An improved estimate for the highest Lyapunov exponent in the method of freezing

Abstract:

Let $\dot x=A(t)x$ and $\lambda _k(t)$ be the eigenvalues of the matrix $A(t)$. The main result of the Method of Freezing states that if $\sup _J \|A(t)\|\leq M$, $\sup _J\max _{1\leq k\leq n}\mathrm {Re} \lambda _k(t)\leq \rho$ and $\sup _J(\|A(t)-A(s)\|/|t-s|)\leq \delta$, then \[ x_{\max }\leq \rho +2M\lambda _\delta ,\] for the highest exponent $x_{\max }$ of the system, where \[ \lambda _\delta =\left (\frac {C_n\delta }{4M^2}\right )^{\frac {1}{n+1}}.\] The previous best known value $C_n=\frac {n(n+1)}{2}$ and the substantially smaller values of $C_n$ are reduced to the still smaller value.