Global attractor

M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75 (1997) 293-317... 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

A system may be in a stable, metastable, unstable, or neutral equilibrium state. In a stable system, a perturbation causes small departures from the original conditions, which are restorable. In an unstable equilibrium, even a small perturbation causes large irreversible changes. A metastable system may be stable or unstable according to the level and direction of perturbation. All thermodynamic equilibria are stable or metastable, but not unstable. This means that all natural processes evolve toward an equilibrium state, which is a global attractor. 

Figure 12.1. Near equilibrium AS is a Lyapunov function and equilibrium is a global attractor.

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. 

The main result in [BWol] is that every solution converges to one of these equilibria. In particular, since at most one population has a nonzero component at equilibrium, no more than one population can survive. As expected, it is also shown that if < A, then Eq is the global attractor. The following result gives sufficient conditions for population X[ to be the winner. 

Lemma 4.3. If the origin is a repeller or a saddle point for IR+ H F, then there is exactly one rest point of (3.2) in the interior of this set and it is a global attractor. 

Figure 4.2. a Case (a)(i) of Theorem 4.5, where E2 is the global attractor, b One of several possible scenarios in case (a)(ii) there could be more rest points, c A possible scenario in case (c) this case cannot be eliminated, but it is believed to be unlikely for biologically reasonable uptake functions. 

Since the [ 0jt(q) basis does not depend on the PCB geometry, the class-1 functions generate Uk i) potentials that are confining, i.e., stationary with respect to -displacements around the vector of a unique minimum, dU , 4>k)/d = 0 at 1=1. In other words, there is a single global attractor in the 91 -space of -configurations that is totally determined by each 0jt(q)-function. This result also applies to the case of asymptotic attractors. 

Thus it was shown by direct numerical experiments that main flow and all the wave regimes except the dominating ones are unstable. They are transformed spontaneously to the dominating wave for chosen wave number s and similarity parameter 6. Hence dominating waves are global attractors of initial value problem for (9) and have to be used for comparison with experiments. 

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