Five-Node Tunnel-Diode Ladder

【MAL-Seeker Circuit Equation Test】A nonlinear ladder with 243 DC operating points

$$I_{TD}(v)=v^3-v$$ $$F_i=v_i^3-v_i+g_s(v_i-b_i)+g_c\sum_{j\in N(i)}(v_i-v_j)=0$$

5-Node Normalized Nonlinear Circuit Model
Expected Isolated Roots: 3⁵ = 243 / Solutions Found: 243

1. A forest of roots built from five tunnel diodes

A tunnel diode is a nonlinear device with a negative-resistance region, where current temporarily decreases as voltage rises. This benchmark represents that feature with the normalized cubic law $I_{TD}(v)=v^3-v$ and weakly couples five nodes into a ladder.

Without coupling, each node has approximately three states: $-1, 0,$ and $1$. Five nodes therefore give $3^5=243$ combinations. The weak coupling and small biases perturb the locations while preserving 243 isolated DC operating points.

B-Source SPICE3f5 Netlist

Five-node tunnel-diode ladder

Vb1 b1 0 -0.06
Vb2 b2 0  0.03
Vb3 b3 0  0.00
Vb4 b4 0 -0.03
Vb5 b5 0  0.06

Rs1 n1 b1 50
Rs2 n2 b2 50
Rs3 n3 b3 50
Rs4 n4 b4 50
Rs5 n5 b5 50

Rc12 n1 n2 66.6666666667
Rc23 n2 n3 66.6666666667
Rc34 n3 n4 66.6666666667
Rc45 n4 n5 66.6666666667

Btd1 n1 0 I=V(n1)*V(n1)*V(n1)-V(n1)
Btd2 n2 0 I=V(n2)*V(n2)*V(n2)-V(n2)
Btd3 n3 0 I=V(n3)*V(n3)*V(n3)-V(n3)
Btd4 n4 0 I=V(n4)*V(n4)*V(n4)-V(n4)
Btd5 n5 0 I=V(n5)*V(n5)*V(n5)-V(n5)

.nodeset V(n1)=1 V(n2)=1 V(n3)=1 V(n4)=1 V(n5)=1

.op
.end

2. Simplified model used in SPICE

Parameter	Value	Description
R_bias = 1 / g_s	50	Resistance from each node to its bias source
R_couple = 1 / g_c	66.6666666667	Resistance between adjacent nodes
b	(−0.06, 0.03, 0, −0.03, 0.06)	Five normalized bias values
I_TD(v)	v³ − v	Normalized tunnel-diode current

Since SPICE does not provide a dedicated tunnel-diode element, each B current source contains only the simplified characteristic $I=v^3-v$. Connections to the bias sources and between adjacent nodes are represented by resistors of $R_{bias}=50$ and $R_{couple}=66.6666666667$, respectively. For SPICE3f5 compatibility, values and cubic expressions are written directly without extended parameter or function syntax. The KCL of this circuit is identical to the system solved by MAL-Seeker.

$$I_{TD}(v)=v^3-v$$ $$F_i=I_{TD}(v_i)+g_s(v_i-b_i)+g_c\sum_{j\in N(i)}(v_i-v_j)=0$$

Boundary nodes have one neighbor; internal nodes have two. The middle node, for example, is described by:

$$v_3^3-v_3+0.02(v_3-0)+0.015\{(v_3-v_2)+(v_3-v_4)\}=0$$

All values are normalized. The netlist values 50 and 66.6666666667 are normalized resistances; no physical scaling is specified, so they are not presented as SI resistance values.

3. 243 paths, 243 roots, nothing missing

243Expected Isolated Roots

243 / 243Homotopy Paths Succeeded

243Real Solutions Found

9.26×10⁻¹⁴Maximum |Imaginary Part|

Run	Method	Expected Roots	Solutions Found	Run Time (sec)
Direct real search	Five real variables	243	243	0.308423
All-root verification	Complex-expanded total-degree homotopy	243	243	0.054863

The direct real search captured 243 valid roots. I then expanded the system into complex variables and applied total-degree homotopy: all 243 paths succeeded and reached the same 243 roots. The largest imaginary component was approximately $9.26\times10^{-14}$, within numerical error, so all 243 solutions are treated as real for the current coefficients.

4. Analysis environment

Analysis EngineMAL-Seeker (Project Antares Ver 3.1.6)
Development ProjectProject Bellatrix V1.0.4
ComputerMacBook Pro (Mac17,2)
OSmacOS Tahoe 26.4 (Build 25E246)
CPUApple M5, 10-core (4P + 6E), arm64
Memory32 GB

Dr.WataWata Insight:

One node has only three states, but connecting five of them immediately creates 243 candidates. The circuit is small while its operating-point count grows exponentially. Verifying the 243 roots from direct search against 243 successful paths in the complex domain moves the result from “many roots found” to a much stronger view of the complete solution set.

1. A forest of roots built from five tunnel diodes

2. Simplified model used in SPICE

3. 243 paths, 243 roots, nothing missing

4. Analysis environment

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