Dynamical Systems with a Cyclic Sign Variation Diminishing Property

Tsuff Ben Avraham, Guy Sharon, Yoram Zarai, Michael Margaliot

Research output: Contribution to journalArticlepeer-review

Abstract

In 1970, Binyamin Schwarz defined and analyzed totally positive differential systems (TPDSs), i.e., linear time-varying systems whose transition matrix is totally positive. He showed that any solution of a TPDS satisfies a sign variation diminishing property with respect to the standard number of sign variations. It has been recently shown that several important results on entrainment [stability] in time-varying [time-invariant] nonlinear tridiagonal cooperative systems follow from the fact that the variational equation associated with these nonlinear systems is a TPDS. Thus, the number of sign variations in the vector of derivatives can be used as an integer-valued Lyapunov function. Here we develop the theory of linear cyclic variation diminishing differential systems (CVDDSs). These are systems whose transition matrix satisfies a variation diminishing property with respect to the cyclic number of sign variations. Thus, the cyclic number of sign variations can be used as an integer-valued Lyapunov function for any vector solution of a CVDDS. We show that several known classes of nonlinear cooperative dynamical systems have a variational equation, which is a CVDDS.

Original languageEnglish
Article number8706539
Pages (from-to)941-954
Number of pages14
JournalIEEE Transactions on Automatic Control
Volume65
Issue number3
DOIs
StatePublished - Mar 2020

Keywords

  • Compound matrices
  • TP differential systems (TPDS)
  • cooperative dynamical systems
  • cyclic sign variation diminishing property (VDP)
  • minor
  • stability analysis
  • totally positive (TP) matrices

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