In advance: Please forgive my lack of adequate vocabulary, I fear my current vague understanding would rather confuse my question than help solve it.
Imagine you are standing at the shore of a river. Your 'system of interest' is a 3m by 3m square of water (or a 2-D numerical model thereof). Now 'split' reality into two branches. At the start of the experiment, you remove a bucket of water in one of these realities, but not in the other. Initially, the two systems will behave differently, but eventually converge towards a similar state. After a few seconds (let alone a day or a century) there will be no significant difference between the two realities - the forcings of the dynamical system (the water table at its boundary) will have compensated and erased the discrepancies.
I am trying to understand this 'system memory' and am looking for literature into if and how this 'memory' can be quantified. I could imagine scenarios in which this memory would be longer - for example if flow within the system was slower or the forcings weaker or even missing. Key words I have so far come up with are 'attractors' and 'dissipation', but the literature I have found seemed more focused on identifying such systems rather than quantifying their properties.
Would you know any key words I should look into, or could refer me to an introductory text or video?