1. G.A. Enciso, H.L. Smith, and E.D. Sontag. Non-monotone systems decomposable into monotone systems with negative feedback. J. of Differential Equations, 224:205-227, 2006. [PDF] Keyword(s): nonlinear stability, dynamical systems, monotone systems. Abstract:

   Motivated by the theory of monotone i/o systems, this paper shows that certain finite and infinite dimensional semi-dynamical systems with negative feedback can be decomposed into a monotone open loop system with inputs and a decreasing output function. The original system is reconstituted by plugging the output into the input. By embedding the system into a larger symmetric monotone system, this paper obtains finer information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. An important new result is the extension of the "small gain theorem" of monotone i/o theory to reaction-diffusion partial differential equations: adding diffusion preserves the global attraction of the ODE equilibrium.

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