Model coupled differential equations with intuitive parameters. See trajectories, null behavior, and system type immediately. Download neat outputs for study, teaching, and verification tasks.
| a | b | c | d | x₀ | y₀ | Total time | Steps | Expected type |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | -2 | -0.5 | 1 | 0 | 10 | 1000 | Stable spiral |
| 2 | 0 | 0 | -1 | 0.4 | 1 | 6 | 600 | Saddle point |
| 0 | -1 | 1 | 0 | 1 | 0 | 12 | 1200 | Center |
This calculator solves the planar system x′ = ax + by + f₁ and y′ = cx + dy + f₂.
The coefficient matrix is A = [[a, b], [c, d]]. Its trace is a + d. Its determinant is ad − bc.
The characteristic equation is λ² − (trace)λ + determinant = 0. The discriminant is trace² − 4(det).
When the determinant is nonzero, the equilibrium point solves A[x, y]ᵀ = −[f₁, f₂]ᵀ.
The trajectory itself is computed numerically with the fourth-order Runge–Kutta method. This method improves accuracy over simple Euler stepping.
A phase plane trajectory plotter helps you study two coupled first-order differential equations. It maps how one state variable changes against another. That view is useful in linear systems, control theory, population models, mechanics, and classroom demonstrations. Instead of reading only formulas, you see geometric behavior directly.
Many mathematical systems depend on stability. A small change in the starting point can decay, grow, or circulate. This calculator gives the trajectory, the equilibrium point, and the matrix-based classification. You can quickly test whether the system behaves like a node, saddle, center, or spiral. That saves time during analysis and homework.
The tool uses a general linear planar model with optional constant terms. That means you can analyze both origin-centered systems and shifted equilibria. The coefficient matrix controls local geometry. The trace and determinant summarize much of the behavior. Eigenvalues reveal whether motion moves inward, outward, or rotates around an equilibrium.
Closed-form solutions are not always convenient for quick exploration. This page therefore uses a fourth-order Runge–Kutta method. It tracks the path over a chosen time span using many small steps. You can increase the step count for smoother curves. The generated table helps you inspect time, x values, and y values together.
The plotted curve is the actual numerical orbit from your initial point. Dashed nullclines mark where one derivative becomes zero. Where the nullclines intersect, an equilibrium may exist. If the orbit approaches that point, the equilibrium is stable. If it moves away, the equilibrium is unstable. If it crosses directions sharply, a saddle is likely.
This phase plane trajectory plotter is useful for lectures, assignments, revision, and quick validation. It also helps compare different parameter sets without manual algebra each time. Because the calculator exports CSV and PDF files, it fits well into notes, reports, and lab-style documentation. It turns abstract linear dynamics into a readable visual workflow.
A phase plane trajectory is the path traced by a two-variable dynamical system. It shows how x and y evolve together, rather than separately over time.
This version handles linear planar systems with optional constant forcing terms. It is ideal for models written as x′ = ax + by + f₁ and y′ = cx + dy + f₂.
The calculator uses trace, determinant, and the discriminant of the coefficient matrix. These values indicate whether the equilibrium behaves like a node, saddle, center, or spiral.
Fourth-order Runge–Kutta is more accurate than a simple Euler step for the same time span. It usually produces smoother and more reliable trajectories for classroom and practical analysis.
Nullclines are curves where one derivative equals zero. They help identify direction changes and possible equilibria. Their intersection often marks an equilibrium point.
Yes. If constant terms are present, the equilibrium can shift away from the origin. This calculator solves for that point automatically when the determinant is nonzero.
Fast growth usually means the system is unstable or your initial point lies on an expanding direction. Large positive real parts of eigenvalues often cause that behavior.
Use any consistent units that match your model. The calculator does not assume fixed physical units. Consistency matters more than the specific unit choice.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.