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)
Adaptive stabilization of uncertain unified chaotic systems with nonlinear input. (English) Zbl 1245.65076
The authors consider the unified chaotic system described by the following set of differential equations $$\dot x= (25\alpha+ 10)(y- x),\quad \dot y= (28- 35\alpha)x+ (29\alpha- 1)y- xz,\quad \dot z= xy- {8+\alpha\over 3} z,$$ where $x$, $y$, $z$ are state variables and the system parameter $\alpha\in[0,1]$. -- A novel representation of nonlinear input function, that is, a linear input with bounded time-varying coefficient, is established. An adaptive control scheme is proposed based on the new nonlinear input model. Numerical simulations are performed to verify the analytical results.

65K10Optimization techniques (numerical methods)
Full Text: DOI
[1] Lü, J.; Chen, G.; Cheng, D.; Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system, International journal of bifurcation and chaos 12, 2917-2926 (2002) · Zbl 1043.37026 · doi:10.1142/S021812740200631X
[2] Lorenz, E. N.: Deterministic non-periods flows, Journal of the atmospheric sciences 20, 130-141 (1963)
[3] Chen, G. R.; Ueta, T.: Yet another chaotic attractor, International journal of bifurcation and chaos 9, 1465-1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[4] Lü, J.; Chen, G.: A new chaotic attractor coined, International journal of bifurcation and chaos 12, 659-661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[5] Lu, J. A.; Huang, B.; Wu, X.: Control of a unified chaotic system with delayed continuous periodic switch, Chaos, solitons & fractals 22, 229-236 (2004) · Zbl 1060.93526 · doi:10.1016/j.chaos.2003.01.001
[6] Yu, Y.; Li, H.; Duan, J.: Chaos synchronization of a unified chaotic system via partial linearization, Chaos, solitons & fractals 41, 457-463 (2009) · Zbl 1198.34125 · doi:10.1016/j.chaos.2008.02.010
[7] Park, J. H.: On synchronization of unified chaotic systems via nonlinear control, Chaos, solitons & fractals 25, 699-704 (2005) · Zbl 1125.93469 · doi:10.1016/j.chaos.2004.11.031
[8] Park, J. H.; Ji, D. H.; Won, S. C.; Lee, S. M.: Adaptive H$\infty $ synchronization of unified chaotic systems, Modern physics letters B 23, 1157-1169 (2009) · Zbl 1179.37123 · doi:10.1142/S021798490901934X
[9] L., J.; Wu, X.; Lü, J.: Synchronization of a unified chaotic system and the application in secure communication, Physics letters A 305, 365-370 (2002) · Zbl 1005.37012 · doi:10.1016/S0375-9601(02)01497-4
[10] Yau, H. T.; Yan, J. J.: Design of sliding mode controller for Lorenz chaotic system with nonlinear input, Chaos, solitons & fractals 19, 891-898 (2004) · Zbl 1064.93010 · doi:10.1016/S0960-0779(03)00255-8
[11] Chiang, T. Y.; Hung, M. L.; Yan, J. J.; Yang, Y. S.; Chang, J. F.: Sliding mode control for uncertain unified chaotic systems with input nonlinearity, Chaos, solitons & fractals 34, 437-442 (2007) · Zbl 1134.93405 · doi:10.1016/j.chaos.2006.03.051
[12] Hung, Y. C.; Liao, T. L.; Yan, J. J.: Adaptive variable structure control for chaos suppression of unified chaotic systems, Applied mathematics and computation 209, 391-398 (2009) · Zbl 1167.65071 · doi:10.1016/j.amc.2008.12.058
[13] Yau, H. T.; Yan, J. J.: Chaos synchronization of different chaotic systems subjected to input nonlinearity, Applied mathematics and computation 197, 775-788 (2008) · Zbl 1135.65409 · doi:10.1016/j.amc.2007.08.014
[14] Yan, J. J.: Design of robust controllers for uncertain chaotic systems with nonlinear inputs, Chaos, solitons & fractals 19, 541-547 (2004) · Zbl 1068.93022 · doi:10.1016/S0960-0779(03)00123-1
[15] Yan, J. J.; Chang, W. D.; Lin, J. S.; Shyu, K. K.: Adaptive chattering free variable structure control for a class of chaotic systems with unknown bounded uncertainties, Physics letters A 335, 274-281 (2005) · Zbl 1123.93308 · doi:10.1016/j.physleta.2004.12.028
[16] Polycarpou, M. M.: Stable adaptive neural control scheme for nonlinear systems, IEEE transactions on automatic control 41, 447-451 (1996) · Zbl 0846.93060 · doi:10.1109/9.486648
[17] Ioannou, P. A.; Sun, J.: Robust adaptive control, (1996) · Zbl 0839.93002