On the stability of $\ddot x(t)+\alpha(t)\dot x(t)+\beta(t) x(t)=0$

Abstract

Our main goal is to understand the stability of second order linear homogeneous differential equations $\ddot x(t)+\alpha(t)\dot x(t)+\beta(t)x(t)=0$ for $C^0$-generic values of the variable parameters $\alpha(t)$ and $\beta(t)$. For that we embed the problem into the framework of the general theory of continuous-time linear cocycles induced by the random ODE $\ddot x(t)+\alpha(\varphi^t(\omega))\dot x(t)+\beta(\varphi^t(\omega))x(t)=0$, where the coefficients $\alpha$ and $\beta$ evolve along the $\varphi^t$-orbit for $\omega\in M$, and $\varphi^t: M\to M$ is a flow defined on a compact Hausdorff space $M$ preserving a probability measure $\mu$. Considering $y=\dot x$, the above random ODE can be rewritten as $\dot X=A(\varphi^t (\omega))X$, with $X=(x,y)^\top$, having a kinetic linear cocycle as fundamental solution. We prove that for a $C^0$-generic choice of parameters $\alpha$ and $\beta$ and for $\mu$-almost all $\omega\in M$ either the Lyapunov exponents of the linear cocycle are equal ($\lambda_1(\omega)=\lambda_2(\omega)$), or else the orbit of $\omega$ displays a dominated splitting. Applying to dissipative systems ($\alpha<0$) we obtain a dichotomy: either $\lambda_1(\omega)=\lambda_2(\omega)<0$, attesting the stability of the solution of the random ODE above, or else the orbit of $\omega$ displays a dominated splitting. Applying to frictionless systems ($\alpha=0$) we obtain a dichotomy: either $\lambda_1(\omega)=\lambda_2(\omega)=0$, attesting the asymptotic neutrality of the solution of the random ODE above, or else the orbit of $\omega$ displays a hyperbolic splitting attesting the \emph{uniform} instability of the solution of the ODE above. This last result implies also an analog result for the 1-d continuous aperiodic Schr\"odinger equation. Furthermore, all results hold for $L^\infty$-generic parameters $\alpha$ and $\beta$.