Unstable saddle

The partial differential equations used to model the dynamic behavior of physicochemical processes often exhibit complicated, non-recurrent dynamic behavior. Simple simulation is often not capable of correlating and interpreting such results. We present two illustrative cases in which the computation of unstable, saddle-type solutions and their stable and unstable manifolds is critical to the understanding of the system dynamics. Implementation characteristics of algorithms that perform such computations are also discussed. 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

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. 

In previous chapters we have dealt only with systems which have one or two independent concentrations. This has been sufficient for a wide range of intricate behaviour. Even with just a single independent concentration (one variable), reactions may show multiple stationary states and travelling waves. Oscillations are, however, not possible. To understand the latter point we can think in terms of the phase plane or, more correctly for a one-dimensional system, the phase line (Fig. 13.1(a)). As the concentration varies in time, so the system moves along this line. Stationary-state solutions are points on the line the arrows indicate the direction of motion along the line, as time increases, towards stable states and away from unstable ones. Figure 13.1(b) shows this motion and phase line behaviour represented in terms of some potential, with stable states a minima and an unstable (saddle) solution as a maximum. 

For yc = 0.6 find the minimum value of Kc, that makes the middle unstable saddle-type steady state unique and stable. 

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 middle two plots show the dynamics of the reaction in the second tank. One steady state of tank 2 lies at (xAi(Tend), XBi(Tend), y Tend)) ss (0.33,0.67,1.28) and another at (xAi Tend), XBi Tend), y (Tend)) ss (0, 0.2,1.87). The latter gives the smaller yield of jg and results from the initial second tank conditions (x,i2(0), xb2(0), 2/2(0)) = (0.95,0,1.3) depicted in black. These two steady states are stable. There is another unstable steady state for this data, but our graphical method does obviously not allow us to find it because it is an unstable saddle-type steady state that will repel any profile that is near to it. It can be easily obtained from the steady-state equations, though. For a method to find all steady states of a three CSTR system, see Section 6.4.3. 

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... 

At Dpu the periodicity of the system changes form period one (PI) to period two (P2). Figure 7.31(A) shows that as D decreases further, the periodic attractor P2 grows in size till it touches the middle unstable saddle-type steady state and the oscillations disappear homoclinically at D bt = 0.041105 hr-1 without completing the Feigenbaum25 period-... 

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

Let Bt be an unstable saddle. Then k, will be a positive eigenvalue of the matrix for the linear approximation at this point. 

Application of this theorem permits analysis of the equilibrium points of the system with a monotone binding curve. If in the equilibrium point P = (y(, cf) we have (jf (y ) < 0, the system is stable. If the feedback curve is assumed to be continuous over a domain of permissible values of receptor occupancy y (r), in the nearest equilibrium point P2 = (2/2 c2) we will have (y2) > 0. This condition is necessary but not sufficient for the instability of the system. But if moreover dc B/dy, one eigenvalue from (11.6) is positive and the system becomes unstable at P2, which is an unstable saddle point or repellor. 

Figure 11.3 State space for different initial conditions. The equilibrium point Pi ( o ) is a stable focus, P2 ( ) is an unstable saddle point, and P3 ( ) is stable.

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... 

For 0.53 < X < 0.72, one has the sequence U —> D —> O —> PR as before [see Fig. 13(b)], however there is an additional bifurcation between PR states. In fact, the limit cycle amplitude of the PR regime, now labeled PRi [curve 2 in Fig. 13(b)], abruptly increases. This results in another periodic rotating regime labeled PR2 with higher reorientation amplitude [curve 3 in Fig. 13(b)]. This is a hysteric transition connected to a double saddle-node structure with the (unstable) saddle separating the PRi and PR2 branches as already found... 

X =[Ri,K2,M3] with IR2 = 1 3 Asymptotically stable/unstable node + half unstable saddle/half stable saddle node 2... 

As in the case of commensalism, the steady-state solution is a stable node (Miura et al., 1980). When competition is accompanied by mutualism (and/or commensalism), two kinds of steady-state solutions are obtained owing to the values of system parameters, one being a stable node and the other a stable focus or an unstable saddle point. Damped oscillations tend to occur when either Xi or x2 has a lower growth ability than the other. 

According to linear stability analysis the trivial stationary point is an unstable saddle point, while the nontrivial stationary point is a marginally stable centre. 

Figure 7.10 Phase plane for fx = fii m Figure 7.9A. Solid curve stable limit cycle trajectories unstable saddle stable steady state (node or focus) O unstable steady state (node or focus) dashed curve separatix. 

A Psuedo Maximum of Periodic Attractor Average on Periodic Attractor H Unstable Saddle Middle Steady Slate Unique Stable ARractor T Psuedo Minimum of Periodic ARractor... 

In the case of no catalyst inflow, the lowest solution corresponding to no conversion is always a stable node (sn). The middle solution is always an unstable saddle point (sp). The nature of the highest extent of conversion varies with the residence time it may be a stable node, stable focus (sf), unstable focus (uf) or unstable node (un). For non-zero catalyst inflow there is further complication as the character along the lowest branch, which now corresponds to non-zero extents of conversion, varies between stable and unstable node or focus. Some of the different sequences found are indicated on the mushrooms in Fig. 3. 

For close systems with instantaneous diffusion, steady states are either stable or unstable depending on whether the solid phase consists in a single grain or in N grains (N>1). For open systems and uniform liquid phases, steady states are unstable saddle-points. For close systems with diffusion, results are similar to the first case ones. For open systems with diffusion, steady states are unstable and uniform. 

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