Multifolded torus chaotic attractors: Design and implementation

013118-12 Yu, Lu, and Chen Chaos 17, 013118 2007
Let R 1 =100 k, R 2 =2 k, R 31 =R 51 =R 71 =R 91 =1 k,
and R 11 =R 21 =R 41 =R 61 =R 81 =10 k. According to a formula
similar to 19 and R n2 =r n R n1 1n9, we can rigorously
calculate all resistors in N R .
Thus, the circuit diagram Fig. 7 can be controlled based
on Tables III and IV, to generate 3-, 5-, 7-, and 9-folded torus
chaotic attractors via the switchings of the switches
K i 1i5. Figure 9 shows the experimental observation
results for 3-, 5-, 7-, and 9-folded torus chaotic attractors.
Remark 3: The real measurement values of r n and R n2
for 1n9 in the circuit experiment may have a small departure
from the theoretically calculated values shown in
Tables III and IV, due to the discrete nature of real circuit
parameters and the measurement errors. These differences
can be corrected via a small adjustment of the resistors R n2
for 1n9 in the circuit experiment. Our experimental results
show that it is technically very difficult to implement a
chaotic attractor with more than nine tori by analog circuits
because a circuit is highly sensitive to small variations of
parameters with the increased number of tori. For example,
the measurement precision of the switching points x 2
=2.0735 and x 3 =3.5735 of the 7-folded torus attractor has to
pinpoint to the fourth digit of the decimal. Here, our circuit
can realize up to a maximum of 9-folded tori in the chaotic
attractor under the real experimental conditions.
V. CONCLUDING REMARKS
This paper has developed a systematic methodology for
generating multifolded torus chaotic attractors from a simple
three-dimensional autonomous circuit. Recursive formulas
for system parameters have been rigorously derived, useful
for improving hardware implementation. Dynamical behaviors
of the multifolded torus system, including symmetry,
bifurcation, eigenspaces, and conditions for chaos generation,
have also been investigated. Our theoretical analysis
shows that multifolded torus chaotic attractors can be generated
via alternative switchings between two linear systems. A
simple circuit diagram has been designed for experimentally
verifying 3-, 5-, 7-, and 9-folded torus chaotic attractors.
This is the first time in the literature that an experimental
realization of a 9-folded torus chaotic attractor has been reported.
The circuit design method developed in this paper outperforms
the existing methods, in the sense that all system
design parameters and physical circuit parameters can be rigorously
derived beforehand by our new techniques. Although,
as is well known, there are many technical reasons
that cause difficulties in hardware implementation of multifolded
torus chaotic attractors, our approach has been physically
implemented via circuitry, which can generate up to
9-fold torus chaotic attractors visible on the oscilloscope,
showing the effectiveness and realizability of the proposed
methodology.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China under Grants No. 60304017, No.
20336040, and No. 60572073, the National Key Basic Research
and Development 973 Program of China under Grant
2006CB708202, the Scientific Research Startup Special
Foundation on Excellent Ph.D. Thesis and Presidential
Award of Chinese Academy of Sciences, the Natural Science
Foundation of Guangdong Province under Grants No. 32469
and No. 5001818, the Science and Technology Program of
Guangzhou City under Grant No. 2004J1-C0291, and the
Strategic Research Grants of the City University of Hong
Kong under Grant No. 7001702/EE.
1 J. Lu, T. Zhou, G. Chen, and X. Yang, Chaos 12, 344 2002.
2 S. M. Yu, J. Lu, W. K. S. Tang, and G. Chen, Chaos 16, 033126 2006.
3 K. Murali and S. Sinha, Phys. Rev. E 68, 016210 2003.
4 S. Sinha and D. Biswas, Phys. Rev. Lett. 71, 2010 1993.
5 S. Sinha and W. L. Ditto, Phys. Rev. Lett. 81, 2156 1998.
6 J. Lu and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 659
2002.
7 J. Lu, G. Chen, D. Cheng, and S. Čelikovský, Int. J. Bifurcation Chaos
Appl. Sci. Eng. 12, 2917 2002.
8 G. Chen and J. Lu, Dynamics of the Lorenz System Family: Analysis,
Control and Synchronization Science Press, Beijing, 2003 in Chinese.
9 M. E. Yalcin, J. A. K. Suykens, and J. P. L. Vandewalle, Cellular Neural
Networks, Multi-Scroll Chaos and Synchronization World Scientific, Singapore,
2005.
10 J. Lu and G. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 775
2006.
11 J. A. K. Suykens and J. P. L. Vandewalle, IEEE Trans. Circuits Syst., I:
Fundam. Theory Appl. 40, 861 1993.
12 J. A. K. Suykens and L. O. Chua, Int. J. Bifurcation Chaos Appl. Sci. Eng.
7, 1873 1997.
13 M. A. Aziz-Alaoui, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 1009
1999.
14 M. E. Yalcin, J. A. K. Suykens, and J. P. L. Vandewalle, Proceedings of
the IEEE Workshop on Nonlinear Dynamics of Electronic Systems, Catania,
Italy IEEE, Piscataway, NJ, 2000, p.25.
15 K. S. Tang, G. Q. Zhong, G. Chen, and K. F. Man, IEEE Trans. Circuits
Syst., I: Fundam. Theory Appl. 48, 13692001.
16 S. Ozoguz, A. S. Elwakil, and K. N. Salama, Electron. Lett. 38, 685
2002.
17 J. Lu, X. Yu, and G. Chen, IEEE Trans. Circuits Syst., I: Fundam. Theory
Appl. 50, 198 2003.
18 M. E. Yalcin, J. A. K. Suykens, J. P. L. Vandewalle, and S. Ozoguz, Int. J.
Bifurcation Chaos Appl. Sci. Eng. 12, 232002.
19 A. S. Elwakil and M. P. Kennedy, IEICE Trans. Fundamentals E82-A,
1769 1999.
20 J. Lu, S. M. Yu, and H. Leung, Dynam. Cont. Dis. Ser. B Sp.ISS.SI1,
324 2005.
21 J. Lu, F. Han, X. Yu, and G. Chen, Automatica 40, 1677 2004.
22 J. Lu, G. Chen, X. Yu, and H. Leung, IEEE Trans. Circuits Syst., I:
Fundam. Theory Appl. 51, 2476 2004.
23 J. Lu, S. M. Yu, H. Leung, and G. Chen, IEEE Trans. Circuits Syst., I:
Fundam. Theory Appl. 53, 149 2006.
24 M. E. Yalcin, J. A. K. Suykens, and J. P. L. Vandewalle, IEEE Trans.
Circuits Syst., I: Fundam. Theory Appl. 47, 4252000.
25 S. M. Yu, Q. H. Lin, and S. S. Qiu, Acta Phys. Sin. 53, 2084 2004.
26 S. M. Yu, J. Lu, H. Leung, and G. Chen, IEEE Trans. Circuits Syst., I:
Fundam. Theory Appl. 52, 1459 2005.
27 M. E. Yalcin, J. A. K. Suykens, and J. P. L. Vandewalle, IEEE Trans.
Circuits Syst., I: Fundam. Theory Appl. 51, 1395 2004.
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