For a [logistic map](https://en.wikipedia.org/wiki/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?

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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