasiperiodically Forced MLC Circuit
,”
Int. J. Pure Appl. Math.
,
119
(
13
), pp.
163
171
.
15.
Ding
,
M.
, and
Scott Kelso
,
J. A.
,
1994
, “
Phase-Resetting Map and the Dynamics of Quasiperiodically Forced Biological Oscillators
,”
Int. J. Bifurcation Chaos
,
4
(
3
), pp.
553
567
.10.1142/S0218127494000393
16.
Srinivasan
,
K.
,
Chandrasekar
,
V. K.
,
Gladwin Pradeep
,
R.
,
Murali
,
K.
, and
Lakshmanan
,
M.
,
2016
, “
Different Types of Synchronization in Coupled Network Based Chaotic Circuits
,”
Commun. Nonlinear Sci. Numer. Simul.
,
39
, pp.
156
168
.10.1016/j.cnsns.2016.03.002
17.
Yang
,
T.
, and
Chua
,
L. O.
,
2002
, “
Secure Communication Via Chaotic Parameter Modulation
,”
IEEE Trans. Circuits Syst.-II
,
43
(
9
), p.
817
.10.1109/81.536758
18.
Chua
,
L. O.
, and
Roska
,
T.
,
1993
, “
The CNN Paradigm
,”
IEEE Trans. Circuits Syst.-I
,
40
(
3
), pp.
147
156
.10.1109/81.222795
19.
Chua
,
L. O.
, and
Yang
,
L.
,
1988
, “
Cellular Neural Networks: Theory
,”
IEEE Trans. Circuits Syst.
,
35
(
10
), pp.
1257
1272
.10.1109/31.7600
20.
Arena
,
P.
,
Caponetto
,
R.
,
Fortuna
,
L.
, and
Manganaro
,
G.
,
1997
, “
Cellular Neural Network to Explore Complexity
,”
Soft Comput.
,
1
(
3
), pp.
120
136
.10.1007/s005000050013
21.
Chua
,
L. O.
,
1998
, “
CNN: A Paradigm for Complexity
,”
World Scientific
,
Singapore
.
22.
Arena
,
P.
,
Baglio
,
S.
,
Fortuna
,
L.
, and
Manganaro
,
G.
,
1995
, “
Chua's Circuit Can Be Generated by CNN Cells
,”
IEEE Trans. Circuits Syst.-I
,
42
(
2
), pp.
123
125
.10.1109/81.372854
23.
Arena
,
P.
,
Baglio
,
S.
,
Fortuna
,
L.
, and
Manganaro
,
G.
,
1995
, “
Hyperchaos From Cellular Neural Networks
,”
Electron. Lett.
,
31
(
4
), pp.
250
251
.10.1049/el:19950189
24.
Arena
,
P.
,
Baglio
,
S.
,
Fortuna
,
L.
, and
Manganaro
,
G.
,
1996
, “
How State Controlled CNN Cells Generate the Dynamics of the Colpitts-Like Oscillator
,”
IEEE Trans. Circuits Syst.-I
,
43
(
7
), pp.
602
605
.10.1109/81.508183
25.
Li
,
Z.
,
Ma
,
M.
,
Wang
,
M.
, and
Zeng
,
Y.
,
2017
, “
Realization of Current-Mode SC-CNN Based Chua's Circuit
,”
Int. J. Electron. Commun.
,
71
, pp.
21
29
.10.1016/j.aeue.2016.11.003
26.
Kilic
,
R.
,
Alci
,
M.
, and
Gunay
,
E.
,
2004
, “
A SC-CNN Based Chaotic Masking System With Feedback
,”
Int. J. Bifurcation Chaos
,
14
(
1
), pp.
245
256
.10.1142/S0218127404009120
27.
Buscarino
,
A.
,
Fortuna
,
L.
, and
Frasca
,
M.
,
2009
, “
A New CNN-Based Chaotic Circuit: Experimental Results
,”
Int. J. Bifurcation Chaos
,
19
(
8
), pp.
2609
2617
.10.1142/S0218127409024359
28.
Gunay
,
E.
,
2010
, “
MLC Circuit in the Frame of CNN
,”
Int. J. Bifurcation Chaos
,
20
, pp.
3267
3274
.10.1142/S0218127410027659
29.
Gunay
,
E.
,
2012
, “
A New Autonomous Chaos Generator From State Controlled Cellular Neural Networks
,”
Int. J. Bifurcation Chaos
,
22
, p.
1250069
.10.1142/S0218127412500691
30.
Swathy
,
P. S.
, and
Thamilmaran
,
K.
,
2013
, “
An Experimental Study on SC-CNN Based Canonical Chua's Circuit
,”
Nonlinear Dyn.
,
71
(
3
), pp.
505
514
.10.1007/s11071-012-0677-7
31.
Swathy
,
P. S.
, and
Thamilmaran
,
K.
,
2014
, “
Dynamics of SC-CNN Based Variant of MLC Circuit: An Experimental Study
,”
Int. J. Bifurcation Chaos
,
24
(
2
), p.
1430008
.10.1142/S0218127414300080
32.
Gottwald
,
G. A.
, and
Melbourne
,
I.
,
2009
, “
On the Validity of the 0-1 Test for Chaos
,”
Nonlinearity
,
22
(
6
), pp.
1367
1382
.10.1088/0951-7715/22/6/006
33.
Inaba
,
N.
, and
Mori
,
S.
,
1991
, “
Chaos Via Torus Break Down in a Piecewise-Linear Forced Van Der Pol Oscillator With a Diode
,”
IEEE Trans. Circuits Syst.-I
,
38
(
4
), pp.
398
409
.10.1109/31.75393
34.
Tuinenga
,
P. W.
,
1988
,
SPICE: A Guide to Circuit Simulation and Analysis Using PSpice
,”
Prentice Hall
,
Upper Saddle River, NJ
.
35.
Chen
,
L.
,
Wu
,
R.
,
Chu
,
Z.
,
Yigang
,
H.
, and
Yin
,
L.
,
2017
, “
Pinning Synchronization of Fractional-Order Delayed Complex Networks With Non-Delayed and Delayed Couplings
,”
Int. J. Control
,
90
(
6
), pp.
1245
1255
.10.1080/00207179.2016.1278268
36.
Yang
,
J. H.
,
Sanjuan
,
M. A. F.
,
Liu
,
H. G.
, and
Cheng
,
G.
,
2015
, “
Bifurcation Transition and Nonlinear Response in a Fractional-Order System
,”
J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061017
.10.1115/1.4029512
You do not currently have access to this content.