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In this paper we introduce complicated oscillating system, namely quotient oftwo multiforms based on Riemann-Siegel formula. We prove that there is aninfinite set of metamorphoses of this system (=chrysalis) on critical line$\sigma=\frac 12$ into a butterfly (=infinite series of M\" obius functions inthe region of absolute convergence $\sigma>1$).

Jacob's ladders, Riemann's oscillators, quotient of two oscillating multiforms and set of metamorphoses of this system