Successive Spectral Sequences

If a chain complex is filtered over a poset I, then for every chain in I we obtain a spectral sequence. In this paper we define a spectral system that contains all these spectral sequences and relates their pages via differentials, extensions, and natural isomorphisms. We also study an analog of exact couples that provides a more general construction method for these spectral systems. This turns out to be a good framework for unifying several spectral sequences that one would usually apply one after another. Examples are successive Leray--Serre spectral sequences, the Adams--Novikov spectral sequence following the chromatic spectral sequence, successive Grothendieck spectral sequences, and successive Eilenberg--Moore spectral sequences.