Semidynamical system

The most basic concept is that of a dynamical (or a semidynamical) system. Let 7T M X R- M be a function of two variables, where M is R" and R denotes the real numbers. (We use M for the first variable or state space to suggest that the results are true in greater generality.) The function 7T is said to be a continuous dynamical system if tt is continuous and has the following properties ... 

To avoid technical conditions, assume that / is such that solutions of initial value problems are unique and extend to [0,oo). Thus (D.l) generates a semidynamical system. Of course, one could assume (as has been done before) that solutions extend to all of K. However, checking the backward continuation of solutions presents a problem in one of the applications, so the results are stated for semidynamical systems. The form of the equations causes the positive cone to be invariant (Proposition B.7) and the coordinate axes and the bounding faces to be invariant (and represent lower-order dynamical systems). 

We review the basic definitions and set up the semidynamical system appropriate for systems of the form (D.l). Let A" be a locally compact metric space with metric d, and let be a closed subset of X with boundary dE and interior E. The boundary, dE, corresponds to extinction in the ecological problems. Let tt be a semidynamical system defined on E which leaves dE invariant. (A set B in A" is said to be invariant if n-(B, t) = B.) Dynamical systems and semidynamical systems were discussed in Chapter 1. The principal difficulty for our purposes is that for semidynamical systems, the backward orbit through a point need not exist and, if it does exist, it need not be unique. Hence, in general, the alpha limit set needs to be defined with care (see [H3]) and, for a point x, it may not exist. Those familiar with delay differential equations are aware of the problem. Fortunately, for points in an omega limit set (in general, for a compact invariant set), a backward orbit always exists. The definition of the alpha limit set for a specified backward orbit needs no modification. We will use the notation a.y(x) to denote the alpha limit set for a given orbit 7 through the point x. 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

Theorem D.2. Let x be a semidynamical system defined on a subset E, the closure of an open set, in a locally compact metric space X. Suppose that dE, the boundary of E, is invariant under w. Assume that tt is dissipative and that the boundary flow ttj is isolated and acyclic with acyclic covering M. Then tt is uniformly persistent if and only if... 

A semidynamic system was used to examine the resin release properties if introduced to natural water. Resin was placed in nylon mesh bags and put in a flask as previously described. 

In order to more closely simulate the dilution effects of a natural water body, a semidynamic system was evaluated using the SNW. As Is evidenced by Figure 5A, In both the rinsed and unrlnsed cases the pellets released approximately 80 percent of their total copper In the first 15 minutes. After 30 minutes very little additional copper was released from the resin. 

The effect of particle size was also examined In the semidynamic system. I (20 x 50 mesh) was exposed under the same conditions (Figure 5 B) and the rinsed cases behave much the same therefore, It Is safe to assume that a good portion of the copper In most of these experiments came from the surface water associated with the moist resin, with more surface water being associated with the smaller size particles. This copper most likely originated from the solutions used to load the polymers. 

Figure 5, Release of copper by I (20 X 50-mesh pellets) in a semidynamic system using synthetic natural -water. Key Q, unrinsed , rinsed A and B, total Cu released (mg/L),...

The fundamentals and practical use of MAE have been described in detail in several review articles (68-70) and books (18, 71, 72). The following text focuses on closed-vessel (pressurized) MAE, which permits extractions at elevated temperatures. A major difference of MAE compared to SEE and PLE, in addition to its unique heating performance, is that the commercially available MAE systems today operate in batch mode. The possibility of built-in clean-up is therefore difficult to perform and related to the design of the instrumentation. Both automated SEE and PLE are most commonly used in a dynamic or semidynamic mode, which simplifies the development of combined extraction/clean-up strategies. 

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