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Summary: Multi-Stability in Monotone Input/Output Systems
David Angeli
Dip. Sistemi e Informatica
University of Florence, 50139 Firenze, Italy
angeli@dsi.unifi.it
Eduardo D. Sontag
Dept. of Mathematics
Rutgers University, NJ, USA
sontag@hilbert.rutgers.edu
Abstract This paper studies the emergence of multi-stability and hysteresis in those systems that
arise, under positive feedback, starting from monotone systems with well-de ned steady-state responses.
Such feedback con gurations appear routinely in several elds of application, and especially in biology.
Characterizations of global stability behavior are stated in terms of easily checkable graphical conditions.
Keywords: monotone systems, bistability, hysteresis, multiple steady-states
1 Introduction
Multi-stability and associated hysteresis e ects form the basis of many models in molecular biology, in
areas such as cell di erentiation, development, and periodic behavior described by relaxation oscillations.
See for instance the classic work by Delbruck [5], who suggested in 1948 that multi-stability could explain
cell di erentiation, as well as references in the current literature (e.g., [4], [11], [12], [13], [15], [19], [23],
[31]).
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