Difference between an ODE and dynamical system?

Question

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?

Answer 1

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).

Answer 2

Most of above answers are beautifully explained, here I am just trying to present one more observation:

LHS of an ODE $\dot{x}=V(x)$ is a local concept as it its involve temporal derivative and hence may generate a velocity field structure, while RHS of the same ODE involves a vector field which is global in nature and may capture characterization of a dynamical system.

Fundamental theorem of ODE says that if V is Lipschitz continuous, then the solution of the ODE is unique and continuously depends on a given initial condition from state space. This attempts to related the both concepts, an ODE and a dynamical system i.e. how the flow of the vector field V matches with the solution map of the ODE?