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System Analysis
Class 10 A/B/C/D · Network diagrams · Exam map
📐 4-Variable A/B/C/D System (Class 10)
The full system
$$A'(t)=\frac{512}{512+B(t)^9}-0.1A(t)$$
$$B'(t)=0.4A(t-10)-0.5B(t)+0.3D(t)$$
$$C'(t)=A(t-4)-\frac{D(t-1)^3}{1+D(t-1)^3}-0.2C(t)$$
$$D'(t)=\frac{C(t)^2}{1+C(t)^2}-0.1D(t)$$
Reading each term
A' equation: $\frac{512}{512+B^9}$ = decreasing sigmoid of $B$ with $n=9$ (steep!) → negative feedback from B on A. $-0.1A$ = degradation. No explicit delay.B' equation: $0.4A(t-10)$ = positive link from A with explicit delay $\tau=10$. $0.3D(t)$ = positive link from D (no delay). $-0.5B$ = degradation.
C' equation: $A(t-4)$ = positive link from A with explicit delay $\tau=4$. $\frac{D(t-1)^3}{1+D(t-1)^3}$ = increasing sigmoid of $D(t-1)$ with $n=3$ and delay $\tau=1$ (negative sign → D inhibits C). $-0.2C$ = degradation.
D' equation: $\frac{C^2}{1+C^2}$ = increasing sigmoid of C with $n=2$ → C stimulates D. $-0.1D$ = degradation. No explicit delay.
The feedback loops
7 feedback loops total (Bennoun confirmed in Class 10):4 self-loops: Each variable has a $-kX$ degradation term → negative self-loop on A, B, C, and D.
A↔B loop (negative): A→B positive (via $+0.4A(t-10)$), B→A negative (via decreasing sigmoid $\frac{512}{512+B^9}$). Product: $(+)(-) = $ negative feedback. Has delay $\tau=10$ and steep $n=9$ → A and B WILL oscillate.
C↔D loop (negative): C→D positive (via increasing sigmoid $\frac{C^2}{1+C^2}$), D→C negative (via $-\frac{D^3}{1+D^3}$). Product: $(+)(-) = $ negative feedback. BUT: delay $\tau=1$ is too small and $n=3$ is not steep enough → C and D do NOT oscillate. Confirmed explicitly in class 2 slides.
A→C→D→B→A loop (positive): A→C positive ($\tau=4$), C→D positive, D→B positive ($+0.3D$), B→A negative. Three positives and one negative = negative overall? Wait — count: $(+)(+)(+)(-) = $ negative feedback loop overall.
What Bennoun tests: Given a network diagram, identify each link and delay, count all loops (including self-loops), and classify each as positive or negative feedback.
📋 Bennoun's Exam Topic Map
The 5 topics explicitly listed (from Class 10 annotations)
| Topic | What it tests | Where in guide |
|---|---|---|
| A — Linearity | Prove/disprove linearity using $f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$. Full formal proof or specific counterexample. | Unit 2 |
| B — Linearity with vectors | Same proof but for vector-valued functions. Also: given known outputs, compute $f(\alpha\mathbf{u}+\beta\mathbf{v})$ using linearity alone. | Unit 2 |
| C — Discrete-time models | Build stage-structured matrix from assumptions. Long-term behavior from eigenvalues. Dominant eigenvector for ratio prediction. | Units 2–3 |
| D — Equilibrium points | Find all EPs via nullclines. Jacobian at each EP. Classify stability. Coexistence analysis. | Unit 4 |
| E — Oscillation | Identify feedback loops and delays. Hopf bifurcation conditions. Finding bifurcation parameter. | Units 1 & 4 |
🔥 What this means for your studying
Topics A and B (linearity proofs) are explicitly listed. Practice writing the full formal proof structure: let $\mathbf{u}=(X_1,Y_1)$, $\mathbf{v}=(X_2,Y_2)$, scalars $\alpha,\beta$ — compute both sides — show they match.Topic D (equilibrium points) had an entire 25-minute review session dedicated to it. Be able to find all 4 EPs of a 2D competition model systematically.
Chaos (Unit 5) is important for the final but was NOT in Bennoun's 5-topic list — it's a separate block.