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I think of discrete dynamical systems

$$x_{t+1}-x_{t}=\Delta x_t= f(x_{t})$$

and continuous dynamical systems

$$\frac {\partial x_t}{\partial t} = f(x_t)$$

as totally separate formalisms. However, sometimes I want to be agnostic about whether a dynamical system I'm considering is discrete or continuous. Is there a formalism that has both of these as special cases? (similar to how the Lebesgue integral has the discrete sum and the continuous (e.g. Riemann) integral as special cases)

bonus question: It would be even better if the formalism had partial differential equations and higher order differential equations also as a special case.

KCd
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user56834
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    Are you looking for something like this: https://en.wikipedia.org/wiki/Dynamical_system_(definition)#General_definition? – Hans Lundmark Jul 23 '19 at 10:57
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    @HansLundmark, The problem with that formalism, is that it basically formalizes what in my mind is the "solution" of a dynamical system problem, namely the trajectory of the system, but it doesn't allow you to state the problem in terms of the system's local changes in time, as we do with $\Delta x_t = f(x_t)$ and $\partial_tx_t=f(x_t)$. – user56834 Jul 23 '19 at 11:17
  • I guess if I wanted to be agnostic I would just write $Dx_t=f(x_t)$, where $D$ may represent a difference operator or a differential operator. –  Jul 23 '19 at 11:23
  • You are looking for https://en.wikipedia.org/wiki/Time-scale_calculus. Regretfully, we need to get used a bit to the formalism to be able to use it really. – John B Jul 23 '19 at 22:52

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