We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $$y^{\left(n\right)}+r\left(t\right)f(y,y^{\text{'}},\cdots ,y^{(n-1)})=0.$$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

We study the large-time behaviour of the nonlinear oscillator$$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$where $m,k\>0$ and $f$ is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$ with $\alpha$ real, $A\>0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

We study the large-time behaviour of the nonlinear oscillator $$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$ where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case  $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.