Differential positivity with respect to cones of rank k ≥ 2

Abstract

© 2017 We consider a generalized notion of differential positivity of a dynamical system with respect to cone fields generated by cones of rank k ≥ 2. The property refers to the contraction of such cone fields by the linearization of the flow along trajectories. It provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank k that results in k-dimensional integral submanifold attractors instead. We further develop the theory in the context of invariant cone fields and invariant differential positivity on Lie groups and illustrate the key ideas with an extended example involving consensus on the space of rotation matrices SO(3)