We review the occurrence of the patterns of the onset of chaos in low-dimensional dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics in and towards the attractors at period-doubling accumulation points and at tangent bifurcations to describe features of glassy dynamics, critical fluctuations and localization transitions. We provide an analytical framework to reproduce rank distributions of large classes of data (including Zipf’s law), and mimic the evolution of high-dimensional ecological models. We extend the discussion to point out a common circumstance of drastic contraction of configuration space driven by the attractors of these nonlinear mappings. We show that generalized entropy expressions describe the process for the period doubling and intermittency routes to chaos. Finally, we refer to additional natural phenomena in complex systems where these conditions manifest.