Strongly monotone dynamical system

Theorem C.8. Let ir bea strongly monotone dynamical system such that E has no points of accumulation in D. Suppose that 7 (x) has compact closure in D for every xeD. Then the set of points x for which w(x, t) does not converge to an equilibrium has Lebesgue measure zero. 

C.l) has the form (B.9) where F and G are independent oft and (B.12) holds in D, then w is a monotone dynamical system with respect to If in addition, the Jacobian matrix of f is irreducible at every point ofD, then w is strongly monotone with respect to... 

Theorem C.4. The stable manifold of an unstable, hyperbolic rest point of a monotone dynamical system cannot contain two points that are related by the strict inequality stable manifold cannot contain two distinct points that are related by stable manifold is unordered. 

Theorem C.5. A compact limit set of a monotone dynamical system cannot contain two points related by [Pg.271]

Sufficient conditions for (C.l) to generate a dynamical system which is monotone (strongly monotone) are given in Appendix B. Corollary B.2, Theorem B.3, Corollary B.5, and Theorem B.6 imply the following result. 

Remark 2. If, in addition, Df(x) is also irreducible in [Xj, X2], then Xq cannot belong to the boundary of [Xi,X2j. In fact, if X dynamical system induced by / implies that Xj < Xq < X2. 

One can learn about the stmeture of a system from following its dynamics. The orientational relaxation dynamics of water confined between mica surfaces has been investigated by MD simulations. The presence of wide heterogeneity in the dynamics of water adjacent to a strongly hydrophilic mica surface has been observed. By analyzing the survival probabilities, a 10-fold increase in the survival times for water that is directly in contact with the mica surface and a non-monotonic variation in the survival times moving away fi om the mica surface to the bulk-like... 

Hirai et al. [97] used the AOT/isooctane/water system and titanium tetrabutoxide (TTBO). Particle diameters on the order of 3 nm were obtained by dynamic light scattering, a dimension that is smaller than the diameters of the reverse micelles (9-19.3 nm). Particle formation was strongly influenced by the water/surfactant molar ratio (R) and by the alkoxide concentration. The reaction kinetics was followed with UV-Vis absorption spectrophotometry. The absorbance of the reaction system increased monotonically with time for R = 9, whereas it went through a peak (after 400 min) for R = 30. These results were rationalized in terms of differences in availability of reactant water molecules. For... 

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