Multiscale Ordinal-Pattern Dynamics and Temporal Symmetries in a Photonic Neuron with Single and Dual Delayed Feedback.

Feedback delays and the coexistence of multiple timescales are central features of complex dynamical systems, ranging from neural networks and ecosystems to electronic and optical devices. Interactions between fast and slow dynamics can give rise to rich emergent behaviors that are absent in single-timescale systems. Here we investigate how these coupled timescales shape the dynamics of a photonic neuron with single and dual delayed feedback. Using ordinal pattern analysis and recent ordinal-based complexity measures, we characterize the temporal correlations and symmetry properties of the fast peaks and slow spikes generated by the system. Our results show that the signatures of determinism exhibited at fast and slow timescales differ markedly, revealing a strongly multiscale organization of the dynamics. Despite these differences, when represented in the symmetry-based Φ-space, all cases, fast peaks and slow spikes under both single and dual feedback, collapse onto a common curve. This universal structure indicates the presence of underlying constraints governing the system's dynamics across temporal scales and feedback configurations. These results highlight the power of ordinal-based approaches to uncover hidden symmetries and multiscale organization in delayed nonlinear systems.