In this paper, we propose a reconfigurable intelligent surface (RIS)-assisted frequency-modulated (FM) differential chaos shift keying (DCSK) scheme with simultaneous wireless information and power transfer (SWIPT), called RIS-FM-DCSK-SWIPT scheme, for low-power, low-cost, and high-reliability wireless communication networks. In particular, the proposed scheme is developed under a non-linear energy-harvesting (EH) model which can accurately characterize the practical situation. The proposed RIS-FM-DCSK-SWIPT scheme has an appealing feature that it does not require channel state information, thus avoiding the complex channel estimation. We further derive the closed-form theoretical expressions for the energy shortage probability and bit error rate (BER) of the proposed scheme over the multipath Rayleigh fading channel. In addition, we investigate the influence of key parameters on the performance of the proposed transmission scheme in two different scenarios, i.e., RIS-assisted access point (RIS-AP) and dual-hop communication (RIS-DH). Finally, we carry out various Monte-Carlo experiments to verify the accuracy of the theoretical derivation, illustrate the performance advantage of the proposed scheme, and give some design insights for future study.