  One of the most useful tools for distinguishing between chaotic and
stochastic time series is the so-called complexity-entropy causality plane.
This diagram involves two complexity measures: the Shannon entropy and the
statistical complexity. Recently, this idea has been generalized by considering
the Tsallis monoparametric generalization of the Shannon entropy, yielding
complexity-entropy curves. These curves have proven to enhance the
discrimination among different time series related to stochastic and chaotic
processes of numerical and experimental nature. Here we further explore these
complexity-entropy curves in the context of the R\'enyi entropy, which is
another monoparametric generalization of the Shannon entropy. By combining the
R\'enyi entropy with the proper generalization of the statistical complexity,
we associate a parametric curve (the R\'enyi complexity-entropy curve) with a
given time series. We explore this approach in a series of numerical and
experimental applications, demonstrating the usefulness of this new technique
for time series analysis. We show that the R\'enyi complexity-entropy curves
enable the differentiation among time series of chaotic, stochastic, and
periodic nature. In particular, time series of stochastic nature are associated
with curves displaying positive curvature in a neighborhood of their initial
points, whereas curves related to chaotic phenomena have a negative curvature;
finally, periodic time series are represented by vertical straight lines.
