Computed Results
The summary appears here after you press the analyze button.
Calculator Inputs
Enter a linear planar system in matrix form. The calculator studies the equilibrium at the origin and numerically traces the selected trajectory.
Example Data Table
Use any example to populate the calculator instantly.
| Example | a | b | c | d | x(0) | y(0) | Expected Type | Notes | Use |
|---|---|---|---|---|---|---|---|---|---|
| Stable Node | -2 | 0 | 0 | -1 | 2 | 1 | Stable node | Two negative real eigenvalues. | |
| Saddle Point | 1 | 0 | 0 | -2 | 1.5 | 1 | Saddle point | Eigenvalues have opposite signs. | |
| Center | 0 | -1 | 1 | 0 | 2 | 0 | Center | Purely imaginary eigenvalues. | |
| Stable Spiral | 0 | 1 | -2 | -0.4 | 1.2 | 0.8 | Stable spiral | Complex eigenvalues with negative real part. |
Formula Used
The calculator studies the linear system:
The coefficient matrix is:
Key invariants used for classification:
Eigenvalues are obtained from the characteristic equation:
Nullclines are the sets where one derivative vanishes:
The trajectory shown on the plot is computed numerically with a fourth-order Runge–Kutta method.
How to Use This Calculator
1. Enter the four coefficients a, b, c, and d for your system matrix.
2. Set the initial condition x(0), y(0) to choose the starting point.
3. Pick a time window and a step size for numerical tracing.
4. Adjust the direction field density if you want more or fewer markers.
5. Leave the plot range blank for automatic scaling, or enter a custom half-range.
6. Press Analyze System to show the result section above the form.
7. Review the equilibrium type, eigenvalues, nullclines, and stability summary.
8. Export the output with CSV or PDF when you need a saved report.
FAQs
What does this calculator analyze?
It analyzes two-variable linear autonomous systems of the form x′ = ax + by and y′ = cx + dy. It classifies the equilibrium at the origin, computes eigenvalues, estimates trajectories, draws nullclines, and plots the phase plane.
Which systems are supported?
This version supports 2×2 linear systems with constant coefficients. It is ideal for trace-determinant analysis, classroom demonstrations, and quick numerical checks before moving into deeper symbolic work.
How is the equilibrium classified?
The classification uses the trace, determinant, and discriminant of the coefficient matrix. Those quantities determine whether the origin behaves like a saddle, node, spiral, center, or degenerate equilibrium.
What do trace and determinant tell me?
The trace controls average expansion or contraction, while the determinant helps separate saddles from non-saddles. Combined with the discriminant, they reveal whether eigenvalues are real, repeated, or complex.
Why might a trajectory stop early?
A trajectory can stop if the numerical path becomes excessively large. That safeguard prevents unstable systems from producing unreadable plots or extremely large export files.
What are nullclines?
Nullclines are curves where one derivative is zero. On x′ = 0, motion is purely vertical. On y′ = 0, motion is purely horizontal. Their intersections identify equilibria.
Can I analyze nonlinear systems here?
Not directly. For nonlinear systems, first linearize near an equilibrium using the Jacobian matrix. Then enter that local linear system here to inspect the nearby phase plane behavior.
Why export CSV or PDF?
CSV is useful for numeric review, reports, or further plotting. PDF is convenient for sharing the computed summary and graph with students, colleagues, or clients.