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We show that the Bose-glass phase of a one-dimensional disordered Bose fluid exhibits a chaotic behavior, i.e., an extreme sensitivity to external parameters. Using bosonization, the replica formalism and the nonperturbative functional renormalization group, we find that the ground state is unstable to any modification of the disorder configuration ("disorder" chaos) or variation of the Luttinger parameter ("quantum" chaos, analog to the "temperature" chaos in classical disordered systems). This result is obtained by considering two copi