Correct phrase for a 'forgetful' natural system?
How would you call a system that displays such 'forgetfulness'?


In one word, dissipative. In multiple words, a system whose autocorrelation function tends to zero as time tends to infinity.

Note that the autocorrelation function may not exist; this concept implicitly assumes a linear (or approximately linear) system. Consider a river. Adding a small impulse of water upstream (e.g., a not-too-large downburst) results in the river flow increasing briefly but eventually stabilizing to some nominal value. Adding a slightly larger impulse results in a similar but slightly larger response. But adding a huge impulse of water upstream (e.g., a very large and rapid snowmelt) may cause the river to carve itself a new channel. With this huge impulse, the flow at some downstream point will increase initially but then drop to zero and stay that way because the river no longer flows past that point.