Unstable set

We now consider, for comparison, fluctuational escape from the Lorenz attractor, which, for a certain range of parameters, is a quasihyperbolic attractor consisting of unstable sets only [161] ... 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

Remark. In the multi-dimensional case where besides the central coordinates there are also the stable ones, the unstable set consists of three curves, whereas the stable set is a bimch consisting of three semi-planes intersecting along the non-leading manifold as shown in Fig. 10.5.4, for the three-dimensional example. 

As already mentioned, problems of this nature had appeared as early as in the twenties in connection with the phenomenon of transition from synchronization to an amplitude modulation regime. A rigorous study of this bifurcation was initiated in [3], under the assumption that the dynamical system with the saddle-node is either non-autonomous and periodically depending on time, or autonomous but possessing a global cross-section (at least in that part of the phase space which is under consideration). Thus, the problem was reduced to the study of a one-parameter family of C -diffeomorphisms (r > 2) on the cross-section, which has a saddle-node fixed point O at = 0 such that all orbits of the unstable set of the saddle-node come back to it as the number of iterations tends to -hoo (see Fig. 12.2.1(a) and (b)). 

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. 

The fourth and last situation corresponds to the blue sky catastrophe , i.e. when both period and length of the periodic orbit go to infinity upon approaching the stability boundary. This boundary is distinguished by the existence of a saddle-node periodic orbit under the assumption that all trajectories of the unstable set W ( ) return to as t -> -hoc, where W C ) n — 0. The tra-... 

The following conditions of general position are needed to define the stability boundary 55 let us denote the unstable set of C by It is locally... 

Both cases have much in common in the sense that the imstable set of both bifurcating equilibrium states is one-dimensional. If the unstable set of the critical equilibrimn state is of a higher-dimension, then the subsequent picture may be completely different. Figure 14.3.1 depicts such a situation. When the imstable cycle shrinks into the equilibrium state we have a dilemma the representative point may jump either to the stable node 0 or to the stable node 02- Therefore this dangerous boundary must be classified as dynamically indefinite. 

Consider the other hypothetical example. Let a two-dimensional diffeo-morphism at e = 0 have a phase portrait as shown in Fig. 14.3.2. Here, O2 and O3 are stable fixed points, and 0 is a saddle. The unstable set Wq of the saddle-node O intersects transversely the stable manifold Wq ... 

We can now assert that a stability boundary is dynamically definite if upon crossing over the boundary the behavior of the representative point is uniquely defined. This situation does occur in the case where the unstable set of the equilibrium state (the periodic trajectory) contains at most one attractor at the critical parameter value. 

In contrast, if the choice of the new regime for the representative point is ambiguously defined, then we can assert that such a boimdary is dynamically indefinite. This occurs if at least two attractors belong to the boundary of the unstable set. It must also contain saddles whose unstable invariant manifolds separate the basins of the attractors. 

Afraimovich, V., Bykov, V. V. and Shilnikov L. P. [1983] On attracting structurally unstable sets of Lorenz attrcator type , Trans. Moscow. Math. Soc. 44, 150-213. 

Dissolve 0 01 mol of the phenohc ether in 10 ml. of warm chloroform, and also (separately) 0 01 mol of picric acid plus 5 per cent, excess (0 -241 g.) in 10 ml. of chloroform. Stir the picric acid solution and pour in the solution of the phenohc ether. Set the mixture aside in a 100 mb beaker and ahow it to crystallise. Recrystahise the picrate from the minimum volume of chloroform. In most cases equahy satisfactory results may be obtained by conducting the preparation in rectified spirit (95 per cent. CjHgOH). The m.p. should be determined immediately after recrystallisation. It must be pointed out, however, that the picrates of aromatic ethers suflFer from the disadvantage of being comparatively unstable and may undergo decomposition during recrystaUisation. 

With batch reactors, it may be possible to add all reactants in their proper quantities initially if the reaction rate can be controlled by injection of initiator or acqustment of temperature. In semibatch operation, one key ingredient is flow-controlled into the batch at a rate that sets the production. This ingredient should not be manipiilated for temperature control of an exothermic reactor, as the loop includes two dominant lags—concentration of the reactant and heat capacity of the reaction mass—and can easily go unstable. 

Fluidized This is an expanded condition in which the sohds particles are supported by drag forces caused by the gas phase passing through the interstices among the particles at some critical velocity. It is an unstable condition in that the superficial gas velocity upward is less than the terminal setting velocity of the solids particles the gas... 

Liquid-Column Breakup Because of increased pressure at points of reduced diameter, the liquid column is inherently unstable. As a result, it breaks into small drops with no external energy input. Ideally, it forms a series of uniform drops with the size of the drops set by the fastest-growing wave. This yields a dominant droplet diameter... 

The material charge of continuous mills called the holdup cannot be set directly but is indirectly determined by operating conditions. There is a maximum throughput rate that depends on the shape of the mill, the flow characteristics of the feed, the speed of the mill, and the type of feed and discharge arrangement. Above this rate the holdup increases unstably. 

These substances contain the -C=NH group and, because they are strong, unstable bases, they are kept as their more stable salts, such as the hydrochlorides. (The free base usually hydrolyses to the corresponding oxo compound and ammonia.) Like amine hydrochlorides, the salts are purified by solution in alcohol containing a few drops of hydrochloric acid. After treatment with charcoal, and filtering, dry diethyl ether (or petroleum ether if ethanol is used) is added until crystallisation sets in. The salts are dried and kept in a vacuum desiccator. 

Figure 21-9 is a stability wind rose that indicates Pasquill stability class frequencies for each direction. For this location, the various stabilities seem to be nearly a set proportion of the frequency for that direction the larger the total frequency for that direction, the greater the frequency for each stability. Since the frequencies of A and B stabilities are quite small (0.72% for A and 4.92% for all three unstable classes (A, B, and C) are added together and indicated by the single line. 

Hence, to give a GM of 2 and a PM of 50°, the controller gain must be set at 1.0. If it is doubled, i.e. multiplied by the GM, then the system just becomes unstable. Check using the Routh stability criterion ... 

If C is a reactive, unstable species, its concentration will never be very large. It must then be consumed at a rate that closely approximates the rate at which it is formed. Under these conditions, it is a valid approximation to set the rate of formation of C equal to its rate of destruction ... 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

Both sets of names are used in the literature. Free imidodisulfuric acid HN(S03H)2 (which is isoelectronic with disulfuric acid H2S2O7, p. 705) and free nitridotrisulfuric acid N(S03H)3 are unstable, but their salts are well characterized and have been extensively studied. 

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