Population protocols (PPs) are a model of passive distributed systems in which a collection of finite-state mobile agents interact with each other to accomplish a common task. Unlike other works, which investigate their computation power, this paper throws light on an aspect of PPs as a model of chemical reactions. Motivated by the wellknown BZ reaction that provides an autonomous chemical oscillator, we address the problem of autonomously generating an oscillatory execution from any initial configuration (i.e., in a self-stabilizing manner). For deterministic PPs, we show that the self-stabilizing leader election (SSLE) and the self-stabilizing oscillator problem (SS-OSC) are equivalent, in the sense that an SS-OSC protocol is constructible from a given SS-LE protocol and vice versa, which unfortunately implies that (1) resorting to a leader is inevitable (although we seek a decentralized solution) and (2) n states are necessary to create an oscillation of amplitude n, where n is the number of agents (although we seek a memory-efficient solution). Aiming at reducing the space complexity, we present and analyze some randomized oscillatory PPs.