Pulsating instabilities in hydrogen/air double-flame partially premixed counterflow flames are numerically investigated with detailed chemistry and transport. The whole stability map on the equivalence ratio-strain rate plane is obtained. With the increase of the equivalence ratio or strain rate, there are two transitions for combustion patterns from stable to unstable, and to stable again. For the one transition presented at the smaller equivalence ratio and strain rate, it is similar to that in the pure premixed flame. After this transition, the flames do not extinguish with the increase of equivalence ratio or strain rate, but transit from unstable to stable. For the second transition appeared at the larger equivalence ratio and strain rate, the effects of equivalence ratio and strain rate on the pulsating instability are completely contrary to those in pure premixed flames. The effective activation energy of the premixed flame of partially premixed flame predicted by a new method is applied to calculate the Zeldovich number. Then the first transition is proved to satisfy Sivashinsky-like criterion, but the critical value is larger than that for pure premixed flame due to the heat transfer between premixed and non-premixed flames. It means that the pulsating instability is more difficult to happen in partially premixed flames. From the first transition to the second transition along the equivalence ratio or strain rate, though the premixed flame becomes weaker, the increase degree of heat transfer from non-premixed flame to premixed flame is much larger than the decrease degree of max heat release of the premixed flame due to decrease of distance and increase of temperature difference of two flames, which induces the premixed flame stable again. The second transition is therefore controlled by the heat transfer between two flames. (C) 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.