Graph unstable

Wlien T < T the graph of H versus m shows a van der Waals like loop, with an unstable region where the... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

The emission of radiation by an atom results in a change in the nature of the atom, which is said to have decayed. The rate at which a quantity of an isotope decays is proportional to the number of unstable atoms present and a graph of... 

In the previous equation, the sum runs over all critical points of the gradient dynamical system. In the Bonding Evolution Theory, the critical points form the molecular graph. In this graph, they are represented according to the dimension of their unstable manifold. Thus, critical points of / = 0, are associated with a dot, these with I = 1 are associated with a line, these with / = 2 by faces, and finally these with 7=3 by 3D cages. 

By plotting a graph between number of neutrons and protons for the nuclei of various elements it has been found that most stable nuclei (non-radioactive nuclei) lie within the shaded area which is called the zone or belt of stability because it contains the stable nuclei. Nuclei that fall above or below this belt are unstable. Nuclei that fall above the stability belt have more neutrons while those lying below have more protons. Such unstable nuclei would attain stability by undergoing change that would produce a nucleus with n/p ratio within the stability zone. 

Fig. 4. Stability of carbon on different sites (A-D) on a pure nickel(l 11) surface (below) and a gold-alloyed nickel(l 11) surface (above). The probability of nucleation of graphite is determined by the stability of the adsorbed carbon atoms. The less stable the adsorbed carbon, the larger the tendency to react with adsorbed oxygen to form CO and the lower the coverage. On the pure nickel) 111) surface, the most stable adsorption configuration of carbon is in the threefold (hep) site (lower curve). The upper graph shows that carbon adsorption in threefold sites next to a gold atom is completely unstable (sites B and C), and even the threefold sites that are next-nearest neighbors (sites A and D) to the gold atoms are led to a substantial destabilization of the carbon. From Reference (79).

When the forcing amplitude is very small and the midpoint of the forcing oscillation scans the autonomous bifurcation diagram, the qualitative response of the forced system for all frequencies can be deduced from the autonomous system characteristics. As the amplitude of the forcing becomes larger, one cannot predict a priori what will occur for a particular system. For this example, the most complicated phenomenon possible is a turning point bifurcation on a branch of periodic solutions where two limit cycles, one stable and one unstable, collide and disappear. This will appear as a pinch on the graph of the map [Fig. 1(d)],... 

It is clear from the graphs that both industrial FCC units have multiple steady states and that the steady state with the most desirable production of gasoline is the middle unstable sad die-type state. 

Regarding stability, the five steady states y, . .., ys depicted in Figure 6 (A-2) alternate in their stability behavior the low temperature y is stable, yi is unstable, ys is stable, 2/4 is unstable and the high temperature steady state ys is stable. This can be deduced as before from the graph. The bifurcation multiple [Pg.557]

Graphs of the differences in the coagulation cascade and circulating biomarkers and unstable angina (open bars) and peripheral arterial disease (.shaded boxes). 

We note two important features on this graph. First, the transition from stable to potentially runway conditions increases dramatically with decreasing e, that is, increasing the reaction constant activation energy in the limit at e — 0 we have explosive conditions. Second, the transition from stable to potentially unstable reactions occurs when the dimensionless at 0 = 7 / 1. Furthermore, for 0 = 10-1 the reaction is stable with 0/(1 + 0) = 1 and for >10 there are is a significant increase (of the order of 104 to 1010) in the dimensionless heat-generation term, denoting the potential of unstable, runway reactions. 

When the proton number is graphed versus the neutron number for a stable nuclei, a belt of stability emerges (Figure 14.2). Nuclei outside this belt tend to be unstable and therefore radioactive. 

Let us consider a stable synthon S (A) e S(S(A)), the transformation S(A) => S (A) is represented by a reaction graph either of the linear or cyclic form. In order to verify that the synthon S (A) belongs to the stable neighborhood of the stable synthon S(A) it is sufficient to verify that each connected subgraph of GR produces the unstable synthon. As this subgraph would be of linear form, at least one nonvirtual atom would gain or loose one valence electron and therefore its valence state would be unstable and the synthon S (A) produced should not be stable. 

Figure 14. An Hel2 surface of section for an unstable trajectory which forms a collision complex. The total energy is —2661.6 cm . Also shown are the reaction separatrix and the intramolecular bottleneck, (a) Graph showing the full dynamics of the trajectory. (b)-(f) Graphs illustrating the trajectory over five consecutive time ranges. These graphs are arranged to demonstrate the manner in which the trajectory moves with respect to the bottleneck and the separatrix. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

More is known about the dynamics generated by P than has been discussed in this chapter. In [DS] and [HaS] it is shown that there is a curve C joining Ei and E2 which is the graph of a strictly decreasing continuous function. This curve C forms the boundary of the unstable manifold of Eq and every fixed point, except Eq must lie on C. Therefore, every orbit of P except Eq is attracted to a fixed point on C. If each fixed point of P is assumed to be hyperbolic, then there are finitely many fixed points. Moving along the curve C, the fixed points alternate between saddle points and attractors. In particular, if the hypotheses of Corollary 5.2 hold then there are an odd number of positive fixed points on C, at least one of which is an attractor. See [S5] for more details. 

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