# Difference between flow and solution of ODE

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

• 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
• @symplectomorphic When you say it "tells you how many solutions evolve", I take it you mean "how an ensemble of solutions evolve" (as opposed to "the number of solutions that evolve")? – binaryfunt Apr 26 at 11:05

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).