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I don’t really understand the difference between an ODE and a dynamical system. The ODE just seems like a way to describe it. Is there more to it?

Are there dynamical systems which cannot be represented by any ODE?

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In general, a dynamical system is defined as a system in which a function (or a set of functions) describes the evolution of a point in a geometrical space.

The point in question may lie in a space where every coordinate is a value you want to track (for example, the current and the voltage drop at the ends of a capacitor, or the population of fish in a lake). I believe your confusion stems from the fact that dynamical systems are introduced when derivatives or integrals are introduced in systems of equations; however, the equations need not be differential in nature.

Now, ODEs are usually the simplest way to describe a dynamical system and its evolution with time. The best example here could be an RLC circuit, where everything could be described in terms of derivatives of voltages and currents. ODEs are also only one of the possible equations that describe a system. There could be Partial DEs, or even something completely different like stochastic processes, random variables, recursive definitions. ODEs are just one of the tools involved in the description of dynamical systems, and often the simplest one to solve - so simple, in fact, that sometimes you may find a closed form for the solution to the equations, but this is not always the case (and in general, a closed, analytical form does not exist for any given dynamical system).

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  • $\begingroup$ So you mean that in general we might have a dynamical system whose points are not related to some function which we can desribe by derivatives or intgrals? $\endgroup$ – user415535 Jan 25 '18 at 9:38
  • $\begingroup$ ah, like discrete dynamical system! $\endgroup$ – user415535 Jan 25 '18 at 9:38
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    $\begingroup$ You haven't mentioned few quite important things. First, there is rigorous definition of dynamical system beyound words "state space" and "evolution operator". Second, when set of times is $\mathbb{R}$ and state space is some manifold or Euiclidean space, dynamical system corresponds to an ODE. Third, not all ODEs define a proper dynamical system. There are two obstructions usually: non-uniqueness of solutions (non-existent for smooth ODEs) and finite-time .... $\endgroup$ – Evgeny Jan 25 '18 at 15:23
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    $\begingroup$ ... blow-up (when trajectories doesn't exist for all $t \in \mathbb{R}$, like in $y' = y^2$). The latter one can be fixed by some transformation that takes trajectories of one ODE to another, but trajectories of the second ODE are defined for all times. So, flows in Euclidean space or manifolds (as a dynamical systems) and ODEs are indistinguishable, they are the same. ODEs are not the one of the ways to describe particular flow. Of course I agree that ODEs are a particular example of dynamical system beside many others. $\endgroup$ – Evgeny Jan 25 '18 at 15:25
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    $\begingroup$ The article you linked is extremely helpful. The only thing I would like to point out is that even with continuous time, if the phase space corresponds to an infinite-dimensional manifold, the system corresponds to a PDE. Thus more in general, ODEs are still a subset of all possible dynamical systems. $\endgroup$ – Niki Di Giano Jan 25 '18 at 15:43

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