This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

References

References
1.
Sauer
,
P. W.
,
Pai
,
M. A.
, and
Chow
,
J. H.
,
2017
,
Power System Dynamics and Stability: With Synchrophasor Measurement and Power System Toolbox
,
Wiley
,
Hoboken, NJ
.
2.
Afshari
,
M.
,
Mobayen
,
S.
,
Hajmohammadi
,
R.
, and
Baleanu
,
D.
,
2018
, “
Global Sliding Mode Control Via Linear Matrix Inequality Approach for Uncertain Chaotic Systems With Input Nonlinearities and Multiple Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
3
), p.
031008
.
3.
Pyragas
,
K.
,
1992
, “
Continuous Control of Chaos by Self-Controlling Feedback
,”
Phys. Lett. A
,
170
(
6
), pp.
421
28
.
4.
Schöll
,
E.
, and
Schuster
,
H. G.
,
2008
,
Handbook of Chaos Control
,
Wiley-VCH
,
Weinheim, Germany
.
5.
Champneys
,
A. R.
,
2006
, “
A Twenty-First Century Guidebook for Applied Dynamical Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
1
(
4
), pp.
279
82
.
6.
Ivancevic
,
T.
,
Jain
,
L.
,
Pattison
,
J.
, and
Hariz
,
A.
,
2009
, “
Nonlinear Dynamics and Chaos Methods in Neurodynamics and Complex Data Analysis
,”
Nonlinear Dyn.
,
56
(
1–2
), pp.
23
44
.
7.
Takens
,
F.
,
2010
, “
Reconstruction Theory and Nonlinear Time Series Analysis
,”
Handb. Dyn. Syst.
,
3
, pp.
345
77
.
8.
Hajiloo
,
R.
,
Salarieh
,
H.
, and
Alasty
,
A.
,
2018
, “
Chaos Control in Delayed Phase Space Constructed by the Takens Embedding Theory
,”
Commun. Nonlinear Sci. Numer. Simul.
,
54
, pp.
453
65
.
9.
Kaveh
,
H.
,
Salarieh
,
H.
, and
Hajiloo
,
R.
,
2018
, “
On the Control of Unknown Continuous Time Chaotic Systems by Applying Takens Embedding Theory
,”
Chaos, Solitons Fractals
,
109
, pp.
53
57
.
10.
Takens
,
F.
,
1981
, “
Detecting Strange Attractors in Turbulence
,”
Dynamical Systems and Turbulence, Warwick 1980
,
Springer-Verlag
,
Germany
, pp.
366
81
.
11.
Liebert
,
W.
,
Pawelzik
,
K.
, and
Schuster
,
H. G.
,
1991
, “
Optimal Embeddings of Chaotic Attractors From Topological Considerations
,”
EPL (Europhys. Lett.)
,
14
(
6
), p.
521
.
12.
Cao
,
L.
,
1997
, “
Practical Method for Determining the Minimum Embedding Dimension of a Scalar Time Series
,”
Phys. D: Nonlinear Phenom.
,
110
(
1–2
), pp.
43
50
.
13.
Buzug
,
T.
, and
Pfister
,
G.
,
1992
, “
Comparison of Algorithms Calculating Optimal Embedding Parameters for Delay Time Coordinates
,”
Phys. D: Nonlinear Phenom.
,
58
(
1–4
), pp.
127
37
.
14.
Narendra
,
K. S.
, and
Annaswamy
,
A. M.
,
2012
,
Stable Adaptive Systems
,
Courier Corporation, Dover Publications
,
Mineola, NY
.
15.
Sun
,
J.
,
2004
, “
Delay-Dependent Stability Criteria for Time-Delay Chaotic Systems Via Time-Delay Feedback Control
,”
Chaos, Solitons Fractals
,
21
(
1
), pp.
143
50
.
16.
Namajūnas
,
A.
,
Pyragas
,
K.
, and
Tamaševičius
,
A.
,
1995
, “
Stabilization of an Unstable Steady State in a Mackey-Glass System
,”
Phys. Lett. A
,
204
(
3–4
), pp.
255
262
.
17.
Leonov
,
G. A.
, and
Moskvin
,
A. V.
,
2017
, “
Stabilizing Unstable Periodic Orbits of Dynamical Systems Using Delayed Feedback Control With Periodic Gain
,”
Int. J. Dyn. Control
,
6
(2), pp.
601
608
.
18.
Liu
,
R.
,
Lu
,
J.
,
Liu
,
Y.
,
Cao
,
J.
, and
Wu
,
Z.-G.
,
2017
, “
Delayed Feedback Control for Stabilization of Boolean Control Networks With State Delay
,”
IEEE Trans. Neural Networks Learn. Syst.
,
29
(
7
), pp.
3283
3288
.
19.
Zhou
,
W.
,
Xu
,
Y.
,
Lu
,
H.
, and
Pan
,
L.
,
2008
, “
On Dynamics Analysis of a New Chaotic Attractor
,”
Phys. Lett. A
,
372
(
36
), pp.
5773
5777
.
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