  Analyzing dynamical data often requires information of the temporal labels,
but such information is unavailable in many applications. Recovery of these
temporal labels, closely related to the seriation or sequencing problem,
becomes crucial in the study. However, challenges arise due to the nonlinear
nature of the data and the complexity of the underlying dynamical system, which
may be periodic or non-periodic. Additionally, noise within the feature space
complicates the theoretical analysis. Our work develops spectral algorithms
that leverage manifold learning concepts to recover temporal labels from noisy
data. We first construct the graph Laplacian of the data, and then employ the
second (and the third) Fiedler vectors to recover temporal labels. This method
can be applied to both periodic and aperiodic cases. It also does not require
monotone properties on the similarity matrix, which are commonly assumed in
existing spectral seriation algorithms. We develop the $\ell_{\infty}$ error of
our estimators for the temporal labels and ranking, without assumptions on the
eigen-gap. In numerical analysis, our method outperforms spectral seriation
algorithms based on a similarity matrix. The performance of our algorithms is
further demonstrated on a synthetic biomolecule data example.
