Advanced Circle Map Calculator

Circle Map Input Panel

Use the standard circle map to iterate angular states, inspect wrapped and unwrapped motion, estimate rotation behavior, and study sensitivity under nonlinear forcing.

Initial Angle θ₀

Starting phase value. Use values like 0.15 or 1.25.

Driving Frequency Ω

Controls average phase advance at each iteration.

Nonlinearity K

Higher values increase distortion and locking effects.

Iterations

Total computed steps for the sequence.

Transient Steps

Early steps ignored for long-run summaries.

Decimal Precision

Controls displayed decimal places in results.

Wrap each new value into the interval [0, 1)

Reset

Example Data Table

Case	θ₀	Ω	K	Iterations	Transient	Use
Baseline	0.15	0.34	0.80	60	10	General study
Weak forcing	0.20	0.25	0.20	80	20	Smoother evolution
Stronger forcing	0.40	0.31	1.20	120	30	Locking sensitivity
Comparison run	0.05	0.45	0.95	100	15	Parameter tuning

Formula Used

Circle map equation:

θ_n+1 = θ_n + Ω − (K / 2π) sin(2πθ_n)

Wrapped form: θ_n+1 mod 1

Estimated rotation number: average increment after transient removal.

Approximate Lyapunov exponent: average of ln|1 − K cos(2πθ_n)| over non-transient steps.

Meaning of terms: θ is the phase angle, Ω is the driving frequency, and K controls nonlinear coupling strength. Wrapped values stay within [0,1), while unwrapped values preserve net drift over time.

How to Use This Calculator

Enter the initial angle, driving frequency, and nonlinearity value.
Select total iterations and how many early steps to treat as transient.
Choose the decimal precision for displayed outputs.
Keep wrapping enabled when you want values constrained to [0,1).
Press Calculate Circle Map to generate metrics, table output, and graph.
Review the rotation number and Lyapunov exponent for behavior clues.
Download the numerical table as CSV or create a printable PDF view.

FAQs

1. What does this calculator compute?

It computes repeated circle-map iterations, showing wrapped and unwrapped values, increments, derivative values, an estimated rotation number, and an approximate Lyapunov exponent.

2. Why is the wrapped value useful?

Wrapped values keep the phase inside one full turn, making periodic behavior easier to compare and plot across many iterations.

3. What is the rotation number?

It is the average net phase advance per iteration after excluding transient steps. It helps identify locking, drift, and long-run motion patterns.

4. What does the Lyapunov exponent indicate?

A negative value suggests nearby states tend to contract, while a positive value suggests stronger sensitivity to initial conditions.

5. Why remove transient steps?

Early iterations can reflect startup effects rather than settled behavior. Removing them gives cleaner long-run summary metrics.

6. Can I use negative inputs?

Yes. Negative initial angles, frequencies, or forcing values are mathematically valid, and the wrapping option will still map results into [0,1).

7. Why does stronger K change the motion so much?

K scales the sinusoidal distortion term. Larger values typically increase locking tendencies, deformation, and sensitivity in the map.

8. What do CSV and PDF exports include?

The CSV file includes the full numeric result table. The PDF option opens the browser print view for saving the page as PDF.