# Technical Document Extraction: Scatter Plot Analysis
## Chart Description
This image is a **scatter plot** comparing three computational methods (ODE, KOL-δ, KOL-m) across varying parameters `C_I` and `C_u`. The y-axis represents a normalized integral value, while the x-axis categorizes data points by `C_I` and `C_u` values.
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### **Axis Labels & Markers**
- **Y-Axis**:
`C_I ∫₀ᵀ I²(t)dt + C_u ∫₀ᵀ u²(t)dt`
(Logarithmic scale: 10⁻⁶ to 10⁻³)
- **X-Axis**:
Discrete categories for `C_I` and `C_u` values:
- `C_I = 1` (repeated for all data points)
- `C_u = 1, 1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6`
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### **Legend**
| Symbol | Method |
|--------|----------|
| 🔵 Circle | ODE |
| 🔴 Triangle | KOL-δ |
| 🟣 Square | KOL-m |
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### **Data Points & Trends**
1. **C_u = 1**
- All methods cluster near **10⁻³** on the y-axis.
- ODE (🔵) and KOL-δ (🔴) overlap closely; KOL-m (🟣) is slightly lower.
2. **C_u = 1e-1 to 1e-3**
- Values decrease logarithmically with increasing `C_u` magnitude.
- ODE and KOL-δ remain tightly grouped; KOL-m lags by ~1–2 orders of magnitude.
3. **C_u = 1e-4 to 1e-6**
- ODE and KOL-δ converge toward **10⁻⁶**, while KOL-m remains significantly lower (~10⁻⁷ to 10⁻⁶).
- At `C_u = 1e-6`, KOL-m is **10× smaller** than ODE/KOL-δ.
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### **Key Observations**
- **ODE vs. KOL-δ**: Nearly identical performance across all `C_u` values.
- **KOL-m**: Consistently underperforms ODE/KOL-δ by 1–2 orders of magnitude.
- **Parameter Sensitivity**: Results degrade (higher integral values) as `C_u` decreases, but KOL-m exhibits greater sensitivity.
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### **Cross-Referenced Legend & Data**
- **C_u = 1e-6**:
- ODE (🔵): ~10⁻⁶
- KOL-δ (🔴): ~10⁻⁶
- KOL-m (🟣): ~10⁻⁷
- **C_u = 1e-1**:
- ODE/KOL-δ: ~10⁻³
- KOL-m: ~10⁻⁴
---
### **Conclusion**
The plot demonstrates that ODE and KOL-δ methods yield comparable results, while KOL-m underperforms significantly, particularly at lower `C_u` values. The logarithmic y-axis highlights exponential differences in computational efficiency or error metrics between methods.
