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 $\ddot{x}(t)+\alpha ({\phi }^t(\omega ))\dot{x}(t)+\beta ({\phi }^t(\omega ))x(t)=0$, where the coefficients $\alpha$ and $\beta$ evolve along the ${\phi }^t$-orbit for $\omega \in M$, and ${\phi }^t:M\to M$ 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 $\dot{X}=A({\phi }^t(\omega ))X$, 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 Schrödinger equation. Furthermore, all results hold for $L^{\infty }$-generic parameters $\alpha$ and $\beta$.

21 pages, 3 figures