Calculator Inputs
Trace-Determinant Plot
This plot places your matrix at the point (trace, determinant). The parabola marks repeated eigenvalues for continuous systems. It still helps visualize matrix structure for discrete systems.
Perturbation Trajectory Plot
This graph follows a small perturbation around the chosen equilibrium. Continuous systems use a forward Euler linearized update. Discrete systems use repeated matrix multiplication.
Example Data Table
| Case | System Type | Matrix | Trace | Determinant | Expected Class |
|---|---|---|---|---|---|
| Example 1 | Continuous | [0, 1; -2, -3] | -3 | 2 | Stable node |
| Example 2 | Continuous | [0, -1; 1, 0] | 0 | 1 | Center |
| Example 3 | Continuous | [2, 0; 0, -1] | 1 | -2 | Saddle point |
| Example 4 | Discrete | [0.5, 0.2; -0.1, 0.4] | 0.9 | 0.22 | Asymptotically stable |
Formula Used
Linear stability analysis starts by linearizing a nonlinear system near an equilibrium point. For a two-variable system, the Jacobian or linearized matrix is:
A = [a b; c d]
The characteristic polynomial is:
λ2 - (tr A)λ + det(A) = 0
where:
- tr A = a + d
- det(A) = ad - bc
- Discriminant = (tr A)2 - 4 det(A)
For continuous systems, stability depends on eigenvalue real parts. If both real parts are negative, the equilibrium is locally asymptotically stable. If one is positive and one is negative, the point is a saddle and unstable.
For discrete systems, stability depends on eigenvalue magnitudes. If both magnitudes are below one, the fixed point is asymptotically stable. If at least one magnitude exceeds one, perturbations grow.
How to Use This Calculator
- Select continuous-time or discrete-time analysis.
- Enter the 2×2 linearized matrix values.
- Provide the equilibrium point coordinates.
- Enter a small initial perturbation.
- Set the horizon and simulation steps.
- Click Analyze Stability.
- Read the classification, eigenvalues, and stability summary.
- Review the trace-determinant and perturbation trajectory plots.
- Download the result as CSV or PDF if needed.
About Linear Stability Analysis
Linear stability analysis studies how small perturbations behave near an equilibrium. It replaces the original nonlinear system with a local linear model. That local model often captures whether nearby solutions decay, grow, rotate, or switch directions.
For two-dimensional systems, the matrix trace and determinant already reveal a large part of the story. They help separate nodes, saddles, spirals, centers, and repeated-root cases. The discriminant adds information about repeated or complex eigenvalues.
In continuous-time problems, negative real parts indicate decay toward equilibrium. Positive real parts indicate growth away from equilibrium. Complex eigenvalues create rotational motion, so nearby points spiral or circulate.
In discrete-time problems, the key test is whether eigenvalue magnitudes lie inside the unit circle. Values below one contract perturbations. Values above one expand them. Mixed magnitudes typically create saddle-like behavior.
This page combines the algebraic tests with two plots. The first plot places the matrix in trace-determinant space. The second simulates a small perturbation using the linearized dynamics, making the local behavior easier to inspect.
FAQs
1. What does this calculator analyze?
It analyzes a 2×2 linearized system near an equilibrium point. It computes trace, determinant, discriminant, eigenvalues, local classification, stability summary, and two supporting plots.
2. When should I use continuous-time mode?
Use continuous-time mode for differential equations such as x′ = Ax. Stability is determined by eigenvalue real parts, not by eigenvalue magnitudes alone.
3. When should I use discrete-time mode?
Use discrete-time mode for recurrence relations or iterated maps such as xₙ₊₁ = Axₙ. Stability depends on whether eigenvalue magnitudes stay below one.
4. Why are trace and determinant useful?
For 2×2 systems, trace and determinant summarize the characteristic polynomial. They quickly indicate saddle regions, repeated roots, and whether eigenvalues are real or complex.
5. What does a negative determinant mean?
In continuous systems, a negative determinant means the eigenvalues have opposite signs. That creates a saddle point, which is always unstable near the equilibrium.
6. Are the trajectory plots exact solutions?
No. The displayed trajectory is a numerical simulation of the linearized model. It is useful for local intuition, but exact formulas may differ from the plotted approximation.
7. What if the eigenvalues lie on the boundary?
Boundary cases need extra care. Zero real parts in continuous systems or unit-modulus eigenvalues in discrete systems can make linear tests inconclusive.
8. Can I use this for nonlinear systems?
Yes, after linearizing the nonlinear system at an equilibrium. Enter the Jacobian matrix evaluated at that point, then interpret the result as a local analysis.