thank you! What if they don't indicate the software and will merely this lead to a bad outcome of the 1st round of review?

@WolfgangBangerth no particular concern, but just wonder why authors usually indicate software when commercial software is used and what will happen if they don't

Thank you. I agree with @whpowell96 that this link in particular relies on time series data. If there is an approach that only relies on the phase plane data rather than time series data, that will resolve my original question

Thanks. In the original question I put a simple example of a linear group of ODEs but my question is more for the purpose of nonlinear ones.

Absolutely. If only this could be more explainable.

I think you made a good point "For t < T_crit it's solutions satisfy the IC and for t > T_crit it's solutions do not satisfy the IC. The loss will be low, but IC information does not get to T > T_crit" in the second comment under this answer. I wonder if there is any existing work similar to that or in support of that.

Thanks for pointing that JCP paper out. That the loss will be low but IC information does not get to T > T_crit, I wonder if this is your viewpoint or is that from the JCP paper?

Thanks for your answer. But what my question assumes is that the loss function is sufficiently low so that both in your terminology Ls are small, then ICs are assumed satisfied.

@BillGreene Thank you very much for pointing that out. I wonder if any elasticity texts discuss a spinning circle with a uniform initial angular velocity released at t=0 (no external force thereafter) and the solution afterwards?

@knl FEM. An initial angular velocity applied. No damping

the change in the volume or cross section area is so big when time goes big that I'm not sure if it's due to numerics at all