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,"

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