answer_intermittent.html

非线性时间学列分析工具

The logistic equation near <font color=blue>a=1.75</font>
 shows so called <b>type-I
intermittency</b>, where a periodic orbit is born out of the chaotic
regime by a tangent bifurcation.
For <font color=blue>a</font>
 larger than the bifurcation value, the graph of the third iterate of the
map intersects the diagonal six times: Three times for the stable
period three orbit, and three times for an unstable counterpart (slope
at the intersection has modulus larger than unity), plus one
intersection  for the unstable genuine fixed point (period one orbit). 
For <font color=blue>a</font> slightly below the bifurcation value,
 none of these 6 
intersections exists, but the graph touches the diagonal almost
tangentially (therefore tangent bifurcation). It thus forms, together
with the diagonal, a very thin channel, through which the trajectory
has to pass in very many iterations. Hence, one observes long episodes
of almost period-three motion, until the trajectory leaves these
channels and performs for some steps chaotic motion. You can convince
 yourselves that this is indeed the case by selecting
a part of the trajectory which starts just before an almost period
 part and finishes just after the end of this periodic part (assumed
 to be stored in the file 
 intermittency.dat) and plotting only every third data point 
 with lines:<br><br>
<font color=green>  plot '&#60; delay intermittency.dat -d3 ' every 3 w
 linespoints, x</font>