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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.

has been cited by the following article:

• TITLE: Limit Cycle Identification in Nonlinear Polynomial Systems

  AUTHORS: Shuqi Zhang, Haotian Liu, Kim Batselier, Ngai Wong

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

  JOURNAL NAME: Applied Mathematics, Vol.4 No.9A, September 6, 2013

  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.