Saddle-type stability

The maximum productivity of the desired product B usually occurs at the middle unstable saddle-type steady state. In order to stabilize the unstable steady state, a simple proportional-feedback-controlled system can be used, and we shall analyze such a controller now. A simple feedback-controlled bubbling fluidized bed is shown in Figure 4.25. 

The control loop affects both the static behavior and the dynamic behavior of the system. Our main objective is to stabilize the unstable saddle-type steady state of the system. In the SISO control law (7.72) we use the steady-state values Yfass = 0.872 and Yrdss = 1.5627 as was done in Figures 7.14(a) to (c). A new bifurcation diagram corresponding to this closed-loop case is constructed in Figure 7.20. 

Point a lies in the middle of the multiplicity region with three steady states. Therefore, this point is an unstable saddle-type steady state. It is clear from Figures 7.21(b) and (c) that this operating point does not correspond to the maximum gasoline yield d-How to alter the operating conditions so that the FCC unit operates at the maximum gasoline yield has been discussed in the previous sections. As explained there, a simple way to stabilize such unstable steady states is to use a negative feedback proportional... 

This is also the Hamiltonian of the activated complex. We will encounter it in Eq. (23) with the customary symbol H. Regardless of its stability properties or the size of the nonlinearity, Eq. (12) is always an invariant manifold. However, we are interested in the case when it is of the saddle type with stable and unstable manifolds. If the physical Hamiltonian is of the form of Eq. (1), then a preliminary, local transformation is not required. The manifold (12) is invariant regardless of the size of the nonlinearity. Moreover, it is also of saddle type with respect to stability in the transverse directions. This can be seen by examining Eq. (1). On qn = Pn = 0 the transverse directions, (i.e., q and p ), are still of saddle type (more precisely, they grow and decay exponentially). 

Complexation of [H2Pc (CF3)8] with FeCU affords the Fe complex containing two coordinated pyridine molecules [Py2FePc (CF3)8] [67], It is oxidized in an acidic medium to Fe" complex isolated as p-oxo dimeric species p-0[FePc (CF3)8]2 Its X-Ray crystal structure (see Fig. 6) evidence that severe saddle-type distortion of the macrocycles due to interaction of a-CF3 groups leads to almost linear r-(FeOFe) bridge [68], The p-oxo species exhibit catalytic activity in epoxidation of olefins with iodosyl benzene to stabilize the Fe" =0 (ferryl) intermediate by electron-accepting CF3 groups [67],... 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

Operating at the middle unstable steady state requires using some means of control for the plant, such as a stabilizing controller or nonadiabatic operation with carefully chosen parameters to stabilize the saddle-point type of the unstable steady state. 

Ceulemans and Fowler [29] have derived the extremal properties of the APES surface of the G <%> (g h) system. There are four types of extrema T minima (with a orbits), D3 minima (with /3 orbits), D3 saddle points (with y orbits) and D2 saddle points (with S orbits). For a dominant JT stabilization from the G mode, the system has T minima (a orbits) only. For a dominant H mode, the system has D3 minima (/3 orbits). The result of the linear problem shows the possibility of a non-degenerate ground state derived from D3 well states. Thus here we consider only the situation when H modes dominate. 

The instability arises and evolves owing to thermodynamic fluctua tion (3.29). Such a fluctuation may cause complete system state decay (see, e.g., region V of unstable saddles in Figure 3.4). Flowever, it may also happen that the arising instability creates a new state of the system to be stabilized in time and space. An example is the formation of the limit (restricted) cycle in a system that involves the exceptional point of the unstable focus type. The orbital stability of such a system means exactly the existence of certain time stabilized variations in the thermody namic parameters (for example, the concentrations of reactants) that are... 

Theoretical calculations of the ab initio type by Radom et a/. (1971, 1972) and at semi-empirical level by Bodor and Dewar (1971 Bodor et al., 1972) did not give consistent results. Recent ab initio calculations including electron correlation by Lischka and Kohler (1978) are inconsistent with earlier ab initio work. Their calculations have confirmed the stability of the 2-propyl cation and the instability of face-protonated cyclopropane [40]. Edge-protonated cyclopropane [41] is found to be a saddle point on the potential energy surface of lower energy than the corner protonated species [42]. 

If Re(A) 0 for both eigenvalues, the fixed point is often called hyperbolic. (This is an unfortunate name—it sounds like it should mean saddle point —but it has become standard.) Hyperbolic fixed points are sturdy their stability type is unaffected by small nonlinear terms. Nonhyperbolic fixed points are the fragile ones. 

Putting it all together, we arrive at the stability diagram shown in Figure 8.5.10. Three types of bifurcations occur homoclinic and infinite-period bifurcations of periodic orbits, and a saddle-node bifurcation of fixed points. 

The complex stabilization method of Junker (7), although it was introduced in a different way, gives practically the same computational prescription as the CESE method, as far as the way of using complex coordinates is considered. Another approach of this type, resembling the CESE method as well as the complex stabilization method, is the saddle-point complex-rotation technique of Chung and Davis (29). These methods provide cleair physical insight into the resonance wave function. They differ in the way the localized paxt of the wave function is expanded in basis sets and how it is optimized. 

The NHIM is of saddle stability type, having 2d - 2)-dimensional stable and unstable manifolds W E) and W (E) that are diffeomorphic to g2d-3 Being of co-dimensiorf one with respect to the energy surface. 

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