  A general method for constructing simplicial complex from observed time
series of dynamical systems based on the delay coordinate reconstruction
procedure is presented. The obtained simplicial complex preserves all pertinent
topological features of the reconstructed phase space and it may be analyzes
from topological, combinatorial and algebraic aspects. In focus of this study
is the computation of homology of the invariant set of some well known
dynamical systems which display chaotic behavior. Persistent homology of
simplicial complex and its relationship with the embedding dimensions are
examined by studying the lifetime of topological features and topological
noise. The consistency of topological properties for different dynamic regimes
and embedding dimensions is examined. The obtained results shed new light on
the topological properties of the reconstructed phase space and open up new
possibilities for application of advanced topological methods. the method
presented here may be used as a generic method for constructing simplicial
complex from a scalar time series which has a number of advantages compared to
the mapping of the time series to a complex network.
