Steady-State Bifurcations

FIGURE I Steady-state bifurcation diagrams for variations in the reactant partial pressures, (a) Partial two-parameter bifurcation diagram representing the projection of turning points, Hopf bifurcation points, and apparent triple points. (b)-(Q One parameter sections of the steady-state ffe surface. The vertical axes are the steady-state (k and range from 0 to I. The horizontal axes correspond to the appropriate axis of the two parameter diagram (a). Steady-states are stable or unstable for solid or dashed curves respectively and periodic branches are denoted by pairs of chained curves which represent the minimum and maximum values of ffe on the limit cycle. The periodic branches all terminate in Hopf bifurcations or, when a saddle is present, homoclinic (infinite period) bifurcations. (b)-(e) a, = 0.017, 0.019, 0.021, 0.025 (f)-(i) oti = 0.031, 0.028, 0.024, 0.022. 

In the case of steady state bifurcations, certain eigenvalues of the linear-approximation matrix reduce to zero. If we consider relaxations towards a steady state, then near the bifurcation point their rates are slower. This holds for the linear approximation in the near neighbourhood of the steady state. Similar considerations are also valid for limit cycles. But is it correct to consider the relaxation of non-linear systems in terms of the linear approximations To be more precise, it is necessary to ask a question as to whether this consideration is sufficient to get to the point. Unfortunately, it is not since local problems (and it is these problems that can be solved in terms of the linear approximations) are more simple than global problems and, in real systems, the trajectories of interest are not always localized in the close neighbourhood of their attractors. 

Olsen, R. J. Epstein, I. R. Bifurcation analysis of chemical reaction mechanisms. 1. Steady-state bifurcation structure. J. Chem. Phys. 1991, 94, 3083-3095 Bifurcation analysis of chemical reaction mechanisms. 2. Hopf bifurcation analysis. J. Chem. Phys. 1993, 98, 2805-2822. 

Equilibrium state —Linear steady state close to equilibrium —Steady state —> Non-linear steady state — Bifurcation phenomena —> Multi-stability —> Temporal and spatio-temporal oscillations —> More complex situations (chaos, turbulence, pattern formation, fractal growth). All these stages have been discussed in different chapters of the book. 

The shape of this torus which can be thought as a sphere with a hole going from north pole to south pole is not as surprising as it seems at first sight. Similar shapes were obtained in theoretical studies by GUCKENHEIMER (4) and lANGFOE D (5) they occur ais the results of the interactions of two instabilities - a Hopf bifurcation and a steady state bifurcation - which are both generic of these chemical systems (6). 

At this point a new branch of nontrivial steady states bifurcates. A bifurcation analysis shows that this new branch can emerge subcritically or supercritically, depending on the value of X. For X>X with... 

R. 2.2.Steady-State Bifurcation Analysis. In reactor dynamics, it is particularly important to bnd out if multiple stationary points exist or if sustained oscillations can arise. 

Experimental information on the early evolution and the beginning of live is extremely difficult to obtain only the arrival of microbial genomics has allowed to reliably retrace subsequent developments. The earth was formed about 4.5 10 years ago. The earliest traces of primitive cells occurred possibly as early as 3.5 10 years ago (Schopf 2006). Nevertheless, this leaves several hundred million years for the actual chemical evolution. The evolution of cells (Woese 2002), however, has taken the major portion of the remaining time. The step from microorganisms (bacteria) to higher live forms occurred much later— less than 1 10 years ago. Even tough the very early steps towards cellular Uve are still in the dark, it is very likely that autocatalytic reactions in conjunction with steady state bifurcations, of which we... 

Figure A3.14.3. Example bifurcation diagrams, showing dependence of steady-state concentration in an open system on some experimental parameter such as residence time (inverse flow rate) (a) monotonic dependence (b) bistability (c) tristability (d) isola and (e) musliroom.

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Furtliennore, since tlie bifurcation must occur from a stable homogeneous steady state we must have D ID < 1 i.e. tlie diffusion coefficient of tlie inhibitor is greater tlian tliat of tlie activator. The critical diffusion ratio at tlie bifurcation is... 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

The next step should clarify why the unstable growth of the variable x occurs through a stable state at the bifurcation point. To determine the stability of the bifurcation point, it is necessary to examine the linear stability of the steady-state solution. For Eq. (1), the steady-state solution at the bifurcation point is given as jc0 = 0. So, let us examine whether the solution is stable for a small fluctuation c(/). Substituting Jt = b + Ax(f) into Eq. (1), and neglecting the higher order of smallness, it follows that... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

GP 9] [R 16] The reaction rate and activation energy of metal catalysts (Rh, Pt or Pd) supported on alumina particles ( 3 mg 53-71 pm) were determined for conversions of 10% or less at steady state (1% carbon monoxide 1% oxygen, balance helium 20-60 seem up to 260 °C) [7, 78]. The catalyst particles were inserted into a meso-channel as a mini fixed bed, fed by a bifurcation cascade of micro-channels. For 0.3% Pd/Al203 (35% dispersion), TOF (about 0.5-5 molecules per site... 

Figure 38, Chapter 3. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel Upon variation of substrate and product saturation (as global parameter, set equalfor all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage r of unstable models, with darker colors corresponding to a higher percentage of unstable models (see colorbar for numeric values).

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Figure 19. The steady state solutions A0 of the pathway shown in Fig. 18 as a function of the influx vi. For an intermediate influx, two pathways exist in two possible stable steady states (black lines), separated by an unstable state (gray line). The stable and the unstable state annihilate in a saddle node bifurcation. The parameters are k.2 0.2, 3 2.0, K] 1.0, and n 4 (in arbitrary units).

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