Non-orthogonal multiple access (NOMA) is considered by the 3GPP as a potential technology for beyond 5G wireless networks to support massive connectivity with robust reliability. However, the massive increase in connected users critically leads to data security problems owing to the high possibility of eavesdropping. In this paper, uplink secure NOMA (sNOMA) schemes based on chaotic physical layer security (PLS) are proposed to achieve two-fold secrecy of integrated channel coding and secret key approaches. Different code-domain and power-domain techniques are employed for sNOMA designs over realistic fading channel environments. Equal power allocation strategy is used for the transmitted chaotic signals in code-domain sNOMA (CD-sNOMA), whereas dynamic power control is employed in power-domain sNOMA (PD-sNOMA) and the hybrid code-power-domain sNOMA (CPD-sNOMA) approaches. The former design adopts joint maximum Likelihood (ML) signal detection while the later scenarios utilize integrated receiver designs based on successive interference cancellation (SIC) and ML techniques. For the proposed sNOMA schemes, power control algorithms are presented to optimize the system performance over constrained total received power and target error rate. Numerical results validate the effectiveness of sNOMA designs compared with the benchmark systems under the worst-case secrecy of unauthorized receiver with complete knowledge of the transmission scheme and associated channel,. Valuable tradeoffs are demonstrated between the achieved error rate, connectivity, security gap, and complexity. Moreover, the utilized chaotic signals offer robust and cost-effective PLS solutions with a huge key-space to combat the most powerful brute-force eavesdropping attacks. INDEX TERMS Chaotic signals, NOMA schemes, physical layer security, Rician channels, wireless communication networks.