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Cross-nonnegativity and monotonicity analysis of nonlinear dynamical systems. (English) Zbl 1478.37036

Summary: Let \(K\) be a closed convex cone in \(\mathbb{R}^n\) and let \(F : \mathbb{R}^n \to \mathbb{R}^n\) be a locally Lipschitz map, not necessarily differentiable. We revisit the problem of analyzing whether the flow generated by an autonomous dynamical system \(\dot{\varphi}(t) = F(\varphi(t))\) is order-preserving with respect to \(K\). This issue has to do with the notion of cross-nonnegativity of a nonlinear map relative to a cone. We study this property in-depth and derive various characterizations of it. We deviate from the classical literature in at least two ways. First of all, \(K\) is allowed to be unpointed and nonsolid. These nontrivial relaxations are needed to cover some interesting examples arising in applications. On the other hand, taking into account the possible lack of differentiability of \(F\), we bring the machinery of F. H. Clarke’s [Pac. J. Math. 64, 97–102 (1976; Zbl 0331.26013)] nonsmooth analysis into the picture.

MSC:

37C60 Nonautonomous smooth dynamical systems
49J52 Nonsmooth analysis
26B10 Implicit function theorems, Jacobians, transformations with several variables

Citations:

Zbl 0331.26013
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Full Text: DOI

References:

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