Autonomous Neural Oscillators: Non-Serial Architecture, Parallel-Path Linearization, and a Quasi-Periodic Attractor
Feedforward neural networks deployed autoregressively act as deterministic oscillators, their period and stability governed by a linear Companion Matrix built from the network’s origin-linearization. This paper stress-tests a serial-baseline “Pyramid Up” topology on a challenging compound signal with a 4:1 ratio of two harmonics; the baseline exhibits 9.39% envelope drift over the test period, leading to the introduction of the Skip-Bottleneck Network (SBN). The SBN topology outperforms the baseline by roughly 50× on our precision metric. Resolving SBN’s non-serial topology requires extending the Companion Matrix framework itself: a three-path sum rule, one term per distinct-depth route through the network’s parallel bypass connections, which to the best of our knowledge is novel. Topological sweeps isolating capacity and bypass routing show that fair capacity alone closes much of the precision and spectral-cleanliness gap; the optimal bypass configuration beyond that depends on which metric is prioritized, but we retain the full architecture for its precision advantage. Finally, we stress-test SBN again against a signal with an irrational period ratio with no periodic solution to settle into, resulting in a lock. We report this as strong evidence for genuine quasi-periodicity: five independent, convergent lines of evidence (a direct chaos test, Companion Matrix eigenvalue clustering, spectral structure, phase-portrait geometry, and violation of an independently derived stability bound) all point in the same direction, reinforced by direct visual evidence. Taken together, these two results—a new mathematical formalism for non-serial autoregressive architectures, and a demonstrated case of genuine quasi-periodic autonomous locking—constitute this paper’s core contribution.