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The paper presents a discussion on an opinion about the stability margin towards an emergency in local climate dynamics from the bifurcation analysis viewpoint. With this purpose we propose to attract the practice-oriented bifurcation analysis, where the conflict-of-units between notions used to understand natural evolution processes and notions used to describe desirable artificial regimes is resolved by integrating analytics on the basis of modified bifurcation diagrams. The discussion focuses on the phenomenon of interannual temperature variability, where local annual maximums and minimums are analyzed with daily details in both time and temperature coordinates. This phenomenon is considered via the probable, periodical and regulator conceptions. Advantages of the regulator conception are verified by results of processing the data of temperature meteorological observations on daily means over the last 135 years. This conception is based on the HDS-hypothesis, in accordance to which local climate dynamics is determined by the natural competition between the amplitude quantization (restricted by the temperature Hysteresis) and time quantization (caused by the Double Synchronization). Thus an alternation between three elementary processes with the same period (year) and different patterns of annual warming-cooling cycles is supposed as a typical behavior for local climate systems, and the idea on high-dynamic local climate ensembles is developed instead of the conventional opinion on quasi-static local climate norms. Mechanisms of temperature changes due to abrupt shifts (so-called change-points) of the HDS-regulator parameters are distinguished from mechanisms of temperature changes due to bifurcations. The notion of a stability margin is used as a distance to an emergency and is visualized in the parametrical space. So, in spite of the mechanisms of temperature changes with/without bifurcations are different, their conflict-free sewing becomes conceptually possible in the context of the stability margin towards emergencies determined relatively bifurcation boundaries in the parametrical space. Since the discussed dynamics is not supposed to exist in terms of the traditional estimations concerning the observed local climate changes, then we believe that the paper would be interesting for scientists in the field of bifurcation analysis as well as for scientists and specialists, activity areas of which relate to the contemporary challenges connected with climate changes.