In this paper, we designed multiple control inputs for Rucklidge oscillator through sliding, adaptive, and backstepping control techniques. Dynamical stability through Lyapunov theory is discussed to check whether the above‐mentioned nonlinear dynamical system is stable or not for defined controller. Based on error dynamics, adaptive and sliding control techniques are used such that solution approaches to its stable state with time. Furthermore, simulation results of nonlinear Rucklidge system are included in this paper to confirm controlled results and to analyze applied techniques. A brief analysis of these techniques for considered dynamical system is an integral part of the paper.

References

References
1.
Volos
,
C. K.
,
Pham
,
V.-T.
,
Vaidyanathan
,
S.
,
Kyprianidis
,
I. M.
, and
Stouboulos
,
I. N.
,
2015
, “
Synchronization Phenomena in Coupled Colpitts Circuits
,”
J. Eng. Sci. Technol. Rev.
,
8
(
2
), pp.
142
151
.
2.
Vaidyanathan
,
S.
,
2015
, “
Adaptive Control of a Chemical Chaotic Reactor
,”
Int. J. PharmTech. Res.
,
8
(
3
), pp.
377
382
.https://www.researchgate.net/publication/280012935_Adaptive_Control_of_a_Chemical_Chaotic_Reactor
3.
Vaidyanathan
,
S.
,
2015
, “
Dynamics and Control of Brusselator Chemical Reaction
,”
Int. J. ChemTech Res.
,
8
(
6
), pp.
740
749
.https://www.researchgate.net/publication/281649757_Dynamics_and_Control_of_Brusselator_Chemical_Reaction
4.
Vaidyanathan
,
S.
,
2015
, “
Anti-Synchronization of Brusselator Chemical Reaction Systems Via Adaptive Control
,”
Int. J. ChemTech Res.
,
8
(
6
), pp.
759
768
.https://www.researchgate.net/publication/281649585_Anti-Synchronization_of_Brusselator_Chemical_Reaction_Systems_via_Adaptive_Control
5.
Vaidyanathan
,
S.
,
2015
, “
Dynamics and Control of Tokamak System With Symmetric and Magnetically Confined Plasma
,”
Int. J. ChemTech Res.
,
8
(
6
), pp.
795
803
.https://www.researchgate.net/publication/281649670_Dynamics_and_Control_of_Tokamak_System_with_Symmetric_and_Magnetically_Confined_Plasma
6.
Garfinkel
,
A.
,
Spano
,
M. L.
,
Ditto
,
W. L.
, and
Weiss
,
J. N.
,
1992
, “
Controlling Cardiac Chaos
,”
Science
,
257
(
5074
), pp.
1230
1235
.
7.
May
,
R. M.
,
1976
, “
Simple Mathematical Models With Very Complicated Dynamics
,”
Nature
,
261
, pp.
259
267
.
8.
Vaidyanathan
,
S.
,
2015
, “
Adaptive Backstepping Control of Enzymes-Substrates System With Ferroelectric Behaviour in Brain Waves
,”
Int. J. PharmTech Res.
,
8
(2), pp.
256
261
.https://www.researchgate.net/publication/278016300_Adaptive_Backstepping_Control_of_Enzymes-Substrates_System_with_Ferroelectric_Behaviour_in_Brain_Waves
9.
Patra
,
A. K.
, and
Rout
,
P. K.
,
2018
, “
Backstepping Sliding Mode Gaussian Insulin Injection Control for Blood Glucose Regulation in Type I Diabetes Patient
,”
ASME J. Dyn. Syst. Meas. Control
,
140
(9), p.
091006
.
10.
Vaidyanathan
,
S.
,
2015
, “
3-Cells Cellular Neural Network (CNN) Attractor and Its Adaptive Biological Control
,”
Int. J. PharmTech Res.
,
8
(4), pp.
632
640
.https://www.researchgate.net/publication/280879428_3-Cells_Cellular_Neural_Network_CNN_Attractor_and_its_Adaptive_Biological_Control
11.
Strogatz
,
S. H.
,
2014
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
,
Westview Press
,
Cambridge, MA
.
12.
Ott
,
E.
,
Grebogi
,
C.
, and
Yorke
,
J.
,
1990
, “
Using Chaos to Direct Trajectories to Targets
,”
Phys. Rev. Lett.
,
65
(26), pp.
349
354
.
13.
Shahzad
,
M.
,
2016
, “
Chaos Control in Three Dimensional Cancer Model by State Space Exact Linearization Based on Lie Algebra
,”
Mathematics
,
4
(2), p.
33
.
14.
Ding
,
X.
, and
Sinha
,
A.
,
2016
, “
Hydropower Plant Frequency Control Via Feedback Linearization and Sliding Mode Control
,”
ASME J. Dyn. Syst. Meas. Control
,
138
(
7
), p.
074501
.
15.
Slotine
,
J. E.
, and
Li
,
W.
,
1991
,
Adaptive Nonlinear Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
16.
Khalil
,
H. K.
,
1992
,
Nonlinear Systems
,
Macmillan Publishing
,
New York
.
17.
Narendra
,
K. S.
, and
Annaswamy
,
A. M.
,
1989
,
Stable Adaptive Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
18.
Chen
,
S.
, and
Lu
,
J.
,
2002
, “
Synchronization of an Uncertain Unified System Via Adaptive Control
,”
Chaos, Solitons Fractals
,
14
, pp.
643
647
.
19.
Kadu
,
C. B.
,
Khandekar
,
A. A.
, and
Patil
,
C. Y.
,
2018
, “
Design of Sliding Mode Controller With Proportional Integral Sliding Surface for Robust Regulation and Tracking of Process Control Systems
,”
ASME J. Dyn. Syst. Meas. Control
,
140
(
9
), p.
091004
.
20.
Jamal
,
M. N.
, and
Ammar
,
N. N.
,
2007
, “
Chaos Control Using Sliding-Mode Theory
,”
Chaos, Solitons Fractals
,
33
, pp.
695
702
.
21.
Chang
,
J. F.
,
Hung
,
M. L.
,
Yang
,
Y.-S.
,
Liao
,
J.-L.
, and
Yan
,
J.-J.
,
2008
, “
Controlling Chaos of the Family of Rossler Systems Using Sliding Mode Control
,”
Chaos Solitons Fractals
,
37
(2), pp.
609
622
.
22.
Yang
,
S. J.
,
Li
,
C. D.
, and
Huang
,
T. W.
,
2014
, “
Impulsive Control and Synchronization of Memristor-Based Chaotic Circuits
,”
Int. J. Bifurcation Chaos
,
24
(
12
), pp.
145
162
.
23.
Zheng
,
Y. G.
, and
Ji
,
Z. L.
,
2016
, “
Predictive Control of Fractional-Order Chaotic Systems
,”
Chaos, Solitons Fractals
,
87
, pp.
307
313
.
24.
Kocamaz
,
U. E.
,
Cevher
,
B.
, and
Uyaroglu
,
Y.
,
2017
, “
Control and Synchronization of Chaos With Sliding Mode Control Based on Cubic Reaching Rule
,”
Chaos, Solitons Fractals
,
105
, pp.
92
98
.
25.
Ablay
,
G.
,
2009
, “
Sliding Mode Control of Uncertain Unified Chaotic Systems
,”
Nonlinear Anal. Hybrid Syst.
,
3
(
4
), pp.
531
535
.
26.
Irfan
,
S.
,
Mehmood
,
A.
,
Tayyab
,
M.
, and
Iqbal
,
J.
,
2018
, “
Advanced Sliding Mode Control Techniques for Inverted Pendulum: Modeling and Simulation
,”
Int. J. Eng. Sci. Technol.
,
21
(
4
), pp.
753
759
.
27.
Bianconi
,
E.
,
Calvente
,
J.
,
Giral
,
P.
,
Petrone
,
G.
,
Paja
,
C. R.
,
Spgnulo
,
G.
, and
Vitelli
,
M.
,
2013
, “
A Fast Current Based MPPT Technique Employing Sliding Mode Control
,”
IEEE Trans. Ind. Electron.
,
60
(3), pp.
1168
1178
.
28.
Yang
,
S. K.
,
Chen
,
C. L.
, and
Yau
,
H. T.
,
2002
, “
Control of Chaos in Lorenz System
,”
Chaos, Solitons Fractals
,
13
, pp.
767
780
.
29.
Yau
,
H. T.
,
Chen
,
C. K.
, and
Chen
,
C. L.
,
2000
, “
Sliding Mode Control of Chaotic Systems With Uncertainties
,”
Int. J. Bifurcation Chaos
,
10
, pp.
1139
1147
.
30.
Wang
,
H.
,
Han
,
Z.
,
Xie
,
Q.
, and
Zhang
,
W.
,
2009
, “
Sliding Mode Control of Chaotic Systems Based on LMI
,”
Int. J. Commun. Nat. Sci. Simul.
,
14
, pp.
1410
1417
.
31.
Zeng
,
Y.
, and
Singh
,
S. N.
,
1998
, “
Adaptive Control in Lorenz System
,”
Int. J. Dyn. Control
,
8
, pp.
255
267
.
32.
Yang
.
T.
,
Yang
,
C. M.
, and
Yang
,
L. B.
,
1998
, “
A Detailed Study of Adaptive Control of Chaotic Systems With Unknown Parameters
,”
Dyn. Control
,
8
(
3
) pp.
255
267
.
33.
Sundarpandian
,
V.
, and
Karthikeyan
,
R.
,
2011
, “
Anti-Synchornization of Hyperchaotic Lorenz and Hyperchaotic Chen System by Adaptive Control
,”
Int. J. Syst. Signal Control Eng. Appl.
,
4
(
2
), pp.
18
25
.
34.
Yau
,
H. T.
,
2004
, “
Design of Adaptive Sliding Mode Controller for Chaos Synchronization With Uncertainties
,”
Chaos, Solitons Fractals
,
22
, pp.
341
347
.
35.
,
J.
, and
Zhang
,
S.
,
2001
, “
Controlling Chen's Chaotic Attractor Using Backstepping Design Based on Parameters Identification
,”
Int. J. Phys. Lett. A
,
286
(
2–3
), pp.
148
152
.
36.
Xiao-Qun
,
W.
, and
Jun-An
,
L.
,
2003
, “
Parametric Identification and Backstepping Control of Uncertain Lü System
,”
Chaos, Solitons Fractals
,
18
, pp.
721
729
.
37.
Yongguang
,
Y.
, and
Suochun
,
Z.
,
2003
, “
Controlling Uncertain Lü Using Backstepping Design
,”
Chaos, Solitons Fractals
,
15
, pp.
897
902
.
38.
Peng
,
C. C.
, and
Chen
,
C. L.
,
2008
, “
Robust Chaotic Control of Lorenz System by Backstepping Design
,”
Chaos, Solitons Fractals
,
37
, pp.
598
608
.
39.
Rucklidge
,
A. M.
,
1992
, “
Chaos in Models of Double Convection
,”
J. Fluid Mech.
,
237
(
1
), pp.
209
229
.
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