Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

2023/01/12 by Dinis Amaro, Amaro, Dinis, Mário Bessa +3
Computer Science · Engineering · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.04909

openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In the present paper we prove that densely, with respect to an L^p-like topology, the Lyapunov exponents associated to linear continuous-time cocycles Φ:ℝ× M→ GL(2,ℝ) induced by second order linear homogeneous differential equations x+α(φ^t(ω)) x+β(φ^t(ω))x=0 are almost everywhere distinct. The coefficients α,β evolve along the φ^t-orbit for ω∈ M and φ^t: M→ M is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation x+β(φ^t(ω))x=0 and for a Schrödinger equation x+(E-Q(φ^t(ω)))x=0, inducing a cocycle Φ:ℝ× M→ SL(2,ℝ).

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