TY - JOUR

T1 - Is My System of ODEs k-Cooperative?

AU - Weiss, Eyal

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 2017 IEEE.

PY - 2021/1

Y1 - 2021/1

N2 - A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called k -positive if its flow maps the set of vectors with up to k-1 sign variations to itself. A nonlinear dynamical system is called k -cooperative if its variational system, which is a time-varying linear dynamical system, is k -positive. These systems have special asymptotic properties. For example, it was recently shown that strong 2-cooperative systems satisfy a strong Poincaré-Bendixson property. Positivity and k -positivity are easy to verify in terms of the sign-pattern of the matrix in the dynamics. However, these sign conditions are not invariant under a coordinate transformation. A natural question is to determine if a given n -dimensional system is k -positive up to a coordinate transformation. We study this problem for two special kinds of transformations: permutations and scaling by a signature matrix. For any ngeq 4 and kin {2, {dots }, n-2} , we provide a graph-theoretic necessary and sufficient condition for k -positivity up to such coordinate transformations. We describe an application of our results to a specific class of Lotka-Volterra systems.

AB - A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called k -positive if its flow maps the set of vectors with up to k-1 sign variations to itself. A nonlinear dynamical system is called k -cooperative if its variational system, which is a time-varying linear dynamical system, is k -positive. These systems have special asymptotic properties. For example, it was recently shown that strong 2-cooperative systems satisfy a strong Poincaré-Bendixson property. Positivity and k -positivity are easy to verify in terms of the sign-pattern of the matrix in the dynamics. However, these sign conditions are not invariant under a coordinate transformation. A natural question is to determine if a given n -dimensional system is k -positive up to a coordinate transformation. We study this problem for two special kinds of transformations: permutations and scaling by a signature matrix. For any ngeq 4 and kin {2, {dots }, n-2} , we provide a graph-theoretic necessary and sufficient condition for k -positivity up to such coordinate transformations. We describe an application of our results to a specific class of Lotka-Volterra systems.

KW - Positive systems

KW - graph theory

KW - invariant sets

KW - k-cooperative systems

UR - http://www.scopus.com/inward/record.url?scp=85091102054&partnerID=8YFLogxK

U2 - 10.1109/LCSYS.2020.2999870

DO - 10.1109/LCSYS.2020.2999870

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AN - SCOPUS:85091102054

SN - 2475-1456

VL - 5

SP - 73

EP - 78

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

IS - 1

M1 - 9107214

ER -