Windows, periodic

is the most conspicuous. Suddenly, against a backdrop of chaos, a stable 3-cycle appears out of the blue. Our first goal in this section is to understand how this 3-cycle is created. (The same mechanism accounts for the creation of all the other windows, so it suffices to consider this simplest case.) 

One can show analytically that the value of r at the tangent bifurcation is 1 =3.8284.. . (Myrberg 1958). This beautiful result is often mentioned in 

Part of the orbit looks like a stable 3-cycle, as indicated by the black dots. But this is spooky since the 3-cycle no longer exists We re seeing theghost of the 3-cycle. 

Eventually, the orbit escapes from the channel. Then it bounces around chaotically until fate sends it back into a channel at some unpredictable later time and place. 

Intermittency is not just a curiosity of the logistic map. It arises commonly in systems where the transition from periodic to chaotic behavior takes place by a saddle-node bifurcation of cycles. For instance. Exercise 10.4.8 shows that intermittency can occur in the Lorenz equations. (In fact, it was discovered there see Pomeau and Manneville 1980). 

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Other periodic windows harbor period triplings, quadruplings, etc., occurring at different sets of di, but all of which scale in the familiar fashion (albeit with different universal 8 8). 

Universality in Unimodal Maps A seminal work on the 2-symbol dynamics of one-dimensional unimodal mappings due to Metropolis, Stein Stein [metro73]. Specifically, they studied the iterates of various mappings within periodic windows, labeling the attractor sequences by strings of the form RLLRL , where R and L indicate whether f xo) falls to the right or left of xq, respectively. Each periodic sequence therefore corresponds to a unique finite length word made up of R s and L s. 

For r > the orbit diagram reveals an unexpected mixture of order and chaos, with periodic windows interspersed between chaotic clouds of dots. The large window beginning near r 3.83 contains a stable period-3 cycle. A blow-up of part of the period-3 window is shown in the lower panel of Figure 10.2.7. Fantastically, a copy of the orbit diagram reappears in miniature ... 

One of the most intriguing features of the orbit diagram (Figure 10.2.7) is the occurrence of periodic windows for r> r . The period-3 window that occurs near... 

In experimental systems, intermittency appears as nearly periodic motion interrupted by occasional irregular bursts. The time between bursts is statistically distributed, much like a random variable, even though the system is completely deterministic. As the control parameter is moved farther away from the periodic window, the bursts become more frequent until the system is fully chaotic. This progression is known as the intermittency route to chaos. 

Figure 10.6.2 shows thatthe qualitative dynamics of the two maps are identical. They both undergo period-doubling routes to chaos, followed by periodic windows interwoven with chaotic bands. Even more remarkably, the periodic windows occur in the same order, and with the same relative sizes. For instance, the period-3 window is the largest in both cases, and the next largest windows preceding it are period-5 and period-6. 

But there are quantitative differences. For instance, the period-doubling bifurcations occur later in the logistic map, and the periodic windows are thinner. ... 

No windows forthe tent map) Prove that, in contrast to the logistic map, the tent map does not have periodic windows interspersed with chaos. 

The final nail in the coffin was the demonstration that the chemical system obeys the U-sequence expected for unimodal maps (Section 10.6). In the regime past the onset of chaos, Roux et al. (1983) observed many distinct periodic windows. As the flow rate was varied, the periodic states occurred in precisely the order predicted by universality theory. 

The Liapunov exponent is necessarily negative in a periodic window. But since A =lnr > 0 for all r > 1, there canbe po periodic windows after the onset of chaos. 

Gestation is the period during which each individual s sexuality is first expressed and shaped. But impact on the organization of fetal sexuality seems to be most effective during certain sensitive gestational periods, windows of vulnerability. The exact periods for human behavioral effects of sex hormones remain unknown. The period 8—24 weeks of gestation may be most critical because that s when testosterone secretion surges in male fetuses, but there may be multiple sensitive... 

The periodic windows in the U-sequence exist in the "chaotic regime (e.g., the regime of Section 3.2) that follows the accumulation point of the period doubling sequence [50]. There are no chaotic intervals, yet the set of bifurcation parameter values for which the behavior is chaotic has positive measure. [For a typical map probably about 85% of the measure (of the bifurcation parameter range in which the U-sequence occurs) corresponds to chaotic states, and 15% to periodic states.]... 

The periodic windows in the chaotic regimes lose stability either by period doubling (towards longer x), or by a tangent bifurcation and intermittency (towards shorter x) Figure 1(a) shows the third iterate of the return map of an intermittent state, for x just less than a value in the period three window in C, The same behavior has been seen just outside a prominent period five window in C2. 

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