2024/04/02 by Bessa, Mario, Vilarinho, Helder
#34A30 #34D08 #37A20 #37H15 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.02066
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)^\top, 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 Schrödinger equation. Furthermore, all results hold for L^∞-generic parameters α and β.