  We describe a methodology that allows us to follow the pulsational behavior
of an RR Lyrae model consistently and automatically along its evolutionary
track throughout the whole instability strip. It is based on the powerful
amplitude equation formalism, and resorts to a judicious combination of
numerical hydrodynamical simulations, the analytical signal time-series
analysis, and amplitude equations. A large-scale survey of the nonlinear
pulsations in RR Lyr instability strip is then presented, and the mode
selection mechanism is delineated throughout the relevant regions of parameter
space. We obtain and examine two regions with hysteresis, where the pulsational
state depends on the direction of the evolutionary tracks, namely a region with
either fundamental (RRab) or first overtone (RRc) pulsations and a region with
either fundamental (RRab) or double-mode (RRd) pulsations. The regions where
stable double-mode (DM, or RRd) pulsations occur are very narrow and hard to
find in astrophysical parameter (L, M, T_eff, X, Z) space with hydrodynamic
simulations, but our systematic and efficient methodology allows us to
investigate them with unprecedented detail. It is shown that by simultaneously
considering the effects of mode selection and of horizontal branch evolution we
can naturally solve one of the extant puzzles involving the topologies of the
theoretical and observed instability strips, namely the slope of the
fundamental blue edge. The importance of the interplay between mode selection
and stellar evolutionary effects is also demonstrated for the properties of
double-mode RR Lyr. Finally, the Petersen diagram of double-mode RR Lyr models
is discussed for the first time.
