The idea of embedding a dynamical system, whose generator has both increasing and decreasing monotonicity properties (positive and negative feedback), into a larger symmetric monotone dynamical system and exploiting the convergence properties of the latter is very old. For a discussion of history of the method, see [4,8]. The method is repeatedly rediscovered and its implications are often unde...

Stability of differential inclusions defined by locally Lipschitz compact valued maps is addressed. It is shown that if such a differential inclusion is globally asymptotically stable, then in fact it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact (not ne...

Cholera is a water and food borne infectious disease caused by the gram-negative bacterium, Vibrio cholerae. Its dynamics are highly complex owing to the coupling among multiple transmission pathways and different factors in pathogen ecology. Although various mathematical models and clinical studies published in recent years have made important contribution to cholera epidemiology, our knowledg...

We prove this conjecture in case (a) and signi cantly extend the range of p and q for which it is known in case (b). In cases (a) and (b), these questions are equivalent to global stability of the xed point (L;L) of the planar map (u; v) = (v; f(u; v)). For as in case (a), we consider natural four dimensional extensions T of 3 and S of . For 0 q < p, we prove that (L;L; L; L) is a global stable...

Recent crisis in the global financial world has generated renewed interests in fragilities of global financial networks among economists and regulatory authorities. In particular, a potential vulnerability of the financial networks is the ‘financial contagion’ process in which insolvencies of individual entities propagate through the ‘web of dependencies’ to affect the entire system. In this pa...

We consider the higher-order nonlinear difference equation xn 1 p qxn−k / 1 xn rxn−k , n 0, 1, . . . with the parameters, and the initial conditions x−k, . . . , x0 are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open probl...

This paper summarizes results of Dai and Vande Vate 15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of two-station open multi-class queueing networks with very general arrival and service processes. The conditions for pathwise global stability arise from two intuitively appealing phenomena: virtual stations an...

The paper considers price adjustment on the plane and derives global stability conditions for such dynamics. First, we examine the well-known Scarf Example, to obtain and analyze a global stability condition for this case. Next, for a general class of excess demand functions, a set of conditions is identified which guarantee not only convergence to some equilibrium but also robustness of these ...

We consider an SIR model of disease transmission which has been formulated with delay in order to give a fixed duration of infectiousness. A Lyapunov functional that has worked for similar models does not work here. Instead, the model is shown to be a consequence of an age-of-infection model for which the same class of Lyapunov functionals has worked, resolving the global stability. In this pap...