Homotopy method for finding the steady states of oscillators

Hans Georg Brachtendorf, Robert Melville, Peter Feldmann, Siegmar Lampe, Rainer Laur

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


Shooting, finite difference, or harmonic balance techniques in conjunction with a damped Newton method are widely employed for the numerical calculation of limit cycles of (free-running, autonomous) oscillators. In some cases, however, nonconvergence occurs when the initial estimate of the solution is not close enough to the exact one. Generally, the higher the quality factor of the oscillator the tighter are the constraints for the initial estimate. A 2-D homotopy method is presented in this paper that overcomes this problem. The resulting linear set of equations is underdetermined, leading to a nullspace of rank two. This underdetermined system is solved in a least squares sense for which a rigorous mathematical basis can be derived. An efficient algorithm for solving the least squares problem is derived where sparse matrix techniques can be used. As continuation methods are only employed for obtaining a sufficient initial guess of the limit cycle, a coarse grid discretization is sufficient to make the method runtime efficient.

Original languageEnglish
Article number6816118
Pages (from-to)867-878
Number of pages12
JournalIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Issue number6
Publication statusPublished - Jun 2014


  • Continuation
  • homotopy
  • oscillator simulation
  • path following method
  • quartz crystal oscillators
  • steady state


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