Transient Length

A bound on the transient length of the one-dimensional majority rule (equation 5.142) follows in straightforward fashion from our more general results of the previous section. 

Suppose we have a finite initial state a = (0°°,. .., 0°°), where i m 

A sharper bound is given by Tchuente [tch82], who showed that Tjd majority r x), where r x) is the cardinality of the set of blocks of consecutive 1-valued sites making up the finite initial state (see also [golesQO]). 

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Transient length increases to about 10 steps periodic structures (with short periodicity) emerge. 

X (j)) 0.35 Transient length grows significantly a new kind of periodic structure with longer period appears, thereby increasing the spectrum of possible dynamical behaviors. 

Transient length increases to about 60 steps periodic structures of period equal to about 40 steps appear. 

Dynamical behavior depends on lattice size N only for A near A transient length, for example, grows exponentially with N for A = Ac. 

As A increases beyond A,., transient lengths decrease until, at A = 3/4, they no longer show any dopendeiu-.e on lattice size. 

A slightly different Lyapunov function from the one defined above can be used to put bounds on the transient length. Let... 

To see how to use this result to bound the transient length, we first note that since L is strictly decreasing in the transient phase, equation 5.129 yields... 

Theorem 6 [goles87b] If A is symmetric, then the transient length of the generalized threshold rule defined in equation 5.121 is bounded by... 

Suppose that the initial state is confined to a rectangle R = (i,nin> max) x (i where imin and max are the minimal and maximal i coordinates of the region containing nonzero sites and j in and jmax are the minimal and maximal j coordinates of that same region. Then according to corollary 1, the transient length T2d majority is bounded by... 

Goles [goles86c] shows that (1) only period T = 4 orbits are allowed, and (2) the transient length ra t,i symmetric <... 

Meiss RA (1993) Persistent mechanical effects of decreasing length during isometric contraction of ovarian ligament smooth muscle. J Muscle Res Cell Motil 14 205218 Meiss RA (1994) Transient length-related mechanical states in smooth muscle. Can J Physiol Pharmacol 72 13251333... 

Figure 11.8. Peak resolution increases in FTMS as the transient length increases. (A) A 0.15-s transient yields aresolving power of 9000 at miz 857 in a 7-T instrument. (B) A resolving power of75,000 is achieved for the same ion, but with a 1.2-s transient. (Adapted from Ref. 161.)...

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