A remark on local activity and passivity

TL;DR

This paper investigates local activity and passivity in linear systems, establishing conditions under which eigenvalues imply local activity and linking system matrix properties to activity, with implications for nonlinear system complexity.

Contribution

It provides a theoretical analysis of local activity and passivity in linear systems and proposes a framework for extending these concepts to nonlinear systems.

Findings

Eigenvalues with positive real part imply local activity.

System matrices that are not dissipative correspond to local activity.

Proposed an abstract scheme for applying local activity concepts to nonlinear systems.

Abstract

We study local activity and its opposite, local passivity, for linear systems and show that generically an eigenvalue of the system matrix with positive real part implies local activity. If all state variables are port variables we prove that the system is locally active if and only if the system matrix is not dissipative. Local activity was suggested by Leon Chua as an indicator for the emergence of complexity of nonlinear systems. We propose an abstract scheme which indicates how local activity could be applied to nonlinear systems and list open questions about possible consequences for complexity.