Abstract

The trajectories of a positive system leave a cone invariant. Positive systems have a rich history both because of the relevance in applications (e.g., when modeling a behavior defined by positive variables) and because positivity significantly restricts the behavior, leading to the Perron-Frobenius theory. Motivated by the importance of positivity in linear system analysis, the talk introduces and explores the concept of differential positivity for nonlinear behaviors. Differential positivity means positivity of  the linearized system along any trajectory. We derive the analog of  Perron-Frobenius theory in this generalized setup and explore its consequences for the asymptotic behavior of the system. We show that, under suitable conditions, differential positivity strongly restricts the asymptotic behavior and provides a novel analysis and design tool for multistable or limit cycle behaviors.  The analysis of a simple bistable model and of a nonlinear pendulum will illustrate the potential of the approach.