Isola varieties

Figure 3.6 shows codimension-1 singularities in a parameter plane spanned by the activation energy of the anodic reaction yA and the activation energy of the electrolyte conductivity yK, when the total cell current I is used as the bifurcation parameter. Hysteresis and isola varieties are found. The nonlinear behavior becomes increasingly complicated for increasing values of yK. Steady-state multiplicities even exist for rather small values of yK, but vanish if the electrolyte s conductivity is temperature-independent, i.e., yK = 0. This confirms that the multiple steady states are caused not by the electrochemical reaction alone, but by the combination of reaction and varying electrical conductivity. 

Eq. (23 has at most 2N+1 solutions and up to (2N+1) bifurcation points. An Isola variety exists in this case so that the bifurcation diagrams are more intricate and contain isolas (isolated branches) in addition to the hysteresis loops. 

It was shown in [6J that the Hysteresis and Isola varieties divide the (a-.o jCX-) space into seven regions with different bifurcation diagrams. A construction of the Hysteresis and Isola varieties of the steady-state Eq. (11) has shown that the seven bifurcation diagrams shown in Figure 4 are the only ones that exist in the global parameter space (a,B,9, y) [4J. 

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... 

Figure 1. Possible forms of transformation of an unstable bifurcation diagram (middle column) into either one of two possible stable forms (left or right column) at the Hysteresis (a), Isola (b, c) and Double limit varieties (d, e).

The Isola and Double Limit varieties do not exist in this case. The Hysteresis variety (a =0) divides the a. space into two regions (a > 0 and a. < 0) corresponding to the two bifurcation diagrams shown in Figures 2.a and 2.b. These two are also the only possible global bifurcation diagrams (0 vs. Da) for Eq. (13) as the Hysteresis variety (B 4) divides the B. space into two regions. 1 1... 

A major breakthrough with regard to the understanding of this phenomenon in the field of chemical reaction engineering was achieved by Ray and co-workers (Uppal et ai, 1974, 1976) when in one stroke they uncovered a large variety of possible bifurcation behaviours in non-adiabatic continuous stirred tank reactors. In addition to the usual hysteresis type bifurcation, Uppal et al. (1976) uncovered different types of bifurcation diagrams, the most important of which is the isola which is a closed curve disconnected from the rest of the continuum of steady states. 

The prototype, cubic autocatalytic reaction (A + 2B 3B) forms the basis of a simple homogeneous system displaying a rich variety of complex behaviour. Even under well-stirred, isothermal open conditions (the CSTR) we may find multi stability, hysteresis, extinction and ignition. Allowing for the finite lifetime of the catalyst (B inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas and mushrooms. Sustained oscillations (stable limit cycles) are also possible. There are strong analogies between this simple system and the exothermic, first-order reaction in a CSTR. 

Despite its simplicity, our model displays an unexpected variety of stationary-state behaviour, with isolas and mushrooms for the dependence of the extent of conversion on residence time or flow-rate. There is also a rich variety of stability and character, with stable and unstable nodes and foci. It is also the simplest isothermal scheme to show sustained oscillatory responses. 

---