Infinite Dimensional Chaotic Synchronization for Randomly Coupling Systems and Dynamical Phase Transition

We rigorously investigate the macroscopic dynamical phase transitions and chaotic synchroniza- tion in infinite-dimensional networks governed by generalized Boole transformations under symmet- ric structural quenched disorder. By leveraging the heavy-tailed statistics of the chaotic mappings, we establish a self-consistent macroscopic renormalization group framework and show that the back- ground fields asymptotically converge to an invariant Cauchy distribution characterized by a stable scale parameter γ∗. Through a single-node deviation analysis in the thermodynamic limit, combined with exact complex residue calculus, we analytically derive the conditional Lyapunov exponent in a parsimonious closed form: λc = 2 ln√α + p1 − α − K E [|εij |]. This enables us to establish the exact, universal synchronization critical coupling threshold: Kc = 2√α(1 − √α)/E|εij |. Crucially, our pure theory demonstrates that the macroscopic synchronization threshold is governed solely by the first absolute moment of the coupling weights, revealing a profound universality class that is fundamentally resilient against microscopic structural fluctuations. Furthermore, we show that this transition marks a coherent macroscopic condensation of infinite chaotic degrees of freedom into a single effective one-dimensional orbit, in close analogy with Bose-Einstein condensation (BEC).