James S. Muldowney

A Geometric Approach to Global-Stability Problems

A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the

Global stability for the SEIR model in epidemiology

C^1

Bendixson-Dulac Criteria for Difference Equations

closing lemma of Pugh. The classical result of Lyapunov is obtained as a special case.

On Invertibility of Linear Ordinary Differential Boundary Value Problems

The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.

Lower bounds for the Hausdorff dimension of attractors

Conditions are given which preclude the existence of a nontrivial periodic orbit for a difference equation in ℝn. The conditions are analogous to those of Bendixson and Dulac for autonomous planar differential equations.

Stability Implications of Bendixson's Criterion

A criterion is given which is necessary and sufficient that certain homogeneous linear boundary value problems have only the trivial solution. The condition includes results of Polya [Traps. Amer. Math. Soc. 24 (1922), pp. 312–324] on disconjugacy and Muldowney [Proc. Amer. Math. Soc., 74 (1979), pp. 49–55] on disfocality. A mean value theorem and a positivity result are also obtained. Analogues of the Sturm comparison principle are established including a generalization of a theorem of Hartman [Amer. J. Math., 91 (1969), pp. 306–362; 93 (1971), pp. 439–451] and Levin [Soviet Math. Dokl. 4 (1963), pp. 121–124] which gives a necessary and sufficient condition for invertibility in terms of the existence of solutions to a family of differential inequalities.

Riccati differential inequalities and dichotomies for linear systems

The existence of certainm-dimensional structures in a dynamical system implies that the Hausdorff dimension of its attractor is at leastm+1. A Bendixson criterion for the nonexistence of periodic orbits for systems in Hilbert spaces is found.

Discontinuous scalar functions and ordinary differential equations

This note presents a proof that the omega limit set of a solution to a planar system satisfying the Bendixson criterion is either empty or is a single equilibrium. The proof involves elementary techniques which should be accessible to senior undergraduates and graduate students.

The influence of host competition and predation on tick densities and management implications

A general notion of dichotomy for linear differential systems is investigated. It is well known that a system x′=A(t)x in which the matrix A(t) is bounded and diagonally dominant by rows or columns has an invariant splitting of its solution space into two subspaces each uniformly asymptotically stable, one for increasing time and the other for decreasing time. Similar results are obtained here where the concept of diagonal dominance is weakened using Riccati inequalities.

Markov systems of vector-valued functions and disconjugacy

(2) V(t, q~(t))V(to, Xo) <_ W(s, q~(s)) ds, t >_ to, ,o where V and W are prescribed real-valued functions. It will usually be assumed tha t f sa t i s f i es the Cara thdodory c o n d i t i o n s : f (t, x) is cont inuous in x for each t, measurable in t for each x and If(t, x)i <_ m(t) uniformly in x where m is locally Lebesgue integrable. Occasionally the stronger condit ion that f is continuous on its domain will be assumed.