Limit Cycle Identification in Nonlinear Polynomial Systems

Affiliation(s)

Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China.

Department of Electrical Engineering (ESAT)-SCD, KU Leuven/IBBT Future Health Department, Leuven, Belgium.

Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China.

Department of Electrical Engineering (ESAT)-SCD, KU Leuven/IBBT Future Health Department, Leuven, Belgium.

ABSTRACT

We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.

KEYWORDS

Limit Cycle Identification; Polynomial Representation; Roots Finding; Macaulay Matrix; Immersion

Limit Cycle Identification; Polynomial Representation; Roots Finding; Macaulay Matrix; Immersion

Cite this paper

S. Zhang, H. Liu, K. Batselier and N. Wong, "Limit Cycle Identification in Nonlinear Polynomial Systems,"*Applied Mathematics*, Vol. 4 No. 9, 2013, pp. 19-26. doi: 10.4236/am.2013.49A004.

S. Zhang, H. Liu, K. Batselier and N. Wong, "Limit Cycle Identification in Nonlinear Polynomial Systems,"

References

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[17] K. Batselier, P. Dreesen and B. De Moor, “Numerical Al gebraic Geometry: The Canonical Decomposition and Nu merical Grobner Bases,” Technical Report, Leuven De partment of Electrical Engineering ESAT/SCD, 2012.

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[1] A. Teplinsky and O. Feely, “Limit Cycles in a Mems Os cillator,” IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 55, No. 9, 2008, pp. 882-886. doi:10.1109/TCSII.2008.923402

[2] R. J. Field and R. M. Noyes, “Oscillations in Chemical Sys tems. Limit Cycle Behavior in a Model of a Real Chemi cal Reaction,” The Journal of Chemical Physics, Vol. 60, No. 5, 1974, p. 1877.

doi:10.1063/1.1681288

[3] M. Agarwal and A. S. Bhadauria, “Mathematical Model ing and Analysis of Tumor Therapy with Oncolytic Vi rus,” Applied Mathematics, Vol. 2, No. 1, 2011, pp. 131-140. doi:10.4236/am.2011.21015

[4] R. M. May, “Nonlinear Phenomena in Ecology and Epi demiology,” Annals of the New York Academy of Scienc es, Vol. 357, No. 1, 1980, pp. 267-281. doi:10.1111/j.1749-6632.1980.tb29692.x

[5] J. C. Frauenthal and K. E. Swick, “Limit Cycle Oscilla tions of the Human Population,” Demography, Vol. 20, No. 3, 1983, pp. 285-298. doi:10.2307/2061243

[6] I. Hiskens, “Stability of Hybrid System Limit Cycles: Ap plication to the Compass Gait Biped Robot,” Proceedings of the 40th IEEE Conference on Decision and Control, Vol. 1, 2011, pp. 774-779.

[7] T. D. Burton, “Non-Linear Oscillator Limit Cycle Analy sis Using a Time Transformation Approach,” Internatio nal Journal of Non-Linear Mechanics, Vol. 17, No. 1, 1982, pp. 7-19. doi:10.1016/0020-7462(82)90033-6

[8] A. Buonomo, C. Di Bello and O. Greco, “A Criterion of Existence and Uniqueness of the Stable Limit Cycle in Second-Order Oscillators,” IEEE Transactions on Circuits and Systems, Vol. 30, No. 9, 1983, pp. 680-683. doi:10.1109/TCS.1983.1085407

[9] M. Hayashi, “On Canard Homoclinic of a Liénard Pertur bation System,” Applied Mathematics, Vol. 2, No. 10, 2011, pp. 1221-1224. doi:10.4236/am.2011.210170

[10] J. Jiang, “Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center,” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 772-777. doi:10.4236/am.2012.37115

[11] J. Moiola and G. Chen, “Computations of Limit Cycles via Higher-Order Harmonic Balance Approximation,” IEEE Transactions on Automatic Control, Vol. 38, No. 5, 1993, pp. 782-790.

doi:10.1109/9.277247

[12] C. Andersen and J. F. Geer, “Power Series Expansions for the Frequency and Period of the Limit Cycle of the van der pol Equation,” SIAM Journal on Applied Mathematics, Vol. 42, No. 3, 1982, pp. 678-693. doi:10.1137/0142047

[13] K. Batselier, P. Dreesen and B. De Moor, “Prediction Er ror Method Identification Is an Eigenvalue Problem,” Pro ceedings of 16th IFAC Symposium on System Identifica tion (SYSID 2012), 2012, pp. 221-226.

[14] P. Dreesen, K. Batselier and B. De Moor, “Back to the Roots: Polynomial System Solving, Linear Algebra, Sys tems Theory,” Proceedings of 16th IFAC Symposium on System Identification (SYSID 2012), 2012, pp. 1203-1208.

[15] H. K. Khalil and J. Grizzle, “Nonlinear Systems,” Pren tice Hall, Upper Saddle River, 2002.

[16] L. Menini, “Symmetries and Semi-Invariants in the Ana lysis of Nonlinear Systems,” Springer, Berlin, 2011. doi:10.1007/978-0-85729-612-2

[17] K. Batselier, P. Dreesen and B. De Moor, “Numerical Al gebraic Geometry: The Canonical Decomposition and Nu merical Grobner Bases,” Technical Report, Leuven De partment of Electrical Engineering ESAT/SCD, 2012.

[18] M. A. Savageau and E. O. Voit, “Recasting Nonlinear Dif ferential Equations as s-Systems: A Canonical Nonlinear Form,” Mathematical Biosciences, Vol. 87, No. 1, 1987, pp. 83-115. doi:10.1016/0025-5564(87)90035-6