Phase Line Diagram
Visualize equilibrium points and stability for autonomous ODEs \(y' = f(y)\). Roots are located by sign-change scanning then refined with high-precision bisection.
What Is a Phase Line?
For an autonomous ODE \(y' = f(y)\), the phase line is the \(y\)-axis annotated with equilibrium points \(f(y^*)=0\) and arrows showing whether solutions increase or decrease in each interval.
Stability Classification
Algorithm
The interval \([y_{\min},y_{\max}]\) is scanned for sign changes in \(f(y)\). Each bracket is refined with a high-precision bisection method using 50-digit arithmetic (Decimal.js), then further polished with Newton's method. This reliably finds roots even for non-smooth or piecewise-defined functions.
How to Use
- Type \(f(y)\) in the equation box. Press Enter or click Update Plot.
- Adjust y min / y max to control the scan range.
- Toggle Show graph of f(y) to overlay the curve on the phase line.
- Click example equations or function names to load them instantly.
Phase line visualization. Filled circles mark equilibrium points. Arrows show direction of flow between equilibria.
Run Update Plot to see equilibrium classification.
Click a name to load it as \(y'=f(y)\) and plot it immediately.
| Code | Notation |
|---|---|
| sqrt(y) | \(\sqrt{y}\) |
| cbrt(y) | \(\sqrt[3]{y}\) |
| abs(y) | \(|y|\) |
| sin(y) | \(\sin y\) |
| cos(y) | \(\cos y\) |
| tan(y) | \(\tan y\) |
| asin(y) | \(\arcsin y\) |
| acos(y) | \(\arccos y\) |
| atan(y) | \(\arctan y\) |
| sinh(y) | \(\sinh y\) |
| cosh(y) | \(\cosh y\) |
| tanh(y) | \(\tanh y\) |
| exp(y) | \(e^{y}\) |
| log(y) | \(\ln y\) |
| log10(y) | \(\log_{10} y\) |
| pi | \(\pi\) |
| e | \(e\) |
| ceil(y) | \(\lceil y\rceil\) |
| floor(y) | \(\lfloor y\rfloor\) |
| round(y) | \(\operatorname{round}(y)\) |
| max(a,b) | \(\max(a,b)\) |
| min(a,b) | \(\min(a,b)\) |
Cite this tool
Kapita, S. (2026). Phase Line Diagram. Math Tools. https://doi.org/10.5281/zenodo.20981338
Kapita, Shelvean. "Phase Line Diagram." Math Tools, 2026, doi.org/10.5281/zenodo.20981338.
@online{kapita2026phaseline,
author = {Shelvean Kapita},
title = {{Phase Line Diagram}},
year = {2026},
organization = {Math Tools},
doi = {10.5281/zenodo.20981338},
url = {https://doi.org/10.5281/zenodo.20981338}
}