I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, +\infty) $.
To provide some background, consider the following examples of linear ODEs:
- $ x' = -x $
- $ x'' = -x $
The solutions to both of these equations are bounded on $[0, +\infty) $. Intuitively, the negative sign on the right-hand side acts as a "negative feedback mechanism," pulling the solutions back towards the origin when they become too large.
Conversely, if we change the sign:
- $ x' = x $
- $ x'' = x $
We can easily see that these equations have unbounded solutions on $ [0, +\infty) $.
However, this intuition does not always hold. For instance, consider the third-order linear ODE:
$$ x''' = -x $$
Despite the negative feedback, the general solution includes terms that grow without bound as $ t \to +\infty $.
My question is:
Can this phenomenon be extended to nonlinear ODEs? Specifically, does there exist a function $ f(x) $ such that the nonlinear third-order ODE:
$$ x''' = -f(x) $$
has all its solutions bounded for $ t \in [0, +\infty) $?
I am aware that my familiarity with techniques for handling nonlinear ODEs is limited, and I would greatly appreciate any insights or references that could help address this problem. Understanding this could have significant implications for the study of dynamical systems and stability analysis.
Thank you in advance for your help!
I have posted it in MSE (link: Bounded solutions of nonlinear third-order ODEs), as the comment of Ayman Moussa suggested.
