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 $J$.


1. 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.
2. 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?
3. 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.



Euler method (and others) for unbounded intervals
Regarding 1 and 2:


Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in **provably (pointwise) convergent schemes**. At least for traditional methods (Runge-Kutta, linear multistep, and many others) the step-by-step nature of the approximation **precludes the possibility of uniform convergence on unbounded intervals**. That is because the local truncation errors from each step accumulate -- not in a catastrophic way, if the method is stable, but still in general they accumulate and the error grows without bound as the number of steps taken goes to infinity. So **uniform pointwise convergence can only be proved for finite intervals**.


So it's not that you magically run into trouble when you go past the end of the chosen interval. After all, the bounds of your interval don't enter into the numerical method, and it knows nothing about them. It's simply that your solution will be less accurate at the end of the interval than at the start, and this effect will continue as you integrate further.


There are some areas where very long time integrations are needed, and special techniques are used, for instance in the study of stability of the solar system.


Perhaps a helpful example is [Exercise 7.1 from this set](https://faculty.washington.edu/rjl/fdmbook/exercises/allexercises.pdf), although you should keep in mind that it involves an intentionally poor choice of integrator for the specified problem, so the accumulation of error will not generally be quite so bad.


Of course, there's also the fact that if "one does not stop the iteration ... after a finite number of steps" then your computation will never terminate.


You didn't fully specify the problem, but based on the fact that you mentioned Euler's method, I have assumed that you're dealing with an initial value problem. If this question were about a boundary value problem, that would of course be a completely different matter.


Regarding your 3rd question:



> 
> "given a discrete dynamical system, does there exist a smooth dynamical system such that the discrete one arises as its [discrete numerical] approximation)?"
> 
> 
> 


This is often known as modified equation analysis and is a form of backward error analysis. For linear PDEs, it is described for instance in Section 10.9 of [LeVeque's book](https://faculty.washington.edu/rjl/fdmbook/exercises/allexercises.pdf). For Runge-Kutta methods applied to arbitrary initial value ODEs, it is much more difficult and is described most fully in Chapter IX of [Hairer, Lubich, & Wanner](https://link.springer.com/book/10.1007/3-540-30666-8). I think the last book is also a good example of one that takes the abstract viewpoint you are wishing for.


