I am reading Wikipedia's entry on Flow and it is not clear the distinction between solution of an ODE and the flow of an ODE. In particular it is clearly written $φ(x_0,t) = x(t)$, then what is the purpose of even defining the flow?

https://en.wikipedia.org/wiki/Flow_(mathematics)

Can someone please concisely explain the difference between the two concepts and provide some examples showing their differences?

Much thanks

  • 3
    The solution to an ODE IVP is a local phenomenon: it tells you how the system evolves from a specific initial condition. A flow, by contrast, is a global phenomenon; it corresponds to solving a whole family of IVPs at once. The time evolution of a flow tells you how many solutions evolve, not just the solution through one point. – symplectomorphic Feb 29 '16 at 4:53
up vote 1 down vote accepted

Simple answer:

Indeed, if you only consider autonomous differential equations, the concept of (local) flow has nothing to add, although as always being an additional point of view it helps in understanding or perhaps even finding properties that otherwise could be missed.

Not so simple answer:

However, it also happens that the concept of flow is much more general and need not be related to a differential equation. It can be associated for example to a stochastic differential equation, a delay equation, a partial differential equation, or even be associated to multidimensional time, etc, etc.

Complicated but more complete answer:

Having said this it may seem that the concept of flow is something more general than the set of solutions of a differential equation. This is also not a good perspective, since there are generalizations of an autonomous differential equation, even general nonautonomous differential equations, that don't lead to obvious concepts of flows.

The trick of adding $t'=1$ is clearly unsatisfactory in many situations (such as when compactness is crucial), leading for example to the study of convex hulls or lifts in the context of ergodic theory (but leading always to infinite-dimensional systems).

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