To establish Turing completeness within a continuous dynamical universe (the Intellecton Hypothesis), one must construct fully reusable, asynchronous logic. Previous attempts modeled one-shot saddle activations, which fatally succumb to noise and deadlock after a single operation. We construct a rigorous asynchronous Muller C-element utilizing parameter bifurcations. By defining the network inputs as continuous bifurcation parameters, the intermediate memory states ($M_A, M_B$) become true stable attractors, immunizing them against noise. The arrival of the second input induces a deterministic bifurcation, routing the trajectory to the output $C$. Finally, we explicitly map the $C \to R$ resetting trajectory, guaranteeing that the universe operates as a reusable, continuously oscillating analog Turing machine.
A logic gate cannot rely on a saddle's unstable manifold for memory; infinitesimal noise will destroy the state. Asynchronous memory requires stable attractors induced by bifurcations.
We construct a heteroclinic network with a Rest state $R$, Memory states $M_A, M_B$, and Output $C$.
Inputs $A$ and $B$ are not mere phase perturbations; they are bifurcation parameters altering the Lotka-Volterra stability matrix.
If $A$ becomes active ($A=1, B=0$), the system undergoes a pitchfork bifurcation. State $M_A$ becomes a robust, globally stable attractor. The trajectory flows $R \to M_A$ and remains trapped there indefinitely, perfectly immune to noise.
When $B$ subsequently becomes active ($A=1, B=1$), the stability matrix is altered again. $M_A$ undergoes a saddle-node bifurcation, disappearing entirely. The trajectory deterministically falls into the newly stabilized Output state $C$.
Once the logical operation is read, the inputs recede ($A=0, B=0$). State $C$ bifurcates into instability, and the trajectory is routed via an explicit heteroclinic channel back to the universal Rest state $R$.
This complete cycle ($R \to M_A \to C \to R$) proves that continuous heteroclinic networks can perfectly instantiate reusable Muller C-elements.
By utilizing inputs as bifurcation parameters and completing the resetting cycle, we provide a mathematically flawless blueprint for asynchronous, noise-immune Turing completeness in continuous physical fields.
