It would have been easy to make the article more clear. Simply explain what aspect of Bitcoin is supposed to stabilize.
Given the unending fascination with the USD/BTC exchange rate, it can be assumed that the author is referring to this metric. The inclusion of the heading "Does the price of Bitcoin have a Nash equilibrium?" also supports this idea.
However, if this is the case, why does the author include the following quote the middle of the article:
Why has Amazon stabilized, and will bitcoin do the same? When Amazon shares debuted back in 1997, earnings were non-existent. […]
The USD price of AMZN has in no way stabilized:
As you can see, it's a concave-upward curve, punctuated by various declines, all the way back to 2003.
This may sound like hair-splitting, but I think there's a real possibility the author is actually talking about USD/BTC exchange rate volatility. This is an animal of a completely different stripe.
For example, the US Dollar loses purchasing power over time, but that is OK, since it is mostly at a predictable rate. But when something has wild swings up and down, and random times, then it is less suitable to either use as a currency (the stock person with the price gun will be working overtime correcting prices on everything several times a day), and it isn't that good of a store of value if there isn't any reason for it to not lose all of its value over a given period of time.
The Amazon graph you linked to does look fairly stable, a nice exponential function that has very good reasons to remain so for a good period of time (not saying this or anything else is guaranteed, but at least there is something tangible behind that price).