Calculator Inputs
Enter a 2×2 system matrix, choose continuous or discrete analysis, then submit to test stability using exact eigenvalues and standard criteria.
Example Data Table
| Case | System Type | Matrix A | Expected Nature | Reason |
|---|---|---|---|---|
| Example 1 | Continuous | [[-2, 1], [-3, -4]] | Asymptotically stable | Both eigenvalues have negative real parts. |
| Example 2 | Continuous | [[1, 2], [0, -1]] | Unstable | One eigenvalue is positive, so trajectories diverge. |
| Example 3 | Discrete | [[0.6, 0.2], [-0.1, 0.5]] | Asymptotically stable | All eigenvalues lie inside the unit circle. |
| Example 4 | Discrete | [[1.1, 0.3], [0.0, 0.9]] | Unstable | Spectral radius exceeds one. |
Formula Used
For a 2×2 matrix A = [[a, b], [c, d]], the basic quantities are:
- Trace: τ = a + d
- Determinant: Δ = ad − bc
- Characteristic polynomial: λ² − τλ + Δ = 0
- Eigenvalues: λ = (τ ± √(τ² − 4Δ)) / 2
Continuous-Time Stability Rule
For the system x′ = Ax, asymptotic stability requires every eigenvalue to have a negative real part. For a 2×2 matrix, this is equivalent to trace < 0 and determinant > 0.
Discrete-Time Stability Rule
For the system xk+1 = Axk, asymptotic stability requires every eigenvalue to satisfy |λ| < 1. For a 2×2 matrix, Jury conditions are:
- 1 − τ + Δ > 0
- 1 + τ + Δ > 0
- 1 − Δ > 0
Response Simulation
Continuous systems are simulated with a fourth-order Runge-Kutta method. Discrete systems are simulated with repeated matrix-state multiplication. The plotted norm is ||x|| = √(x₁² + x₂²).
How to Use This Calculator
- Select whether the system is continuous-time or discrete-time.
- Enter the four entries of the 2×2 system matrix.
- Provide initial conditions for x1 and x2.
- Set a time horizon or number of steps, plus plot points.
- Choose a tolerance for near-boundary decisions.
- Click Check Stability to generate the result above the form.
- Review eigenvalues, conditions table, and both Plotly graphs.
- Export the summary as CSV or PDF when needed.
FAQs
1) What does this calculator test?
It tests the stability of a 2×2 linear system using exact eigenvalues, trace, determinant, spectral radius, and standard continuous or discrete criteria.
2) What is asymptotic stability?
Asymptotic stability means trajectories move toward equilibrium over time. In continuous systems, real parts must be negative. In discrete systems, eigenvalue magnitudes must stay below one.
3) Why are trace and determinant useful?
For 2×2 systems, trace and determinant summarize the characteristic polynomial. They make stability checks fast and help identify nodes, spirals, saddles, and boundary cases.
4) What does the spectral radius show?
The spectral radius is the largest eigenvalue magnitude. In discrete-time systems, it directly indicates contraction, boundary behavior, or divergence around equilibrium.
5) Why can a system be marginally stable?
Marginal stability appears when eigenvalues lie exactly on the boundary, such as zero real part or unit magnitude. Motion neither clearly decays nor clearly grows.
6) What does the response graph represent?
The response graph shows x1, x2, and the state norm from the chosen initial condition. It helps confirm whether the numerical trajectory decays, oscillates, or grows.
7) Can I use this for larger matrices?
This page is designed for exact 2×2 analysis. Larger systems need higher-order characteristic equations, broader numerical routines, and more advanced stability tests.
8) Is this suitable for nonlinear systems?
It is best for linear systems. Nonlinear systems often require linearization around an equilibrium or completely different methods, depending on the model and operating point.