Consider the system $\dot{x}=f(x,t)$, where $f$ is continuously differentiable in both $x$ and $t$.
And let $f(x,t)=f(x,t+T)$.
The Poincare Map $P$ maps the value of $x(t=0)$ to $x(t=T)$, i.e,
$$P(x_0)=x_T$$
Is $P(x_0)$ continuous in $x_0$? If not, what are the conditions sufficient for it to be continuous?
I wanted an answer to this question, and the answer given there, as far as I can tell, assumes the Poincare Map is continuous.
Thanks in advance!
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SweepingBishops
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1There is such a thing as the continuous dependence on parameters, and the initial value is such a parameter. In general, the flow $\phi(t;t_0,x_0,p)$ inherits the smoothness of $f$, in the $t$ dependence with one order higher. – Lutz Lehmann Jun 10 '23 at 12:22