Overdamping Condition of Nonlinear Equation

[LagrangeEuler]

In case of equation
$$\alpha y''(x)+\beta y'(x)+\gamma y(x)=0$$
where $\alpha>0$, $\beta>0$, $\gamma>0$, characteristic equation is
$$\alpha r^2+\beta r+\gamma=0$$
and characteristic roots are
$$r_{1,2}=\frac{-\beta \pm \sqrt{\beta^2-4\alpha \gamma}}{2 \alpha}$$
If $\beta^2<4\alpha \gamma$ system is underdamped, and
if $\beta^2>4\alpha \gamma$ system is overdamped.
What in the case of equation
$$\alpha y''(x)+\beta y'(x)+\gamma \sin[y(x)]=0$$
when equation is nonlinear? How to find when system is overdamped? Thanks a lot for your help in advance.

 

[haushofer]

How do you define underdamping in such a situation? Is such an equation even solvable? And does it have periodic solutions (i.e. imaginary exponential parts in its solution)?

 

[LagrangeEuler]

Overdamping is when characteristic roots are real and negative, I suppose. But I am not sure how to see that in case of nonlinear equations. I saw in literature that people discuss overdamped limit in case of nonlinear equations, but I am not sure how to do that. Because of that I asked here in the forum.

This limit is to my mind important to see when term $\beta y'(x)$ dominates over $\alpha y''(x)$.

 

[LagrangeEuler]

And just one more question, but very similar that the mentioned. What if we have equation
$$\alpha y''(x)+\beta y'(x) +\gamma y(x)=cz+d$$
If $c$ and $d$ are not functions of $x$? What's happening with condition in that case?