Calculator Input
Use coefficients for the autonomous model dx/dt = ax³ + bx² + cx + d. Equilibrium points occur when the expression equals zero.
Example Data Table
| Model | Equation | Real Equilibria | Typical Stability Pattern |
|---|---|---|---|
| Logistic-style cubic | dx/dt = x³ - 6x² + 11x - 6 | x = 1, 2, 3 | Unstable, stable, unstable |
| Quadratic balance | dx/dt = x² - 4x + 4 | x = 2 | Semi-stable repeated point |
| Linear decay model | dx/dt = -2x + 8 | x = 4 | Stable |
Formula Used
Equilibrium condition: Set the autonomous expression equal to zero.
g(x) = ax³ + bx² + cx + d = 0
The calculator reduces automatically to quadratic or linear form when higher coefficients are zero. For cubic models, it uses a depressed cubic transformation and Cardano-style real-root handling.
Local stability test: Evaluate the derivative at each equilibrium point.
g'(x) = 3ax² + 2bx + c
If g'(x*) < 0, nearby solutions tend to move toward the equilibrium. If g'(x*) > 0, nearby solutions move away. When g'(x*) = 0, the calculator checks sign changes around the point to label semi-stable or inconclusive behavior.
How to Use This Calculator
- Enter coefficients for the model dx/dt = ax³ + bx² + cx + d.
- Add a short label if you want the report to name your system.
- Select decimal precision between 2 and 10 places.
- Press the calculate button to solve real equilibrium points.
- Review multiplicity, derivative, and stability classification in the result table.
- Download the output as CSV or PDF for records.
Frequently Asked Questions
1. What is an equilibrium point?
An equilibrium point is a real x-value where the rate equation becomes zero. At that state, the modeled quantity stops changing instantaneously.
2. Which equations does this calculator support?
It supports autonomous polynomial models up to cubic form. If the cubic or quadratic coefficient is zero, it automatically solves the simpler equation.
3. Why are only real roots shown?
Equilibrium points in one-dimensional real models are real states. Complex roots may exist algebraically, but they usually do not represent physical or mathematical steady states here.
4. What does multiplicity mean?
Multiplicity tells you how many times the same root repeats. Repeated roots often create semi-stable or neutral behavior rather than simple stable or unstable patterns.
5. How is stability classified?
The tool first checks the derivative at each equilibrium. When that derivative is near zero, it inspects nearby sign changes to classify semi-stable or inconclusive behavior.
6. Can I use this for teaching examples?
Yes. The example table, interpretive labels, and downloadable reports make it useful for classroom demonstrations, assignments, and quick verification work.
7. What happens if all coefficients are zero?
The model becomes 0 = 0 for every x-value. In that special case, every real number acts like an equilibrium point.
8. Can I export my results?
Yes. After calculating, use the CSV or PDF buttons above the results table to save a structured copy of your equilibrium analysis.