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In wireless ad hoc networks, IEEE 802.11 power management may completely fail if power-saving (PS for short) stations are out of synchronization. To fix this problem, Haas and Perlman (2001) proposed various cyclic quorum-based power management (CQPM) protocols, which, however, may also completely fail if some PS stations have different schedule repetition intervals (SRIs). To conquer all these problems, in this paper, we propose the AAPM (adaptive asynchronous power management) protocol, which has the following attractive features, (i) By using the novel AA-quorum space, AAPM ensures that any two asynchronous PS neighbors can discover each other in finite time regardless of their individual SRIs. (ii) The idle duty cycles of AAPM for all eligible SRI values are minimal, (iii) The time complexity of AAPM neighbor maintenance is constant, (iv) Two cross-layer SRI adjustment schemes are proposed such that a PS station in AAPM can dynamically vary its SRI according to the residual battery power or traffic QoS requirements. Primary numerical results show that AAPM achieves better energy efficiency than existing CQPM protocols.