FFT and Spectral Analysis

Generate time-series outputs and compute Fast Fourier Transforms to compare frequency content in simulated trajectories.

Steps

1. Open the FFT workflow in the toolbox.
2. Select the same system and parameters used for the trajectory you want to analyze.
3. Use Runge-Kutta 4, a small time step, and a total time long enough to leave the startup transient behind.
4. Choose which state variable or visible coordinate will be transformed.
5. Compute the normalized spectra for the retained state variables.
6. Compare broadband regions against narrow spectral peaks before describing an orbit as chaotic or periodic.
7. Export the figure after confirming the frequency range is not dominated by the initial transient.

Theory

Spectral analysis converts a time-domain signal into frequency content. Periodic trajectories display sharp peaks at a fundamental frequency and its harmonics. Chaotic trajectories often distribute energy over broader bands because many time scales are mixed.

In the toolbox, the FFT is computed from retained state data after centering and normalizing the signal. The choice of variable matters: x(t), y(t), and z(t) can emphasize different frequencies because they observe different coordinates of the same orbit.

FFT is useful for comparison, not final classification. Broadband energy can support an interpretation of irregular motion, but the same plot must be checked against phase portraits, transients, bifurcation structure, and Lyapunov diagnostics.

How Parameters Change This Figure

• Sampling step: dt controls the highest frequency that can be represented without aliasing.
• Total time: longer windows improve frequency resolution; short windows blur peaks.
• Transient removal: startup behavior can leak energy across frequencies and hide the final regime.
• Observed variable: x, y, and z can produce different spectral emphasis for the same trajectory.
• Windowing and normalization: scaling choices change visual contrast but not the underlying sampled signal.
• Frequency range: zooming too far out can compress useful structure near low frequencies.