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How can I determine by hand if the following expressions can be the flow of a 1D-system $\dot{x}=f(x)$.

Expressions:

$\phi_t(x)=x+t$

$\phi_t(x)=t*x$

$\phi_t(x)=e^tx$

$\phi_t(x)=e^xt$

Jannesemak
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  • The maps $\phi_t$ need to be a representation of the additive group $(\Bbb R,+)$. – Lutz Lehmann Oct 04 '22 at 14:17

1 Answers1

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You have to check that

  1. $\phi_0(x)=id(x)$

  2. $(\phi_t \circ \phi_s) (x)=\phi_{t+x}(x)$

For example, the second and fourth cases do not satisfy requirement 1. On the contrary, the first and third cases satisfy both.

A. J. Pan-Collantes
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