  Detailed analysis of the log-periodic structures as precursors of the
financial crashes is presented. The study is mainly based on the German Stock
Index (DAX) variation over the 1998 period which includes both, a spectacular
boom and a large decline, in magnitude only comparable to the so-called Black
Monday of October 1987. The present example provides further arguments in
favour of a discrete scale-invariance governing the dynamics of the stock
market. A related clear log-periodic structure prior to the crash and
consistent with its onset extends over the period of a few months. Furthermore,
on smaller time-scales the data seems to indicate the appearance of analogous
log-periodic oscillations as precursors of the smaller, intermediate decreases.
Even the frequencies of such oscillations are similar on various levels of
resolution. The related value $\lambda \approx 2$ of preferred scaling ratios
is amazingly consistent with those found for a wide variety of other complex
systems. Similar analysis of the major American indices between September 1998
and February 1999 also provides some evidence supporting this concept but, at
the same time, illustrates a possible splitting of the dynamics that a large
market may experience.
