The simplest normal form and its application to bifurcation control. (English) Zbl 1136.34042
Normal form theory is one of the useful tools in the study of nonlinear dynamical systems. The main idea is to apply successive coordinate transformations in order to get a form of the original system as a simple as possible. The new form is qualitatively equivalent to the original system and thus the dynamical analysis is greatly simplified. A conventional normal form is in general not unique and can be further simplified using similar near-identity transformations, leading to the simplest normal form. This paper is concerned with the computation of the simplest normal forms with perturbation parameters associated with codimension-one singularities, and applications to control systems. It is shown that, unlike the classical normal forms, the simplest normal forms for single zero and Hopf singlarities are finite up to an arbitrary order. Symbolic programs have been developed using Maple.
Reviewer: Weinian Zhang (Sichuan)
MSC:
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 37G05 | Normal forms for dynamical systems |
| 93C10 | Nonlinear systems in control theory |
Software:
MapleReferences:
| [1] | Kang, W., Bifurcation and normal form of nonlinear control systems, parts I and II, SIAM J Contr Optim, 36, 193-232 (1998) · Zbl 0965.93022 |
| [2] | Chen, G.; Moiola, J. L.; Wang, H. O., Bifurcation control: theories, methods, and applications, Int J Bifur Chaos, 10, 511-548 (2000) · Zbl 1090.37552 |
| [3] | Chen, D.; Wang, H. O.; Chen, G., Anti-control of Hopf bifurcation, IEEE Trans Circ Sys-I, 48, 661-672 (2001) · Zbl 1055.93037 |
| [4] | (Chen, G., Controlling chaos and bifurcations in engineering systems (1999), CRC Press: CRC Press Boca Raton, FL) |
| [5] | Yu P. Bifurcation dynamics in control systems. In: Bifurcation control: Theory and applications. Berlin: Springer-Verlag; 2003. p. 99-126.; Yu P. Bifurcation dynamics in control systems. In: Bifurcation control: Theory and applications. Berlin: Springer-Verlag; 2003. p. 99-126. · Zbl 1032.93527 |
| [6] | Yu, P.; Chen, G., Hopf bifurcation control using nonlinear feedback with polynomial functions, Int J Bifur Chaos, 14, 5, 1673-1704 (2004) · Zbl 1129.37335 |
| [7] | Chen, Z.; Yu, P., Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions, Chaos, Solitons & Fractals, 26, 4, 1231-1248 (2005) · Zbl 1093.37508 |
| [8] | Chen, Z.; Yu, P., Hopf bifurcation control for an internet congestion model, Int J Bifur Chaos, 15, 8, 2643-2651 (2005) · Zbl 1092.34562 |
| [9] | Ushiki, S., Normal forms for singularities of vector fields, Jpn J Appl Math, 1, 1-37 (1984) · Zbl 0578.58029 |
| [10] | Baider, A.; Churchill, R., Unique normal forms for planar vector fields, Math Z, 199, 303-310 (1988) · Zbl 0691.58012 |
| [11] | Chua, L. O.; Kokubu, H., Normal forms for nonlinear vector fields—Part I: Theory and algorithm, IEEE Trans Circ Syst, 35, 863-880 (1988) · Zbl 0683.58021 |
| [12] | Chua, L. O.; Kokubu, H., Normal forms for nonlinear vector fields—Part II: Applications, IEEE Trans Circ Syst, 36, 51-70 (1988) · Zbl 0702.58047 |
| [13] | Sanders, J. A.; van der Meer, J. C., Unique normal form of the Hamiltonian 1:2-resonance, (Broer, H. W.; Takens, F., Geometry and analysis in nonlinear dynamics (1990), Longman: Longman Harlow), 56-69 · Zbl 0766.58051 |
| [14] | Yu, P., Simplest normal forms of hopf and generalized hopf bifurcations, Int J Bifur Chaos, 9, 1917-1939 (1999) · Zbl 1089.37528 |
| [15] | Yu, P.; Yuan, Y., The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue, Dynamics of continuous, discrete and impulsive systems, series B: Applications & algorithms, 8b, 219-249 (2000) · Zbl 0985.34029 |
| [16] | Wang, D.; Li, J.; Huang, M.; Jiang, Y., Unique normal forms of Bogdanov-Takens singularity, J Differential Equation, 163, 223-238 (2000) · Zbl 0957.34041 |
| [17] | Chen, G.; Dora, J. D., An algorithm for computing a new normal form for dynamical systems, J Symb Comput, 29, 393-418 (2000) · Zbl 0973.34029 |
| [18] | Yu, P.; Yuan, Y., The simplest normal forms associated with a triple zero eigenvalue of indices one and two, Nonlinear Anal: Theor Methods Appl, 47, 2, 1105-1116 (2001) · Zbl 1042.34538 |
| [19] | Algaba, A.; Freire, E.; Gamero, E., Characterizing and computing normal forms using Lie transforms: a survey, Dynamics of continuous, discrete and impulsive systems series A: Mathematical analysis, 8a, 449-475 (2001) · Zbl 1001.34030 |
| [20] | Yuan, Y.; Yu, P., Computation of the simplest normal forms of differential equations associated with a double-zero eigenvalues, Int J Bifur Chaos, 11, 5, 1307-1330 (2001) · Zbl 1090.37539 |
| [21] | Sanders, J. A., Normal form theory and spectral sequences, J Differential Equation, 192, 536-552 (2003) · Zbl 1039.34032 |
| [22] | Yu, P., Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling, J Comput Appl Math, 144, 2, 359-373 (2002) · Zbl 1019.65043 |
| [23] | Yu, P., A simple and efficient method for computing center manifold and normal forms associated with semi-simple cases, Dynamics of continuous, discrete and impulsive systems, series B: Applications & algorithms, 10, 1-3, 273-286 (2003) · Zbl 1036.34056 |
| [24] | Yu, P.; Yuan, Y., An efficient method for computing the simplest normal forms of vector fields, Int J Bifur Chaos, 13, 1, 19-46 (2003) · Zbl 1067.34041 |
| [25] | Yu, P.; Yuan, Y., A matching pursuit technique for computing the simplest normal forms of vector fields, J Symb Comput, 35, 5, 591-615 (2003) · Zbl 1021.37033 |
| [26] | Yu, P.; Leung, A. Y.T., A perturbation method for computing the simplest normal forms of dynamical systems, J Sound Vib, 261, 1, 123-151 (2003) · Zbl 1237.34074 |
| [27] | Carr, J., Applications of center manifold theory (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0464.58001 |
| [28] | Yu, P., Computation of normal forms via a perturbation technique, J Sound Vib, 211, 19-38 (1998) · Zbl 1235.34126 |
| [29] | Yu, P.; Leung, A. Y.T., The simplest normal form of Hopf bifurcation, Nonlinearity, 16, 1, 277-300 (2003) · Zbl 1086.34033 |
| [30] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1993), Springer-Verlag: Springer-Verlag New York |
| [31] | Chua, L. O., Introduction to nonlinear network theory (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0221.94039 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
