This study explores a novel neural optimisation model that integrates Legendre polynomial-based neuron activation with a chaotic neural network architecture incorporating a self-feedback memory mechanism. Building on previous work that introduced self-feedback memory into chaotic dynamics, this study investigates the potential impact of combining these dynamics with the functional flexibility of Legendre polynomials. The resulting model is applied to a series of continuous function optimisation problems, each characterised by complex and multi-modal energy landscapes. We evaluate whether the combined mechanisms are capable of reliably navigating such landscapes and converging on global minima across a range of parameter settings. Results indicate that the model achieves performance comparable to established chaotic optimisation methods, exhibiting structured chaotic motion with trajectories spanning complex energy landscapes in a grid-like manner.