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)
Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems. (English) Zbl 1136.93430
Summary: We study chaos (lag) synchronization of a new LC chaotic system, which can exhibit not only a two-scroll attractor but also two double-scroll attractors for different parameter values, via three types of state feedback controls: (i) linear feedback control; (ii) adaptive feedback control; and (iii) a combination of linear feedback and adaptive feedback controls. As a consequence, ten families of new feedback control laws are designed to obtain global chaos lag synchronization for $\tau < 0$ and global chaos synchronization for $\tau = 0$ of the LC system. Numerical simulations are used to illustrate these theoretical results. Each family of these obtained feedback control laws, including two linear (adaptive) functions or one linear function and one adaptive function, is added to two equations of the LC system. This is simpler than the known synchronization controllers, which apply controllers to all equations of the LC system. Moreover, based on the obtained results of the LC system, we also derive the control laws for chaos (lag) synchronization of another new type of chaotic system.

93D15Stabilization of systems by feedback
34D99Stability theory of ODE
93B52Feedback control
93C40Adaptive control systems
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] Lorenz, E. N.: Deterministic nonperiodic flow. J atmos sci 20, 130-141 (1963)
[2] Li, T. Y.; Yorke, J. A.: Period three implies chaos. American math monthly 82, 985-992 (1975) · Zbl 0351.92021
[3] Rössler, O. E.: An equation for continuous chaos. Phys lett A 57, 397 (1976)
[4] Rössler, O. E.: Continuous chaos-four prototype equations. Ann New York acad sci 316, 376-392 (1979) · Zbl 0437.76055
[5] Chen, G.; Ueta, T.: Yet another chaotic attractor. Int J bifurcat chaos 9, 1465-1466 (1999) · Zbl 0962.37013
[6] Chen, G.; Celikovsky, S.: On a generalized Lorenz canonical form of chaotic systems. Int J bifurcat chaos 12, 1789-1812 (2002) · Zbl 1043.37023
[7] Matsumoto, T.; Chua, L. O.; Kobayashi, K.: Hyperchaos: laboratory experiment and numerical confirmation. IEEE trans circuits syst 33, 1143-1147 (1986)
[8] Li, Y.; Tang, S. K.; Chen, G.: Generating hyperchaos via state feedback control. Int J bifurcat chaos 15, No. 10, 3367-3376 (2005)
[9] Yan, Z.: Controlling hyperchaos in the new hyperchaotic Chen system. Appl math comput 168, 1239-1250 (2005) · Zbl 1160.93384
[10] Chen, G.; Dong, X.: From chaos to order: methodologies, perspectives and applications. (1998) · Zbl 0908.93005
[11] Boccaletti, S.: The synchronization of chaotic systems. Phys rep 366, 1-101 (2002) · Zbl 0995.37022
[12] Liu, W. B.; Chen, G.: A new chaotic system and its generation. Int J bifurcat chaos 13, 261-267 (2003) · Zbl 1078.37504
[13] Liu, W. B.; Chen, G.: Can a three-dimensional smooth autonomous quadratic chaotic system generate a single four-scroll attractor?. Int J bifurcat chaos 14, 1395-1403 (2004) · Zbl 1086.37516
[14] Pecora, L. M.; Caroll, T. L.: Synchronization in chaotic systems. Phys rev lett 64, 821-824 (1990) · Zbl 0938.37019
[15] Ott, E.; Grebogi, C.; Yorke, J. A.: Controlling chaos. Phys rev lett 64, 1196-1199 (1990) · Zbl 0964.37501
[16] Kapitaniak, T.: Synchronization of chaos using continuous control. Phys rev E 50, 1642-1644 (1994)
[17] Parmananda, P.: Synchronization using linear and nonlinear feedbacks: a comparison. Phys lett A 240, 55-59 (1998)
[18] Liao, X. X.: Absolute stability of nonlinear control systems. (1993) · Zbl 0817.93002
[19] Liao, X. X.: Stability theory and application of dynamical systems. (2002)
[20] Liao, T. L.; Tsai, S. H.: Adaptive synchronization of chaotic systems and its application to secure communications. Chaos, solitons & fractals 11, 1387-1396 (2000) · Zbl 0967.93059
[21] Jiang, G. P.; Chen, G.; Tang, K. S.: A new criterion for chaos synchronization using linear state feedback control. Int J bifurcat chaos 13, 2343-2351 (2002) · Zbl 1064.37515
[22] Wang, Y.: Feedback and adaptive control for the synchronization of Chen system via a single variable. Phys lett A 312, 34-40 (2003) · Zbl 1024.37053
[23] Liao, X. X.; Yu, P.: Analysis of the global exponent synchronization of Chua’s circuit using absolute stability theory. Int J bifurcat chaos 15, No. 12, 3687-3881 (2005) · Zbl 1094.37502
[24] Liao XX, Yu P. Study of globally exponential synchronization for the family of Rössler systems. Int J Bifurcat Chaos 2006:16. · Zbl 1192.37042
[25] Yu P, Liao XX. Globally attractive and positive invariant set of the Lorenz system. Int J Bifurcat Chaos 2006:16. · Zbl 1141.37335
[26] Yan, Z.: Q-S synchronization in 3D hénon-like map and generalized hénon map via a scalar controller. Phys lett A 342, 309-317 (2005) · Zbl 1222.37093
[27] Yan, Z.: Q-S synchronization backstepping scheme in a class of continuous-time hyperchaotic systems--a symbolic-numeric computation approach. Chaos 15, 023902-023909 (2005) · Zbl 1080.93008
[28] Sun, J. T.: Impulsive control of a new chaotic system. Math comput simulat 64, 669-677 (2004) · Zbl 1076.65119
[29] Luo H, Jian J, Liao X. Chaos synchronization for a 4-scroll chaotic system via nonlinear control. In: International conference on intelligent computing, China, August 23-26, 2005. p. 797-806.
[30] Yassen, M. T.: Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, solitons & fractals 26, 913-920 (2005) · Zbl 1093.93539
[31] Lü, J.; Chen, G.; Cheng, D.: A new chaotic system and beyond: the generalized Lorenz-like system. Int J bifurcat chaos 14, 1507-1537 (2004) · Zbl 1129.37323