This letter introduces a framework for the automatic generation of hardware cores for Artificial Neural Network (ANN)-based chaotic oscillators. The framework trains the model to approximate a chaotic system, then performs design space exploration yielding potential hardware architectures for its implementation. The framework then generates the corresponding synthesizable High-Level Synthesis code and a validation testbench from a selected solution. The hardware design primarily targets FPGAs. The proposed framework offers a rapid hardware design process of candidate architectures superior to manually designed works in terms of hardware cost and throughput. The source code is available on GitHub.