Limit set

Abstract: A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The principal result is that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension. In fact orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane. Limit sets are nowhere dense, unknotted and unlinked. In dimension 2 every trajectory is eventually monotone. In dimension 3 a compact limit set which does not contain an equilibrium is a closed orbit or a cylinder of closed orbits. Introduction. One of the most interesting questions to ask about a dynamical system is: what is the long-run behavior of its trajectories? In many systems it is natural to expect, or at least hope, that almost all trajectories either converge to an equilibrium or asymptotically approach a closed orbit (= periodic trajectory). Unfortu- nately there are many systems that not only lack this convenient property, but cannot even be approximated by systems that have it. Such systems are often said to be "chaotic" or to possess "strange attractors". To make matters worse, it is very hard to discover the long-run behavior of any but the simplest systems. Research on this problem has bifurcated into two quite different methodologies. A great deal of recent work has gone toward exploring the consequences of various assumptions about the large scale structure of the system, e.g., hyperbolicity of the nonwandering set, structural stability, ergodicity, and so forth. The basic examples come from geometry and physics; the mathematical tech- niques tend to be topological. For a recent overview of this work see Smale (15, Chapt. I).

Abstract: A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of “normal form” or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

Abstract: In this paper, the behavior of a continuous flow in the vicinity of a closed positively .invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equiyalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider rarity of ecological models.

Abstract: The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion. The limit set theorem of Benaim and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems. Applications to game theory are given, in particular to Blackwell's approachability theorem and the convergence of fictitious play.