2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62948The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.Updates and replaces math.OC/0112021 See http://www.math.rutgers.edu/~sontag/ for related workOptimization and ControlDynamical Systems93B52;93D20;35B35text