  We review the occurrence of the patterns of the onset of chaos in
low-dimensional nonlinear dissipative systems in leading topics of condensed
matter physics and complex systems of various disciplines. We consider the
dynamics associated with the attractors at period-doubling accumulation points
and at tangent bifurcations to describe features of glassy dynamics, critical
fluctuations and localization transitions. We recall that trajectories
pertaining to the routes to chaos form families of time series that are readily
transformed into networks via the Horizontal Visibility algorithm, and this in
turn facilitates establish connections between entropy and Renormalization
Group properties. We discretize the replicator equation of game theory to
observe the onset of chaos in familiar social dilemmas, and also to mimic the
evolution of high-dimensional ecological models. We describe an analytical
framework of nonlinear mappings that reproduce rank distributions of large
classes of data (including Zipf's law). We extend the discussion to point out a
common circumstance of drastic contraction of configuration space driven by the
attractors of these mappings. We mention the relation of generalized entropy
expressions with the dynamics along and at the period doubling, intermittency
and quasi-periodic routes to chaos. Finally, we refer to additional natural
phenomena in complex systems where these conditions may manifest.
