Homotopy reservoir computing: Harnessing chaos for computation.

Reservoir computing (RC) has traditionally relied on tuning systems toward the edge of chaos to optimize their computational capability. In contrast, we propose a novel method that starts from a fully chaotic system and systematically tames it into a trainable reservoir using homotopy. Our approach constructs adaptive reservoirs whose internal dynamics evolve in real time with the input, yielding a new class of computational models: Homotopy Reservoir Computing (Homotopy RC). We demonstrate the effectiveness of this method across several canonical chaotic systems-including coupled Lorenz networks, the Lorenz-96 model, and the Kuramoto-Sivashinsky system-showing high performance in computational tasks. Furthermore, we explore how the complexity of the underlying chaotic system correlates with computational performance, revealing that both moderate coupling and node heterogeneity enhance RC capabilities. This work establishes a general and adaptive framework for utilizing chaotic dynamics in real-time computation.