Structurally unstable

The following case study demonstrates the convergence behavior for the LH mechanism (50) with irreversible first stage (i.e. r i = 0). In this case the kinetic polynomial (51) always has (structurally unstable with respect to feasibility) zero root whereas three other roots could be found from the cubic equation... 

TaCl2(NMe2)3]2 X-Ray structure, unstable towards redistribution 4... 

The static bifurcation characteristics of the resulting closed loop system have also been discussed in the previous section and we have seen that the bifurcation diagram of the reactor dense-phase dimensionless temperature, namely a plot of Yrd versus the controller gain Kc is a pitchfork. Such bifurcations are generally structurally unstable when any of the system parameters are altered, even very slightly. 

Now consider the case of smoother interaction between structures. Two structures may interact readily and smoothly with one another to form a composite structure, a system whose properties are additive properties of the individual structures, as well as gestalt properties unique to the combination. Or, two or more structures may interact with one another in such a way that the total system alters some of the properties of the individual structures to various degrees, producing a system with gestalt properties that are not simple additive properties of the individual structures, unstable interactions may also occur between two or more structures that compete for energy, producing an unstable, shifting relationship in the composite system. 

The majority of the above examples are non-rough (structurally unstable) systems. The rough dynamic systems on the plane cannot demonstrate the properties shown by the above examples. If Tt is specified by a rough individual (without parameters) system on the plane, there cannot exist th, rj2 slow relaxations and rh 2,3 and tj3 slow relaxations can take place only simultaneously. This can be confirmed by the results given below and the data of some classical studies concerning smooth rough two-dimensional systems [20, 21],... 

Structurally stable systems are not dense this is the title of Smale s study [41] that has opened a new period in understanding dynamics. Structurally stable (rough) systems are those whose phase patterns undergo no qualitative changes at small perturbations (for accurate definitions with comprehensive motivation, see ref. 11). Smale constructed such a structurally unstable system that any system sufficiently close to it is also structurally unstable. This result has destroyed any hope of the possibility of classifying "almost all dynamic systems. Such hopes were associated with the advance in the classification of two-dimensional systems, among which the structurally stable ones are dense. 

The other attempt to get rid of the "Smale nightmare (the presence of domains of structurally unstable systems) was to consider the e-motion (with or without further examination of the limit e - 0). The picture obtained is... 

If the ratio were less than this critical value, anions would touch (increasing repulsive forces). Also, the cation and anion would be separated (decreasing attractive forces). Both effects would tend to make the structure unstable. 

Lotka-Volterra reaction scheme A + X— 2X X y — 2Y Y— Z structurally unstable oscillation... 

Using the MCY potential at constant pressure and temperature the system became structurally unstable as described in ref. [74], even though the first nearest neighbour distance was preserved at about 2.9 A. A considerable distribution was found for the local tetrahedral symmetry. This behaviour is reasonable since a simple 6-12 potential has no preference for a tetrahedral ly bonded structure. However, with a fixed cell volume the simulation became stable. Nearest neighbour molecules move within the energy minimum created by the pair-potential and the pair-wise additive electrostatic forces. At low temperatures, these molecules only sample the parabolic part of the potential... 

The above examples throw light on the problem of unfolding (problem 2) and the related problem of structural stability of a function of one variable. Although the function V x) = x3 is structurally unstable, one may hope that its modification having the form... 

Structural stability of the above families of functions, containing structurally unstable functions with degenerate critical points at zero values of parameters, should now be examined. It will appear that embedding of structurally unstable functions (cases IIIA-IIIC, Table 2.1) in parametrized families of functions increased their structural stability the functions given in Table 2.2 are structurally stable. 

III) the point x = 0 is a structurally unstable critical point (degenerate critical point) ... 

The simplest functions compiled in Table 2.4, having degenerate critical points of a given type (D4, ..., E6), are structurally unstable. As with to the case of functions of one variable, structurally stable functions having... 

When condition (3) is not fulfilled, we deal in the case of system (5.2) with a structurally unstable centre. In real physical systems this kind of... 

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