Harry's answer was along the lines: "Think of the period doubling as the progression of fractions: 1 + 1/2 + 1/4 + 1/8 + 1/16 ... = 2. You know that the progression is infinite and is doesn't go beyond 2." (I hadn't thought of this, but it makes sense). "So you can draw a line and say 'chaos starts here', but the doubling goes on right up to the line".
Well, the explanation worked for me. You may get different milage.
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