Yu, Yongguang; Zhang, Suochun Adaptive backstepping synchronization of uncertain chaotic system. (English) Zbl 1062.34053 Chaos Solitons Fractals 21, No. 3, 643-649 (2004). The paper is devoted to the synchronization of systems, which can be represented in so-called strict-feedback form, i.e., \(\dot x_1=f_1(x_1,x_2)\), \(\dot x_2 = f_2 (x_1,x_2,x_3)\), \(\dots\), \(\dot x_{n-1}=f_{n-1}(x_1,\dots,x_n)\), \(\dot x_{n}=f_{n}(x_1,\dots,x_n)+g_1(t)\), where \(f_1\) is a linear function. The response system is coupled via the last \(n\)th component \( \dot {\bar x}_{n}=f_{n}(\bar x_1,\dots,\bar x_n)+g_2(t) +u\).The authors suggest an adaptive design of the coupling term \(u\), including a law for the parameter adaptation such that the systems are synchronized, that is, \(| x(t)-\bar x(t)| \to \infty\) as \(t\to \infty\) for all initial conditions \(x(0)\) and \(\bar x(0)\). Reviewer: Sergiy Yanchuk (Berlin) Cited in 49 Documents MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations Keywords:adaptive backstepping; synchronization; chaos PDF BibTeX XML Cite \textit{Y. Yu} and \textit{S. Zhang}, Chaos Solitons Fractals 21, No. 3, 643--649 (2004; Zbl 1062.34053) Full Text: DOI OpenURL References: [1] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys. rev. lett., 64, 821-824, (1990) · Zbl 0938.37019 [2] Carroll, T.L.; Pecora, L.M., Synchronizing chaotic circuits, IEEE trans. circ. syst. I, 38, 453-456, (1991) [3] Chua, L.O.; Itah, M.; Kosarev, L.; Eckert, K., Chaos synchronization in chua’s circuits, J. circ. syst. comput., 3, 93-108, (1993) [4] Agiza, H.N.; Yassen, M.T., Synchronization of Rössler and Chen chaotic dynamical systems using active control, Phys. lett. A, 278, 191-197, (2001) · Zbl 0972.37019 [5] Bai, E.W.; Lonngran, K.E., Synchronization of two Lorenz systems using active control, Chaos, solitons & fractals, 8, 51-58, (1997) · Zbl 1079.37515 [6] Cuomo, K.M.; Oppenheim, A.V.; Strogatz, S.H., Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE trans. circ. syst. I, 40, 626-633, (1993) [7] Bai, E.W.; Lonngran, K.E., Synchronization and control of chaotic systems, Chaos, solitons & fractals, 10, 1571-1575, (1999) · Zbl 0958.93513 [8] Liao, T.L., Adaptive synchronization of two Lorenz systems, Chaos, solitons & fractals, 9, 1555-1561, (1998) · Zbl 1047.37502 [9] Liao, T.L.; Tsai, S.H., Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, solitons & fractals, 11, 1378-1396, (2000) · Zbl 0967.93059 [10] Lü, J.H.; Zhou, T.S.; Zhang, S.C., Chaos synchronization between linearly coupled chaotic systems, Chaos, solitons & fractals, 14, 529-541, (2002) · Zbl 1067.37043 [11] Tan, X.H.; Zhang, J.Y.; Yang, Y.R., Synchronizing chaotic systems using backstepping design, Chaos, solitons & fractals, 16, 37-45, (2003) · Zbl 1035.34025 [12] Ge, S.S.; Wang, C., Adaptive control of uncertain chua’s circuits, IEEE trans. circ. syst. I, 47, 9, 1397-1402, (2000) · Zbl 1046.93506 [13] Ge, S.S.; Wang, C.; Lee, T.H., Adaptive backstepping control of a class of chaotic systems, Int. J. bifurcat. chaos, 10, 5, 1149-1156, (2000) · Zbl 1090.34555 [14] Wang, C.; Ge, S.S., Adaptive backstepping control of uncertain Lorenz system, Int. J. bifurcat. chaos, 11, 4, 1115-1119, (2001) · Zbl 1090.93536 [15] Yu, Y.G.; Zhang, S.C., Adaptive backstepping control of the uncertain Lü system, Chin. phys., 11, 12, 1249-1253, (2002) [16] Rössler, O.E., Continuous chaos-four prototype equations, Ann. N.Y. acad. sci., 316, 376-392, (1979) · Zbl 0437.76055 [17] Lorenz, E.N., Deterministic non-periodic flows, J. atmos. sci., 20, 130-141, (1963) · Zbl 1417.37129 [18] Chen, G.R.; Ueta, T., Yet another chaotic attractor, Int. J. bifurcat. chaos, 9, 1465-1466, (1999) · Zbl 0962.37013 [19] Lü, J.H.; Chen, G.R., A new chaotic attractor coined, Int. J. bifurcat. chaos, 12, 659-661, (2002) · Zbl 1063.34510 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.