zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Design of adaptive sliding mode controller for chaos synchronization with uncertainties. (English) Zbl 1060.93536
Summary: In this paper an adaptive sliding mode controller is presented for a class of master-slave chaotic synchronization systems with uncertainties. Using an adaptive technique to estimate the switching gain, an adaptive sliding mode controller is then proposed to ensure that the sliding condition is maintained in finite time. The proposed adaptive sliding mode control scheme can be implemented without the requirement that the bounds of the uncertainties and the disturbances should be known in advance. The concept of extended systems is used such that continuous control input is obtained using a sliding mode design scheme. By comparing with the results in the existed literatures, the results show that the master-slave chaotic systems with uncertainties can be synchronized accurately by this controller. Illustrative examples of chaos synchronization for uncertain Duffing-Holmes system are presented to demonstrate the superiority of the obtained results.

93C10Nonlinear control systems
37D45Strange attractors, chaotic dynamics
93C40Adaptive control systems
93B12Variable structure systems
Full Text: DOI
[1] Nayfeh, A. H.: Applied nonlinear dynamics. (1995) · Zbl 0848.34001
[2] Chen, G.; Dong, X.: From chaos to order: methodologies, perspectives and applications. (1998) · Zbl 0908.93005
[3] Kapitaniak, T.: Chaotic oscillations in mechanical systems. (1991) · Zbl 0786.58027
[4] Astakhov, V. V.; Anishchenko, V. S.; Kapitaniak, T.; Shabunin, A. V.: Synchronization of chaotic oscillators by periodic parametric perturbations. Physica D 109, 11-16 (1997) · Zbl 0925.58055
[5] Blazejczyk-Okolewska, B.; Brindley, J.; Czolczynski, K.; Kapitaniak, T.: Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators. Chaos, solitons & fractals 12, 1823-1826 (2001) · Zbl 0994.37044
[6] Yang, X. S.; Duan, C. K.; Liao, X. X.: A note on mathematical aspects of drive-response type synchronization. Chaos, solitons & fractals 10, 1457-1462 (1999) · Zbl 0955.37020
[7] Wang, Y.; Guan, Z. H.; Wen, X.: Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos, solitons & fractals 19, 899-903 (2004) · Zbl 1053.37528
[8] Chua, L. O.; Yang, T.; Zhong, G. Q.; Wu, C. W.: Adaptive synchronization of Chua’s oscillators. Int. J. Bifurc. chaos 6, No. 1, 189-201 (1996)
[9] Liao, T. L.: Adaptive synchronization of two Lorenz systems. Chaos, solitons & fractals 9, 1555-1561 (1998) · Zbl 1047.37502
[10] Lian, K. Y.; Liu, P.; Chiang, T. S.; Chiu, C. S.: Adaptive synchronization design for chaotic systems via a scalar driving signal. IEEE trans. Circuits syst. I 49, No. 1, 17-27 (2002)
[11] Wu, C. W.; Yang, T.; Chua, L. O.: On adaptive synchronization and control of nonlinear dynamical systems. Int. J. Bifurc. chaos 6, 455-471 (1996) · Zbl 0875.93182
[12] Fang, J. Q.; Hong, Y.; Chen, G.: Switching manifold approach to chaos synchronization. Phys. rev. E 59, 2523-2526 (1999)
[13] Yin, X.; Ren, Y.; Shan, X.: Synchronization of discrete spatiotemporal chaos by using variable structure control. Chaos, solitons & fractals 14, 1077-1082 (2002) · Zbl 1038.37506
[14] Yu, X.; Song, Y.: Chaos synchronization via controlling partial state of chaotic systems. Int. J. Bifurc. chaos 11, No. 6, 1737-1741 (2001)
[15] Wang, C.; Ge, S. S.: Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, solitons & fractals 12, 199-206 (2001) · Zbl 1015.37052
[16] Lu, J.; Zhang, S.: Controlling Chen’s chaotic attractor using backstepping design based on parameters identification. Phys. lett. A 286, 145-149 (2001)
[17] Suykens, J. A. K.; Curran, P. F.; Vandewalle, J.: Robust nonlinear synchronization of chaotic Lur’e systems. IEEE trans. Circuits syst. I 44, No. 10, 891-904 (1997)
[18] Slotine, J. E.; Li, W.: Applied nonlinear control. (1991) · Zbl 0753.93036
[19] Chen, C. L.; Lin, W. Y.: Sliding mode control for non-linear systems with global invariance. Proc. inst. Mech. engrs. 211, 75-82 (1997)
[20] Utkin, V. I.: Sliding modes in control optimization. (1992) · Zbl 0748.93044