The Phase Transition of the Ibaguner Fractal Operator: From Stability to Chaos

<p>This paper reports the discovery of a dynamical phase transition within the parameter space of the Ibaguner Fractal Operator (IFO). By analyzing the iterated map x n+1 = tanh(Re(Li s (x n))), we identify a critical polylog parameter s c ≈ −2 that separates two distinct dynamical regimes. For s > s c , the IFO possesses a stable fixed point attractor, functioning as a non-linear stabilizer. For s < s c , the fixed point destabilizes via a period-doubling cascade into a chaotic attractor, and the IFO functions as a complexity generator. This dual nature establishes the IFO as a meta-operator capable of both resolving and generating complexity.</p>