Unique stationary point

Clearly, the solutions of nonlinear gap equations are not unique. In numerical calculations we separated the physical solutions by observing the sign of 4>q and that of the effective potential at the stationary point W//( o)- The temperature dependence of these two quantities are presented in Fig. 2. It is seen that 4>q (solid line) is positive in the large range of r and goes to zero when r is close to r = 1. Similarly, the depth of the effective potential at the stationary point, Veff(4>o), becomes shallow when r —> 1 and vanishes at T = Tc. 

The are unique and referred to as normal coordinates . Stationary points for which all second derivatives (in normal coordinates) are positive are energy minima. 

The discussion of IL-based stationary phases up to this point has centered around ILs that are either coated as a thin film on a capillary wall or on a solid support. Although ILs exhibit a variety of properties that allow them to be unique stationary phases, their most significant drawback lies with their drop in viscosity with increasing temperature. This results in an increased propensity for flowing of the IL within the capillary, which often produces pooling of the stationary phase and nonuniform film thickness throughout the column. These factors often contribute to diminished analyte retention time reproducibility as well as detrimental effects on separation efficiency. 

This depends on fcls a0, and b0. Points of intersection of these two curves on the flow diagram correspond to conditions where R = L, and hence to stationary-state solutions. If R and L have just one intersection, as shown in Fig. 1.12(a) or (e), there is a unique stationary state. If L cuts R three times, as... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

Even for the present simple case, for which the inflow does not contain the autocatalyst, we have seen a variety of combinations of stable and unstable stationary states with or without stable and unstable limit cycles. Stable limit cycles offer the possibility of sustained oscillatory behaviour (and because we are in an open system, these can be sustained indefinitely). A useful way of cataloguing the different possible combinations is to represent the different possible qualitative forms for the phase plane . The phase plane for this model is a two-dimensional surface of a plotted against j8. As these concentrations vary in time, they also vary with respect to each other. The projection of this motion onto the a-/ plane then draws out a trajectory . Stationary states are represented as points, to which or from which the trajectories tend. If the system has only one stationary state for a given combination of k2 and Tres, there is only one such stationary point. (For the present model the only unique state is the no conversion solution this would have the coordinates a,s = 1, Pss = 0.) If the values of k2 and tres are such that the system is lying at some point along an isola, there will be three stationary states on the phase... 

For the present scheme, when there is a unique stationary state, we find Apr < 0 and local stability. Under circumstances with multiple solutions, the highest and lowest states always have Apr < 0 and hence are stable the middle branch of solutions has Apr > 0 and hence is a branch of unstable saddle points. 

Of particular interest is the special case of a complex pair of principal eigenvalues whose real parts are passing through zero. This is the situation which we have seen corresponding to a Hopf bifurcation in the well-stirred systems examined previously. Hopf bifurcation points locate the conditions for the emergence of limit cycles. Using the CSTR behaviour as a guide it is relatively easy to find conditions for Hopf bifurcations, and then locally values of the diffusion coefficient for which a unique stationary state is unstable. Indeed the stationary-state profile shown in Fig. 9.5 is such a... 

For the particular example in Fig. 9.10, the Hopf point occurs for /i0 1.105. The two turning points are located at n0 = 0.72 and 0.636. This means that for reactant concentrations in the range 0.72 < fi0 < 1.105, the system has a unique stationary-state profile which is unstable. Under such conditions, the reaction will exhibit time-dependent as well as spatially dependent solutions, i.e. there is a limit cycle. Some representative non-stationary profiles are shown in Fig. 9.12. 

Due to their characteristic course, there is always one unique intersection point of both curves for any given temperature and voltage. At this intersection point, both reaction rates become zero. Once the gas has reached it, both reactions stop and the gas composition does not change anymore along the spatial coordinate, unless the temperature or the cell voltage is changed. Thus, this is a stationary point. Because the reactions always run towards their equilibrium, this stationary point is an attractor. 

Starting at a saddle point, a path of steepest descent can be defined on the potential energy surface by using the gradient function 8W/8Qj the path of steepest descent is uniquely determined by extremal values of the gradient unless a stationary point is reached (55). Besides the minima corresponding to the reactant and product asymptotes, a potential energy surface may exhibit some additional minima due to, e.g., van der Waals (59) complexes or intermediates (see later). In such cases, the reactant and product asymptote can be interconnected by several steepest descent paths and the construction... 

Corollary 2. If the projections of Xy) V5. X define concave curves and the projections of (X. ) vs. X define concave trajectories, then the projected curves of X<. vs. Xy) are concave beyond the stationary point in this projection, and this stationary point is unique if the projected curve is strictly... 

Traditional reaction kinetics has dealt with the large class of chemical reactions that are characterised by having a unique and stable stationary point (i.e. all reactions tend to the equilibrium ). The complementary class of reactions is characterised either by the existence of more than one stationary point, or by an unstable stationary point (which could possibly bifurcate to periodic solutions). Other extraordinarities such as chaotic solutions are also contained in the second class. The term exotic kinetics refers to different types of qualitative behaviour (in terms of deterministic models) to sustained oscillation, multistationarity and chaotic effects. Other irregular effects, e.g. hyperchaos (Rossler, 1979) can be expected in higher dimensions. 

The analysis of dimensionaUty of sections trajectory separatrix bundles shows that for splits with one distributed component trajectory of only one section in the mode of minimum reflux goes through corresponding stationary point or (there is one exception to this rule, it is discussed below). The dimensionality of bundle 5 - A4+ is equal to A - 2, that of bundle — iV+ is equal to n — A — 1. The total dimensionality is equal to n - 3. Therefore, points x/ i and Xf cannot belong simultaneously to minimum reflux bundles at any value of LlV)r. If only one of the composition points at the plate above or below the feed cross-section belongs to bundle 5 - A + and the second point belongs to bundle 5 - 5 - A+, then the total dimensionality of these bundles will become equal n - 2 therefore, such location becomes feasible at unique value oi(LjV)r (i.e., in the mode of minimum reflux). 

For linear displacements from a stationary point, separable coordinates are uniquely defined for small displacements by normal mode coordinates [62], which simultaneously diagonalize the kinetic energy to infinite order and the potential energy to second order (i.e., through quadratic terms in the potential). Thus, to the extent that one stays in a region where the quadratic expansion of the potential is trustworthy, these coordinates separate the physical motion, and they are not just an artificial mathematical imposition. Not only is the motion separable in normal coordinates, but the coordinates themselves are very convenient for calculations, since they are rectilinear. 

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