Apologies if this has already been discussed, but I searched the site and I couldn't find an answer.
For the sake of simplicity, consider only ODEs, possibly depending on some vector of parameters $\mathbf{p}$. The definition of a well-posed problem in the Hadamard sense is:
https://en.wikipedia.org/wiki/Well-posed_problem
The third point says that "the solution's behavior changes continuously with the initial conditions". I'm not sure what's the rigorous meaning of this. I interpret it as "the integral operator which associates the forcing term, the parameters and the initial condition(s) to the solution of the ODE is continuous", which seems to agree with the usual interpretation that "small changes in the data lead to small changes in the solution".
However, the Lorenz system is usually considered ill-posed: see
Doesn't the solution depend continuously on data, though? Sure, changes $\epsilon$ in the initial conditions or in $\mathbf{p}$ will cause changes of order $\epsilon\exp(\lambda t)$ in time $t$, which may be exponentially large, but they still depend continuously on $\epsilon$, don't they? So, why would it be ill-posed?
Also more recent paper seem to consider the Lorenz system ill-posed, though admittedly it's not stated explicitly:
https://pdfs.semanticscholar.org/85e5/908a9e21fb1c017c4206cf4df4475c89bdd8.pdf