On the stability of x(t)+α(t) x(t)+β(t) x(t)=0

Our main goal is to understand the stability of second order linear homogeneous differential equations x(t)+α(t) x(t)+β(t)x(t)=0 for C⁰-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 x(t)+α(φᵗ(ω)) x(t)+β(φᵗ(ω))x(t)=0, where the coefficients α and β evolve along the φᵗ-orbit for ω∈ M, and φᵗ: M→ M is a flow defined on a compact Hausdorff space M preserving a probability measure μ. Considering y= x, the above random ODE can be rewritten as X=A(φᵗ (ω))X, with X=(x,y)^, having a kinetic linear cocycle as fundamental solution. We prove that for a C⁰-generic choice of parameters α and β and for μ-almost all ω∈ M either the Lyapunov exponents of the linear cocycle are equal (λ₁(ω)=λ₂(ω)), or else the orbit of ω displays a dominated splitting. Applying to dissipative systems (α<0) we obtain a dichotomy: either λ₁(ω)=λ₂(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 λ₁(ω)=λ₂(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 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^∞-generic parameters α and β.