p 41

Demonstration of footnote 13

A sketch of an explanation of the relation between the Lyapunov exponent $\lambda$ and that $\lambda’$ of the corresponding discrete system is as follows.

 Let $\lambda_i$ be the logarithm of the expansion rate of the trajectory spacing between the $i-1$th and $i$th earthquakes. Then during the period between 0th and the $N$th earthquakes the small difference in the initial conditions may be expanded by the ratio $e^{\lamda’_1} e^{\lambda’_2} \cdots e^{\lambda’_N}$. That is, the average expansion rate (see (2.52) in the book) reads

\[

 \frac[1}{N} \sum_{i=1}^N \lambda’_i \rightarrow \lambda’

\]

The Lyapunov exponent for the original continuous system is calculated as \

\[

 \frac[1}{t} \sum_{i=1}^N \lambda’_i \rightarrow \lambda 

\]

with $t = \sum \tau_i$, where $\tau_i$ is the time between the $i-1$th and the $i$th earthquakes. If the average time between two adjacent earthquakes is $\tau$, then, since $t = N\tau$ asymptotically, $N\lambda’ = N\tau \lambda$ asymptotically.