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In this paper, we show an ideal analog min-sum decoder, in the log-likelihood ratio domain, can be considered as a piecewise linear system. Many theoretical aspects of these decoders, thus, can be studied analytically. It is also shown that the dynamic equations can become singular for codes with cycles. When it is non-singular, the corresponding dynamic equations can be solved analytically to derive outputs of the decoder. We study the relationship between singularity and error floor and prove that absorption sets with degree two check nodes are singular graphs and under specific conditions the dynamic equations of an analog min-sum decoder can be reduced to that of an absorption set. The proposed approach paves the way for further analytical analysis on the dynamics of analog min-sum decoders and error floor in low-density parity-check codes.