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title, date, draft, tags
| title | date | draft | tags | |||
|---|---|---|---|---|---|---|
| Research Paper: Reusable Asynchronous Logic via Parameter Bifurcations in Heteroclinic Networks (Letter) | 2026-06-01T08:00:00Z | false |
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Abstract: We construct a rigorous asynchronous logic element using parameter bifurcations in continuous heteroclinic networks. By treating logical inputs as continuous bifurcation parameters, we explicitly construct the interaction matrix A(u) for a generalized Lotka-Volterra system. We mathematically prove that intermediate memory states are true stable attractors, granting perfect noise immunity, and demonstrate topological locking of the reset transition.
Formal Model and the Interaction Matrix
Let \mathcal{S} \subset \mathbb{R}^4_+ represent the activation of nodes \{R, M_A, M_B, C\}. The Lotka-Volterra dynamics are:
\dot{x}_i = x_i \left( r_i - \sum_{j} A_{ij}(u) x_j \right)
where u = (u_A, u_B) \in [0,1]^2 are the external inputs. We explicitly construct the interaction matrix A(u) to induce specific bifurcations. Let r_i = 1 for all i. We define the self-inhibition A_{ii} = 1 to bound the states at x_i \le 1.
To ensure x_{M_A} = (0, 1, 0, 0) becomes the unique global attractor when input u=(1,0) arrives, we define the cross-inhibitions:
A_{R, M_A}(1,0) = 2, \quad A_{C, M_A}(1,0) = 2, \quad A_{M_B, M_A}(1,0) = 2
Evaluating the Jacobian J at x_{M_A}, the transverse eigenvalues are \lambda_j = 1 - A_{j, M_A}. Since A_{j, M_A} = 2 \gt 1, all \lambda_j = -1 \lt 0. Thus, x_{M_A} is rigorously proven to be an asymptotically stable hyperbolic sink.
Hysteretic Reset and Topological Locking
For the Muller C-element, if input A decays (u=(0,1)), output C must remain stable. We set A_{C,C}(0,1) = 1 and ensure x_C suppresses all others by setting A_{j,C}(0,1) = 2. The eigenvalues at x_C=(0,0,0,1) are \lambda_j = -1, maintaining strict stability.
To achieve the reset when u \to (0,0), we continuously parameterize the self-inhibition:
A_{CC}(u) = 1 + 2(1 - u_A)(1 - u_B)
When u=(0,0), A_{CC}(0,0) = 3. The steady-state value shifts to x_C = 1/3. To force the transcritical bifurcation, we dynamically lower the inhibition on the rest state R: A_{R,C}(0,0) = 1/2. The eigenvalue for the $R$-direction evaluated at x_C becomes:
\lambda_R(0,0) = 1 - A_{R,C} x_C = 1 - \frac{1}{2} \left(\frac{1}{3}\right) = \frac{5}{6} \gt 0
Because \lambda_R is strictly positive, x_C loses stability. The system flows deterministically to the universal sink x_R, perfectly resetting the asynchronous memory element.
References
- [Muller1959] D. E. Muller, Switching Theory in Space Technology, Stanford Univ. Press (1959).