Understanding Invariant Sets
An invariant set is a region that keeps mapped points inside itself. In nonlinear dynamics, that idea is useful because curves may bend, stretch, or fold. This calculator studies a two dimensional discrete map. It checks whether an ellipse is forward invariant after one step. The test is numerical, so it gives evidence, not a formal proof.
Why Boundary Sampling Matters
For many smooth maps, the boundary is a practical place to test. The tool places points around the candidate ellipse. Each point is sent through the nonlinear update rule. The new point is measured with the same ellipse equation. When every sampled value is less than or equal to one, the set looks invariant. A positive margin gives more confidence. A negative margin warns that some mapped point escapes.
Local Stability Insight
The linear part near the origin also matters. Its eigenvalues describe local motion. When the spectral radius is below one, nearby orbits tend to contract for the linearized system. Nonlinear terms can still change behavior farther away. That is why the boundary check and orbit simulation are both included.
Practical Use Cases
Students can compare stable and unstable parameter sets quickly. Researchers can explore candidate Lyapunov regions before writing a proof. Engineers can tune nonlinear maps for bounded responses. Teachers can create examples where a set passes locally but fails globally.
Reading The Output
The maximum mapped level is the main number. Values below one indicate forward containment. The invariant margin equals one minus that maximum. The worst boundary point shows where the ellipse is most stressed. The orbit test starts from your chosen point. It reports escape time, final state, and largest orbit level.
Limits Of The Method
Sampling can miss thin escape channels between angles. Increase the sample count for stronger numerical evidence. Use smaller tolerance when precision matters. For rigorous work, combine these results with interval arithmetic, Lyapunov inequalities, or symbolic bounds.
Suggested Workflow
Begin with the linear coefficients and small ellipse radii. Confirm the local spectral radius first. Then raise nonlinear coefficients gradually and watch the margin. If the margin becomes negative, reduce the radii or adjust coefficients. Save and review the report after each useful trial for later comparison.