×

zbMATH — the first resource for mathematics

Existence and uniqueness of limit cycle for a class of nonlinear discrete-time systems. (English) Zbl 1142.39307
Summary: In this paper, the definition of the exponentially stable limit cycle for nonlinear discrete-time systems is firstly introduced. The limit cycle phenomenon for a class of nonlinear discrete-time systems is investigated. Using analytic method, the existence and uniqueness of limit cycle for such systems can be guaranteed. Besides, the exponentially stable limit cycles, the period of oscillation, and guaranteed convergence rate can be correctly estimated. Finally, a numerical example is provided to illustrate the use of the main result.

MSC:
39A11 Stability of difference equations (MSC2000)
92B05 General biology and biomathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cheng, Z.; Lin, Y.; Cao, J., Dynamical behaviors of a partial-dependent predator – prey system, Chaos, solitons & fractals, 28, 67-75, (2006) · Zbl 1083.37532
[2] Hirano, N.; Rybicki, S., Existence of limit cycles for coupled van der Pol equations, J differ equations, 195, 194-209, (2003) · Zbl 1045.34010
[3] Hwang, C.C.; Hsieh, J.Y.; Lin, R.S., A linear continuous feedback control of chua’s circuit, Chaos, solitons & fractals, 8, 1507-1515, (1997)
[4] Huang, X.; Zhu, L.; Cheng, A., Limit cycles in a general two-stroke oscillation, Nonlinear anal, 64, 22-32, (2006) · Zbl 1096.34021
[5] Hwang, T.W., Uniqueness of limit cycles of the predator – prey system with beddington – deangelis functional response, J math anal appl, 290, 113-122, (2004) · Zbl 1086.34028
[6] Jiang, W.; Wei, J., Bifurcation analysis in a limit cycle oscillator with delayed feedback, Chaos, solitons & fractals, 23, 817-831, (2005) · Zbl 1080.34054
[7] Lin, C.; Wang, Q.G.; Lee, T.H., Local stability of limit cycles for MIMO relay feedback systems, J math anal appl, 288, 112-123, (2003) · Zbl 1109.93330
[8] Ohta, H.; Ueda, Y., Blue sky bifurcations caused by unstable limit cycle leading to voltage collapse in an electric power system, Chaos, solitons & fractals, 14, 1227-1237, (2002) · Zbl 1050.34070
[9] Ohta, H.; Ueda, Y., Unstable limit cycles in an electric power system and basin boundary of voltage collapse, Chaos, solitons & fractals, 12, 159-172, (2001)
[10] Palmor, Z.J.; Halevi, Y.; Efrati, T., A general and exact method for determining limit cycles in decentralized relay systems, Automatica, 31, 1333-1339, (1995) · Zbl 0923.93039
[11] Ramos, J.I., Piecewise-linearized methods for oscillators with limit cycles, Chaos, solitons & fractals, 27, 1229-1238, (2006) · Zbl 1108.34030
[12] Sun YJ. Prediction of limit cycles in feedback bilinear systems: Lyapunov-like approach. In: Proceeding of 2002 National Symposium on Automatic Control; 2002. p. 801-5.
[13] Torrini, G.; Genesio, R.; Tesi, A., On the computation of characteristic multipliers for predicting limit cycle bifurcations, Chaos, solitons & fractals, 9, 121-133, (1998) · Zbl 0940.34023
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.