In the course of my research I'm confronted with performing a numerical approximation of the solution of an initial value problem $$\begin{cases} y'=f(y,t),\\ y(t_{0})=y_{0} \end{cases}\quad\quad(1)$$ with $y:I\rightarrow \mathbb{R}^d,\quad I\subseteq\mathbb{R}$ and where $f$ satisfies standard regularity assumptions that guarantee unique solutions. One particular twist is that $I$ in my setup is known to be infinite, e.g., $I=[0,\infty)$ or $I=\mathbb{R}$.

I am familiar with standard approximation schemes, such as the Euler method to name a simple one. The issue is that these are only valid for bounded intervals $J$ of $I$, such as $J=[-T,T]$ for some $T>0$. Searching further on the internet, all texts and books on numerics of ODEs that I have consulted so far also solely develop various numerical approximation schemes that approximate the original solution on a predetermined, bounded interval $J$.

Can you point me to text, or mention results, where approximations schemes are developed for unbounded intervals? I am interested in obtaining an explicit formula for the recurrence equation(s) that determine the approximation (though I do not need a closed-form solution of the recurrence equation(s)).

This is because I have a strong feeling that this discretization is actually equivalent (perhaps after rewriting it in a clever way) to a model that already is well-known. Thus only qualitative information about the solution, such as whether it approaches a periodic orbit, is not sufficient.Conversely, are there (collections of) results (counterexamples) about how fast one runs into trouble, when one takes a standard approximation scheme, valid on $[-T,T]$, but where one does not stop the iteration once $T$ has been hit after a finite number of steps, but rather where one simply continues iterating?

Lastly, if $f$ in $(1)$ does no depend on $t$ and solutions exist globally, it is straight-forward to see how $(1)$ induces a flow of a dynamical system. In this case, and supposing that solutions on unbounded intervals exist, any approximation scheme can be thought of inducing a discrete dynamical system that approximates the original. Do you know of any text that treats these objects abstractly?

E.g. by considering a "discretization operator" that maps the flow of a smooth dynamical system such as (1) to its discrete approximation?

All books on numerics of ODE I have consulted so far do not take an abstract view and I think it would be very enlightening if the relationship of a smooth dynamical system and the discrete system that arises as its approximation would by discussed abstractly.

(Then questions such as "given a discrete dynamical system, does there exist a smooth dynamical system such that the discrete one arises as its Euler approximation?" would be very interesting to investigate.)

Any information or help is much appreciated.

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