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The definition of a flow map given
here: http://www.math.sjsu.edu/~simic/Fall05/Math134/flows.pdf states that if $$\phi_t(X_0) = \phi(t, X_0)$$ is a flow of some ODE $X' = F(X)$ then "because of uniqueness of solutions" $\phi_{s+t} = \phi(s) \circ \phi(t)$.

I don't see how that result follows from uniqueness of solutions. Can someone make this more explicit?

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It is the same as saying that $x(t)$ is a solution, then also $x(t+a)$ is a solution for any fixed $a$. The initial value of that shifted solution is $x(a)$, so that another way to write the above using the flow function is $$ϕ(t,x(a))=ϕ(t+a,x(0)),$$ or using the subscript for the time, $$ϕ_t(ϕ_a(x(0)))=ϕ_{t+a}(x(0)).$$

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