Calculator Inputs
Enter a two dimensional autonomous system. The page evaluates a linear affine model and draws a phase portrait.
Interactive Phase Portrait
The graph shows the direction field, nullclines, equilibrium point, and simulated trajectories. Use the input fields above, then calculate again to refresh all values.
Example Data Table
| Example | System Matrix A | Trace | Determinant | Expected Portrait |
|---|---|---|---|---|
| Stable spiral | [[0, 1], [-1, -0.3]] | -0.3 | 1 | Rotates inward |
| Saddle | [[1, 0], [0, -2]] | -1 | -2 | Attracts and repels |
| Center | [[0, 1], [-1, 0]] | 0 | 1 | Closed orbits |
| Unstable node | [[2, 0], [0, 1]] | 3 | 2 | Moves outward |
Formula Used
Planar system:
x′ = ax + by + e
y′ = cx + dy + g
Matrix: A = [[a, b], [c, d]]
Equilibrium: A[X, Y]T = [-e, -g]T
x equilibrium: x* = (bg − de) / (ad − bc)
y equilibrium: y* = (ce − ag) / (ad − bc)
Trace: T = a + d
Determinant: D = ad − bc
Discriminant: Δ = T² − 4D
Eigenvalues: λ = (T ± √Δ) / 2
Trajectory method: The graph uses fourth order Runge Kutta steps.
How to Use This Calculator
- Enter the coefficients for x′ and y′.
- Set constants if the equilibrium is shifted.
- Add an initial point for the main trajectory.
- Adjust graph limits to zoom into the needed region.
- Choose a step size for trajectory smoothness.
- Click the calculate button.
- Read the equilibrium, eigenvalues, and stability class.
- Export the table, report, or graph when needed.
Article: Understanding Phase Portraits
What a Phase Portrait Shows
A phase portrait is a visual map of a differential equation system. It shows how two dependent variables move together. Each point in the plane has a direction. That direction comes from the derivatives at that point. When many arrows are drawn, the motion becomes clear. The graph can show attraction, repulsion, rotation, or mixed behavior.
Why Equilibrium Matters
An equilibrium occurs when both derivatives are zero. At that point, the system stops changing. Many phase portraits are studied around this point. The nearby arrows often reveal the long term pattern. A stable equilibrium pulls paths inward. An unstable equilibrium pushes paths outward. A saddle point does both at once.
Trace and Determinant Test
Linear systems can be classified with the trace and determinant. The trace is the sum of the diagonal terms. The determinant measures area scaling and orientation behavior. A negative determinant means a saddle. A positive determinant can create nodes, spirals, or centers. The discriminant separates real eigenvalues from complex eigenvalues.
Eigenvalues and Motion
Eigenvalues explain the local shape of the flow. Real eigenvalues usually create node behavior. Complex eigenvalues create rotation. Negative real parts mean inward motion. Positive real parts mean outward motion. Zero real parts need careful interpretation. The calculator reports these values after each run.
Using the Graph
The direction field gives the broad pattern. The nullclines show where one derivative becomes zero. Their intersection gives the equilibrium when it is unique. Trajectory curves show possible solutions from starting points. You can change limits to inspect a small area. You can reduce step size for smoother curves.
Practical Value
Phase portraits help in physics, biology, economics, and engineering. They make invisible dynamics easier to see. They also support quick stability checks. A table export helps document calculations. A graph export helps with reports and lessons. Always treat linear results as local information. Nonlinear models may change behavior far from equilibrium.
FAQs
1. What is a phase portrait?
A phase portrait is a graph of system motion in the x-y plane. It shows direction fields, solution paths, and equilibrium behavior for differential equations.
2. What type of system does this calculator use?
It uses a two variable linear affine system. The equations are x′ = ax + by + e and y′ = cx + dy + g.
3. What is an equilibrium point?
An equilibrium point is where x′ and y′ both equal zero. At that location, the system has no instant movement.
4. How is stability decided?
Stability is decided from trace, determinant, discriminant, and eigenvalues. These values classify nodes, spirals, centers, saddles, and repeated cases.
5. What does a saddle point mean?
A saddle point attracts paths in one direction and repels them in another. It appears when the determinant is negative.
6. Why are nullclines useful?
Nullclines show where one derivative is zero. Their intersections help locate equilibrium points and explain directional changes in the field.
7. Can I export my results?
Yes. Use the CSV button for spreadsheet data, the PDF button for a report, and the graph button for a plot image.
8. Does this replace full nonlinear analysis?
No. It gives strong linear insight. Nonlinear systems may need extra checks, especially far from the equilibrium point.