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?

  • 1
    some key words might be 'transient' and 'steady state' – spaceisdarkgreen Aug 8 at 19:37
up vote 2 down vote accepted

The attractor is a dynamical behaviour which the system will perpetually exhibit if unperturbed. This usually is a constant state (fixed-point dynamics), a regular oscillation (periodic dynamics), or a chaotic dynamics. After sufficiently small perturbations, the dynamics will return to the attractor’s dynamics (note that for many systems, any perturbation is sufficiently small).

What quantifies what you desire is the largest Lyapunov exponent of the attractor. It describes how quickly perturbations of a system (such as your removed bucket of water) grow or shrink. In general, there are three relevant cases:

  • The largest Lyapunov exponent is negative. In this case, you have a fixed-point dynamics, i.e., there is one state to which the dynamics converges. This is the case in your example, where the fixed point corresponds to a certain water level. Here, the more negative the largest Lyapunov exponent (at your fixed point) the quicker the system will return to the fixed point after a perturbation. In this case, the largest Lyapunov exponent is equivalent to the largest eigenvalue of the Jacobian of the phase-space flow at your fixed point.

  • The largest Lyapunov exponent is zero. In this case, you usually have a periodic dynamics. In such a dynamics, a perturbation usually has a permanent effect (ignoring external noise and similar), namely shifting the phase of the periodic oscillation.

  • The largest Lyapunov exponent is positive. In this case, your dynamics subject to the butterfly effect: A tiny perturbation will completely change the system’s behaviour on the long run. In some sense, the system has an eternal memory as you will always see the consequences of your perturbation. In another sense, the memory is very short, since it is impossible to deduce the past state of the system from observation (due to measurement and dynamical noise).

  • Thank you very much, amazing answer! I will look into this topic! – J.Galt Aug 10 at 15:28

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