Small lobe condition

Let us now consider the question concerning what happens when is non-smooth. For the first time, this question was studied in [3] where it was discovered that the possibility of the breakdown of the invariant manifold causes an onset of chaos at such bifurcations. In particular, sufficient conditions (the so-called big lobe and small lobe conditions) were given in [3] for the creation of infinitely many saddle periodic orbits upon the disappearance of a saddle-node in the non-smooth case. Subsequent studies have shown that these conditions may be further refined so we may reformulate them as follows. 

The small lobe condition was originally introduced for the case where is a torus (m = 1), but it makes sense at m = 0 and m = — 1 as well. In terms of the essential map, the small lobe condition is satisfied if there exist

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Theorem 12.6. If the small lobe condition is satisfied then on the pi-axis there is a sequence of intervals Aj which accumulate at p +0, such that the system has infinitely many saddle periodic orbits for any fx G Ai. 

Let us now consider briefly the question on what may happen if is a non-smooth Klein bottle. Since Theorems 12.5 and 12.6 are applicable in this case, it follows that when /x > 0, chaos may appear when the big lobe or the small lobe condition is satisfied. However, a direct analogue of Theorem 12.7 does not exist here because of the following possibility ... 

So, the results of Theorems 12.3, 12.5 and 12.7 are summarized as follows IfW is a smooth toruSy then a smooth attracting invariant torus persists after the disappearance of the saddle-node L. If is homeomorphic to a torus but it is non-smoothy then chaotic dynamics appears after the disappearance of L, Herey either the torus is destroyed and chaos exists for all small /i > 0 the big lobe condition is sufficient for that)y or chaotic zones on the parameter axis alternate with regions of simple dynamics. 

Water Wheels.—The older types of wheel give maximum efficiencies up to 70 to 90 per cent when used under the particular conditions for which they are suitable . c., for heads up to 70 ft. and powers not exceeding 75 hp. Important wheels are turbines. These are of the tangential (impulse) or reaction type. Tangential turbines use the velocity of water to produce power, the head being wholly converted to velocity in open-end nozzles. They run at low peripheral speeds for given heads and are therefore the type best adapted to high heads (300 to 3,000 ft.) and small quantities of water. Their construction is simple. The runners carry double-lobed buckets toward which the jets are directed and the stream is deflected about 90 on both sides of the splitter. ... 

Anticipated collapse modes are examined for contrasting small (200 m) and large (350 m) endogenous dome geometries, for varied inclinations of the detachment plane, and under different conditions of interior water- and gas-pressurization, and for lobe failures resulting from only rainfall induced overpressures [Figure I). 

A relatively new shape for molecular sieve pellets is the TRISIV adsorbent offered by UOP. This is an extrusion with a trilobal (cloverleaf) cross-section. The diameter of each lobe is typically Ai, in. however, since the lobes are joined to give a much larger effective diameter for the extrusion, the pellet length is about 3 to 4 times the lobe diameter. The impact of particle shape on performance was analyzed by Ausikaitis (1983). He compared TRISIV with beads and pellets for the same dehydration service and concluded that, for the specific conditions and assumptions of the study, the TRISIV design offered a small but significant economic advantage. 

Line emission from [FdQ] is produced by dectron collisions from fast shocks and possibly hard photoionization (PJ.). As wdl as the correct ionization conditions, we seem to require an enhanced abundance of Iron in the gas phase (Greenhouse ei aX. 1991). Such a grain destruction medianism would result from a SN blast wave, or other fast shocks. The non-thermal radio emission from starburst galaxies is due to SNRs and superwinds, fri Seyfert galaxies this emission is often associated with jets or lobes powered by processes at very small scales (possibly a black hole). It can be difficult to separate these different mechanisms by radio emission or spectral index alone. 

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