For a logistic map using $f_r(x)=rx(1-x)$, what values of $r$ and starting $x$ are guaranteed (i.e., with an accepted proof) to be chaotic? I mean "chaotic" in the loosest sense: The limiting sequence of $x$ does not tend to a finite number of points.

I am currently using $r=3.8$ and starting $x=0.501234567890123456789$, but have only tested through 10,000 iterations. What is the probability that I am chaotic?

For a logistic map using $f_r(x)=rx(1-x)$, what values of $r$ and starting $x$ are guaranteed (i.e., with an accepted proof) to be chaotic? I mean "chaotic" in the loosest sense: The limiting sequence of $x$ does not tend to a finite number of points.

I am currently using $r=3.8$ and starting $x=0.501234567890123456789$, but have only tested through 10,000 iterations. What is the probability that I am chaotic?

EDIT: Below are new results (with 4,000,000 bits of precision to avoid any rounding problems) for 2,000,000 iterations (showing matches to "0.72224", the end's most significant digits). So, I believe it is fair to say that there are 3 possible cases:

1. There is no limit cycle (through infinity),

2. There is a limit cycle of at least 1,105,578 points, or

3. There is a smaller limit cycle but any "two points chosen from the first 2,000,000 points" are not both within one limit point's attractor zone.

#2 seems the most unlikely. #3 seems unlikely simply because I chose such round numbers from the start. According to answers here, however, it does seem like the probability for #1 is not 100%. Maybe someone can put my statements here into proper mathematical language and clarify this better.

      n: x_n
-----------------------
  53951: 0.7222489331
  66539: 0.7222408270
  68976: 0.7222441979
  75138: 0.7222495664
 120428: 0.7222473699
 134963: 0.7222441673
 235912: 0.7222411119
 395643: 0.7222459509 closest greater value
 417062: 0.7222404139
 462528: 0.7222468852
 472142: 0.7222408308
 645137: 0.7222474275
 679584: 0.7222492244
 731458: 0.7222410420
 761284: 0.7222468048
 891274: 0.7222442328
 894423: 0.7222448046 closest lower value
 935412: 0.7222498698
1110025: 0.7222446506
1220483: 0.7222447341
1222255: 0.7222485044
1269796: 0.7222407187
1301786: 0.7222439936
1422147: 0.7222488714
1431959: 0.7222457998
1503338: 0.7222445272
1509878: 0.7222404127
1568206: 0.7222447453
1569439: 0.7222415020
1612039: 0.7222497768
1634269: 0.7222406207
1642044: 0.7222450907
1791569: 0.7222487370
1865739: 0.7222420900
1879844: 0.7222427753
1902889: 0.7222493257
2000000: 0.7222453893 end