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Positive systems play an important role in systems and control theory and have found applications in multiagent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the non-positive orthant to itself). In other words, they map the set of vectors with zero sign variation to itself. In this article, discrete-time linear systems that map the set of vectors with up to k-1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k-1 sign variations. It is also shown that these systems induce a positive dynamics of k-dimensional parallelotopes.

In this thesis, the recognition problem and the properties of eigenvalues and eigenvectors of matrices which are strictly sign-regular of a given order, i.e., matrices whose minors of a given order have the same strict sign, are considered. The results are extended to matrices which are sign-regular of a given order, i.e., matrices whose minors of a given order have the same sign or are allowed to vanish. As a generalization, a new type of matrices called oscillatory of a specific order, are introduced. Furthermore, the properties for this type are investigated. Also, same applications to dynamic systems are given.

In this paper, rectangular matrices whose minors of a given order have the same strict sign are considered and sufficient conditions for their recognition are presented. The results are extended to matrices whose minors of a given order have the same sign or are allowed to vanish. A matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that A^k has all its minors positive. As a generalization, a new type of matrices, called oscillatory of a specific order, is introduced and some of their properties are investigated.