Transient Chaos, Bifurcation Structure, and Emergent Computation: How Dynamical Instability Organizes Collective Behavior Across Coupled Systems

Version 2 — revised in response to an external structural review and an automated critique pass. See "Response to Review" appendix in the PDF for the change log.

A structural pattern appears across several recent preprints in nonlinear dynamics and complex systems: transient dynamical instability — rather than asymptotic chaos or equilibrium synchronization — serves as the organizing substrate for collective behavior, computation, and pattern selection. This synthesis draws on six to eight findings from the nlin.AO, nlin.CD, and math.DS corpus to argue, as a *heuristic reading rather than a derivation*, that the interplay between bifurcation structure, finite-time Lyapunov instability, and network topology determines not just whether a system synchronizes, but *what it computes and how it patterns itself* during the approach to any attractor. Specific findings examined include: an exact analytical expression for transient maximum Lyapunov exponents (MLEs) in computation-capable networks [corpus:arxiv:2605.21174]; the role of higher-order hypergraph interactions in generating bistability and explosive synchronization transitions [corpus:arxiv:2605.24701]; dead-zone-induced phase drifting as a mechanism for flexible phase organization beyond classical Kuramoto dynamics [corpus:arxiv:2605.29167]; the finite-wavelength instability selecting spatial scale in adaptive transport networks [corpus:arxiv:2605.16130]; symmetry-breaking routes to high-dimensional extensive chaos in sparse neural networks [corpus:arxiv:2605.15872]; the bifurcation analysis of load imbalance in softmax mixture-of-experts routers [corpus:arxiv:2605.29121]; and the energetic characterization of transient clustering in aggregation-diffusion systems [corpus:arxiv:2605.30243]. The unifying claim is that the *transient regime before attractor settling* is where functional differentiation, spatial organization, and information processing are primarily encoded. The falsification path is explicit: if the MLE during transients in computation-capable networks were shown to be consistently negative (i.e., the hypothesized positive transient MLE is an artifact of the specific network class studied), the central thesis would be substantially weakened. ---

Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename.

Cited arXiv preprints: 2605.07614, 2605.11713, 2605.15872, 2605.16130, 2605.17490, 2605.21174, 2605.24701, 2605.27058, 2605.28297, 2605.28997, 2605.29121, 2605.29167, 2605.29231, 2605.29323, 2605.30243, 2605.30432