Poincaré Map Calculator

Explore periodic and chaotic motion using clean Poincaré sampling tools right now. Choose classic systems, set sections, then compare point clouds across sweeps quickly.

Inputs
Use the form to generate a Poincaré point set.
Pick a model family and map style.
Smaller dt improves accuracy but increases runtime.
Number of points saved into the map.
Safety cap for long chaotic runs.
For flows, this is the x start value.
Initial velocity or angular rate.
Initial y value for section-based systems.
Initial z value for section-based systems.
Discard early cycles before sampling.
Discard early time before section crossings.
Collect points when the trajectory crosses this plane.
Controls which crossings enter the map.
Linear stiffness coefficient.
Cubic stiffness coefficient.
Velocity damping strength.
Drive amplitude.
Sampling uses period T = 2π/ω.
Angular damping coefficient.
Periodic torque strength.
Prandtl-like parameter.
Rayleigh-like parameter.
Geometric factor.
Linear coupling term.
Offset term in z equation.
Nonlinear feedback strength.
Results appear above this form after submission.
Example Data Table
Sample stroboscopic points (illustrative only):
#uvt
10.1352-0.0167523.599
20.1318-0.0189528.835
30.1283-0.0210534.071
40.1247-0.0231539.307
50.1210-0.0251544.543
Your values depend on parameters, dt, and transients.
Formula Used

A Poincaré map reduces continuous dynamics to a discrete map by sampling a trajectory on a repeating event. For periodically forced systems, the stroboscopic map is:

xn+1 = ΦT(xn)

Here ΦT is the flow after one forcing period T = 2π/ω. This calculator integrates the state using the classical fourth‑order Runge–Kutta method (RK4):

sk+1 = sk + (Δt/6)(k1 + 2k2 + 2k3 + k4)

For autonomous 3D flows, the map stores points whenever the trajectory crosses a plane x = xsec, with linear interpolation to estimate the crossing location.

How to Use This Calculator
  1. Select a system. Forced models use stroboscopic sampling automatically.
  2. Set dt and transient removal. Smaller dt improves numerical fidelity.
  3. For section maps, choose x = xsec and a crossing direction.
  4. Increase samples to reveal invariant curves or scattered chaotic sets.
  5. Export CSV or PDF to visualize points in external tools.
Professional Notes
  • If the map drifts or smears, reduce dt or raise transient removal.
  • For section maps, choose a plane that the attractor crosses frequently.
  • A very large max‑steps may slow servers; keep it reasonable.
  • Use identical settings across sweeps to compare structures fairly.

Model dynamics clearly, then export maps for deeper insight.

Professional Article

1) Why Poincaré Maps Matter

Many nonlinear systems evolve in continuous time, yet their long‑term behavior is easiest to compare in a discrete snapshot. A Poincaré map samples the state once per cycle (stroboscopic) or whenever a trajectory crosses a chosen surface (section). The resulting point set exposes periodic orbits, quasiperiodic tori, and chaotic attractors with far less clutter than a full time series.

2) What This Calculator Produces

The calculator integrates the selected model and records intersection points after transients are discarded. It reports the number of accepted points, basic statistics, and a downloadable table for plotting. For stroboscopic maps, the sampling period is T = 2π/ω; for section maps, crossings are detected by a sign change across the plane with linear interpolation to estimate the intersection.

3) Default Models and Typical Ranges

Three common benchmarks are included: the Duffing oscillator, the driven pendulum, and the Rössler flow. Reasonable starting ranges are δ≈0.05–0.3, α≈−1 to 1, β≈0.1–1, γ≈0–0.6 for Duffing; damping b≈0.05–0.5 and drive A≈0–2 for the pendulum; and a=0.2, b=0.2, c≈4–10 for Rössler.

4) Time Step and Accuracy Guidance

Because the map is sensitive to phase, choose dt small enough to resolve the fastest oscillation. A practical rule is dt ≤ T/500 for stroboscopic sampling, then tighten if points smear. For flows like Rössler, dt of 0.001–0.01 often works; if the solver hits max‑steps, reduce the total time or increase max‑steps cautiously to protect performance.

5) Transient Removal and Point Count

Discarding early iterations is essential: many systems need hundreds of cycles to settle onto an attractor. A transient fraction of 20–50% is common. For clear structure, aim for 200–2000 Poincaré points. Too few points can hide period‑doubling; too many points can slow rendering and inflate downloads without adding insight.

6) Reading the Geometry

A single repeating point implies a period‑1 orbit; two or four clustered points indicate period‑doubling. A closed curve suggests quasiperiodicity, while a fractal cloud signals chaos. In section maps, check that crossings are consistently oriented; mixing both directions can mirror the set and complicate interpretation.

7) Parameter Sweeps and Reproducibility

To study bifurcations, sweep one parameter while keeping dt, transient fraction, and sampling rules fixed. Export each run to CSV and plot overlays or bifurcation diagrams in external tools. Record the exact initial conditions and seed settings; small changes can shift the observed set when the dynamics are chaotic.

8) Practical Uses in Physics

Poincaré maps support stability analysis in driven oscillators, beam dynamics, and celestial mechanics. They help identify resonances, locking regions, and transport barriers. In experiments, the same concept appears as stroboscopic imaging or phase‑locked sampling, making the calculator useful for planning measurements and validating simulations.

Summary: Reliable maps come from careful sampling and consistent settings.

FAQs

What is the difference between stroboscopic and section maps?

Stroboscopic maps sample once every forcing period. Section maps sample when the trajectory crosses a chosen surface, such as y=0. Stroboscopic maps suit periodically driven systems; section maps suit autonomous flows.

Why do my points look smeared or noisy?

Usually dt is too large, transients are not removed, or the crossing surface is rarely hit. Reduce dt, increase transient removal, and choose a section plane with frequent, consistent crossings.

How many points should I generate?

Start with 500 points after transients. Increase to 2000 for fine structure, or reduce to 200 if you only need a quick diagnostic. Extremely large point counts mainly increase computation and file size.

What does a single point or a few points mean?

One point indicates a period‑1 orbit in the sampled map. Two, four, or more repeating clusters often indicate period‑doubling. A closed curve suggests quasiperiodicity; a scattered cloud suggests chaos.

How is the section crossing location computed?

The calculator checks for a sign change across the section plane between integration steps. It then linearly interpolates within that step to estimate the intersection point and time, improving accuracy over raw step endpoints.

Can I use this for bifurcation studies?

Yes. Sweep a parameter (e.g., drive amplitude) while keeping dt, transient fraction, and sampling rules fixed. Export each run to CSV and plot the sampled coordinate versus the parameter to reveal period‑doubling and chaos windows.

Why does changing initial conditions change the map?

Chaotic dynamics amplify tiny differences, so nearby starts can land on different parts of the attractor. For multistable systems, different initial conditions can converge to different attractors, producing distinct Poincaré sets.

Explore, export, and compare maps to reveal hidden dynamics.

Related Calculators

linearization calculatorvector field plottercritical slowing downspectrogram generatorperiod estimatorsymplectic integratornullcline calculatorphase response curvephase portrait generatorfrequency response calculator

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.