A fixed point theorem for bounded dynamical systems

Abstract

We show that a continuous map or a continuous flow on R n with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in R n intersects W then there is a fixed point in W . Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point.

Set-Theoretic Methods in Control

“…It is quite an easy exercise to show that the considered We finally would like to stress that the existence of equilibria can be inferred from other assumptions. For instance, it turns out that if all the solutions x(t), with x(0) ∈ IR n of a dynamical systemẋ = f (x), with sufficiently regular f are ultimately bounded inside a compact set, then there is necessarily an equilibrium pointx, (i.e., f (x) = 0) in such a set (see, for instance, [Srz85,Hal88,RW02]). …”

“…Boundedness implies the existence of an equilibrium [Srz85,Hal88,RW02] and it can be seen that the system admits a single equilibrium point. Indeed, equatinġ x i = 0 and eliminating x 1 and x 3 , we get the following conditions…”

Positively expansive homeomorphisms of compact spaces

“…This work relies heavily on the notion of bounded dynamical systems (see [11], [12]). A dynamical system is bounded if there exists a compact set W with the property that the forward orbit of every point in X intersects W .…”

“…Below we state several properties that are equivalent to boundedness; the theorem is proved in [11], but since the proof is short we include it again here. We note that the theorem is also true for flows or semiflows and the proof is nearly identical to the one given below.…”

Replicator Dynamics of Symmetric Ultimatum Game

“…that there must be a stationary point. Alternatively, it can be proved by an iterated use of the Poincaré-Bendixson theorem and a fact that a polynomial vector field can have only a finite number of limit cycles or by a direct use of the result from[9,10].…”