Abstract

A dynamical system is said to be differentially positive if its linearization along trajectories is positive in the sense that it infinitesimally contracts a smooth cone field. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. In this talk we consider differentially positive systems defined on Lie groups and outline the mathematical framework for studying differential positivity with respect to invariant cone fields. We motivate the use of this analysis framework with examples from nonlinear consensus theory. We also introduce a generalized notion of differential positivity with respect to an extended notion of cone fields of higher rank k>=2. This 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 which results in k-dimensional integral submanifold attractors.