TY - JOUR

T1 - Dynamical Systems with a Cyclic Sign Variation Diminishing Property

AU - Ben Avraham, Tsuff

AU - Sharon, Guy

AU - Zarai, Yoram

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2020/3

Y1 - 2020/3

N2 - 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.

AB - 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.

KW - Compound matrices

KW - TP differential systems (TPDS)

KW - cooperative dynamical systems

KW - cyclic sign variation diminishing property (VDP)

KW - minor

KW - stability analysis

KW - totally positive (TP) matrices

UR - http://www.scopus.com/inward/record.url?scp=85071679855&partnerID=8YFLogxK

U2 - 10.1109/TAC.2019.2914976

DO - 10.1109/TAC.2019.2914976

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AN - SCOPUS:85071679855

SN - 0018-9286

VL - 65

SP - 941

EP - 954

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 3

M1 - 8706539

ER -