I've heard that dynamical systems have "memory". In discrete time this is represented by the current state depending on the prior states, so that makes sense. But in the continuous time an Linear, Time-Invariant (LTI) system is also said to have "memory". I know that this somehow relates to the fact that convolution is used to solve it, but I'd like to know if there is a deeper definition.
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Using you wording, an LTI system clearly has no memory in any reasonable sense.
In mathematical terms, this LTI notion (which, without exaggeration, corresponds to basically $0$% of the theory) means that you have a linear dynamics of the form $x\mapsto e^{At}x$ for some matrix $A$.
In order to have examples with memory in continuous time, we should consider other types of dynamics. For example, something such as $$x'(t)=x(t-1).$$ Note that the equation is linear (first property of an LTI), but it has memory because you need to know what happens in $[-1,0]$ to find out what happens at $0$.
These ideas go back to Vito Volterra (unfortunately not always recognized), and marvelously put in the context of dynamical systems by Jack Hale with the consequent exponential development.
And going back to your question: we usually say that a dynamics has memory if one cannot determine it at time $t$ if we don't know it at some former time.
John B
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Wow, how easily (and how customary) the whole era of delay equations between Volterra and Hale was swept under the rug... – Artem May 09 '18 at 13:08
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@Artem This was a rather simple sentence, on purpose, as you saw it is not even mentioned in the OP question. But I completely agree with you. Would you say that it is better to erase the sentence? I'm not a big fan of the lines of Bellman and Cooke, or of Halanay, even though I studied them in detail, somewhat archaic by now..., I rather prefer Krasovskii (incidentally a bit earlier than Bellman and Cooke's main work) which as you may know was instrumental to Hale's original work. – John B May 09 '18 at 13:41
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My (limited but existing) experience tells me that the line Volterra -- Hale is very widely accepted among several recent PhDs working in the area of delay equations. Since I personally knew Anatoli Myshkis (who I suspect you are aware of given the fact you so highly put Krasovkii) I decided to make my comment. In retrospect, I should have made my comment more constructive. (+1 for your answer). – Artem May 09 '18 at 18:15
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@Artem Indeed, I regret that not personally. Myshkis' book appeared a bit before the one of Krasovskii, as far I understand it is for example the first place where one can see the classification of linear delay equations into retarded, neutral and advanced. In his original writings Hale never forgot to praise Krasovskii for the origin of his investigations. – John B May 09 '18 at 18:37