Localization attractor

If the omega limit set is particularly simple - a rest point or a periodic orbit - this gives information about the asymptotic behavior of the trajectory. An invariant set which is the omega limit set of a neighborhood of itself is called a (local) attractor. If (3.1) is two-dimensional then the following theorem is very useful, since it severely restricts the structure of possible attractors. 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

Let p = x(0) be an arbitrary initial point with Jr,(0) > 0. Then the initial data do not belong to either stable manifold. Hence w p) is not equal to either Eq or E2, but it does lie on E = 0. Since it is invariant and since every solution of (5.2) on E = 0 converges to an equilibrium, u(p) contains an equilibrium. By the Butler-McGehee theorem, Eq uIp) since M (Eq) is unbounded. If o p) contains E2, then o p) also contains either Eq or an unbounded orbit, again by the Butler-McGehee theorem (see Figure 5.2). Since this is impossible, E must be in 0 p). However, is a local attractor, so u p) = Ey This completes the proof. ... 

However, if D is slightly larger than D then A] < ii < A2 < 2 and Q is disconnected if either (a) A2 < S < 1x2 or (b) 1x2 < S hold. If (a) holds, then El and E2 are local attractors and the complement of the union of their basins of attraction has zero Lebesgue measure. In this case, the winner depends on how the chemostat is charged at / = 0 - that is, on the initial conditions. If (b) holds, then Eq,Ei,E2 are all local attractors and the complement of the union of their basins has zero Lebesgue measure. In this case, washout of all populations, competitive exclusion of X2 by Xi, and competitive exclusion of Xi by X2 are all possible outcomes, depending on the initial conditions. 

Hence there is no rest point with positive coordinates if the origin is a (local) attractor. 

For case (c) to hold, one inequality in each part of (5.2) must be reversed and (by Proposition E.2) both Ei and Ei cannot be stable. Thus, one of El and E2 is a local attractor, say Ei- (Lemma 5.1 provides the explicit conditions for determining which is stable.) Similarly, if case (d) holds, both El and E2 must be unstable. 

The arguments in the four cases of Theorem 6.1 are very similar, so we present only one case, the last and most interesting one. When Eq, Ei, and E2 exist and S[Pg.205]

The definitions that are currently used in the classification of chemical bonds are often imprecise, as they are derived fi om approximate theories. Based on the topological analysis local, quantum-mechanical functions related to the Pauli Exclusion Principle may be formulated as localization attractors of bonding, non-bonding, and core t5qres. Bonding attractors lie between nuclei core attractors and characterize shared electron interactions. The spatial arrangement of bond attractors allows for an absolute... 

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