Global unstable

State X(t) remains in some vicinity of Xg for x(f) < critical and moves away from Xg for x(t) > critical. This represents a locally stable but globally unstable state Xg. 

The thermodynamic critical field above with the SmA phase becomes globally unstable to the N phase at some temperature N -smA lower than To follows from Eq. 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

Let us now consider the case where the global unstable set of the saddle-node periodic orbit L is not a manifold, but has the structure like shown in Fig. 12.4.1. This means that the integer m which determines the homotopy class of the curve fl jSq in the cross-section 5q x = —d is... 

Theorem 12.9. Consider a one-parameter family of dynamical systems which has a saddle-node periodic orbit L at = 0 such that all orbits in the global unstable set tend to L as t -foo, but do not lie in W[. Let the essential map satisfy m = 0 and fo (p) < 1 for all (p. Then after disappearance of the saddle-node for /i > 0, the system has a stable periodic orbit non-homotopic to L in U) which is the only attractor for all trajectories in U. 

Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75 (1997) 293-317... 

Transition Regime the kinks become unstable and are able to propagate but remain localized global intermittent pattern cannot yet be formed. The patterns arc reniiniscent of class c4 behavior. 

The large energy differences between the global minimum structure of C2v symmetry and the other isomers indicate that equilibrium sulfur vapor will contain only minute amounts of the latter, even at very high temperatures. However, under non-equilibrium conditions as in electrical discharges or by illumination with a laser as in Raman spectroscopy unstable isomers may be formed and detected. 

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. 

Overview Batch processes are mostly suited to low volume high value added products that are usually characterised by common recipes, which render them amenable to sharing of equipment units. Due to their intrinsic adaptation to sudden changes in recipe, they are processes of choice in volatile or unstable conditions that have become regular in global markets. This chapter provides the background information on batch chemical processes, which constitutes the basis for the forthcoming chapters. Only the essential elements of batch plants are captured with references, where necessary, to further sources of information for the benefit of the reader. 

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).

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