On the stability of $\ddot x(t)+α(t)\dot x(t)+β(t) x(t)=0$

Abstract

Our main goal is to understand the stability of second order linear homogeneous differential equations $\ddot x(t)+α(t)\dot x(t)+β(t)x(t)=0$ for $C^0$-generic values of the variable parameters $α(t)$ and $β(t)$. For that we embed the problem into the framework of the general theory of continuous-time linear cocycles induced by the random ODE $\ddot x(t)+α(φ^t(ω))\dot x(t)+β(φ^t(ω))x(t)=0$, where the coefficients $α$ and $β$ evolve along the $φ^t$-orbit for $ω\in M$, and $φ^t: M\to M$ is a flow defined on a compact Hausdorff space $M$ preserving a probability measure $μ$. Considering $y=\dot x$, the above random ODE can be rewritten as $\dot X=A(φ^t (ω))X$, with $X=(x,y)^\top$, having a kinetic linear cocycle as fundamental solution. We prove that for a $C^0$-generic choice of parameters $α$ and $β$ and for $μ$-almost all $ω\in M$ either the Lyapunov exponents of the linear cocycle are equal ($λ_1(ω)=λ_2(ω)$), or else the orbit of $ω$ displays a dominated splitting. Applying to dissipative systems ($α<0$) we obtain a dichotomy: either $λ_1(ω)=λ_2(ω)<0$, attesting the stability of the solution of the random ODE above, or else the orbit of $ω$ displays a dominated splitting. Applying to frictionless systems ($α=0$) we obtain a dichotomy: either $λ_1(ω)=λ_2(ω)=0$, attesting the asymptotic neutrality of the solution of the random ODE above, or else the orbit of $ω$ 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 $α$ and $β$.