Welch PSD and Amplitude Spectrum

Estimate Welch power spectral densities or amplitude spectra for all displayed state components while preserving the physical frequency axis.

Difficulty Level Intermediate

Question Which frequencies are present?

Objective

Estimate and compare the frequency content of all displayed state components using either Welch power spectral densities or amplitude spectra.

Workspace

GUI tab

Espectro

Procedure

Generate a compatible trajectory or configure the system directly in Espectro. The tab analyzes and plots every displayed state component; it does not provide a variable selector.
Set the transient, integration step, total duration, and frequency limits before choosing the spectral output.
Use Welch PSD for an averaged power-per-frequency estimate or Amplitude spectrum for Fourier-component magnitude; do not relabel one as the other.
Generate the graph and repeat with a longer retained window or smaller dt to check whether the main features persist.

Result

Expected result

A frequency axis consistent with the trajectory sampling step and one Welch PSD or amplitude curve for each displayed state component.
Readable dominant peaks, harmonics, or broadband regions within the requested frequency limits.

Interpretation

Sharp peaks can support a periodic interpretation, whereas broader content can accompany irregular motion; compare the component curves because each coordinate can emphasize different frequencies.

Spectral content is supporting numerical evidence and must be interpreted with trajectories, transient checks, and other diagnostics.

Export

Export and reproducibility

Save the graph after confirming the mode, units, component labels, and frequency limits shown in the figure.
Record the system, parameters, initial state, integration method, dt, total time, discarded transient, reused-trajectory state, spectral mode, displayed components, and frequency window.

Applications

Comparing dominant frequencies across parameters, state components, or numerical settings.
Characterizing periodic, quasiperiodic-looking, and irregular signals before deeper analysis.
Preparing reproducible spectral figures for teaching and numerical research reports.

Limits

Finite windows limit frequency resolution, while a large dt can alias high-frequency content.
Welch PSD and amplitude spectrum are different quantities and cannot be compared without preserving their definitions and units.
A peak pattern or broadband spectrum alone does not prove chaos, attraction, or hiddenness.

Toolbox Chaos spectral tab displaying a Welch power spectral density — Current Spectrum tab with Welch PSD selected. The graph includes every displayed state component, while the controls retain the simulation step, transient, and frequency limits needed to interpret the axis and units.

Steps

Generate a trajectory in 3D Attractor, then open Spectrum and select Use last trajectory; alternatively, configure the same system directly in this tab.
Set a transient long enough to exclude startup motion. The tab calculates and plots all displayed state components together; there is no observed-variable selector.
Select Welch PSD (recommended) when comparing distributed power, or Amplitude spectrum when the magnitude of discrete frequency components is the quantity of interest.
Confirm that dt and total duration support the frequency range and resolution required by the study.
Generate the graph, inspect peaks and broadband regions, then repeat with a longer window or smaller step to assess numerical sensitivity.
Save the figure only after recording the system, parameter values, initial state, integration method, dt, total time, transient, displayed components, and spectral mode.

Theory

Spectral analysis converts a sampled time-domain signal into frequency content. Periodic trajectories commonly display sharp peaks at a fundamental frequency and its harmonics, while irregular trajectories may distribute power over broader bands.

Toolbox Chaos offers two distinct outputs. The Welch power spectral density averages overlapping windowed segments to reduce the variance of the estimate; the amplitude spectrum reports Fourier-component magnitude. They do not represent the same physical quantity and should not be labeled interchangeably. The tab draws a curve for every displayed coordinate because each one observes a different projection of the orbit.

A spectrum is a comparison and characterization tool, not a proof of chaos. Broadband energy can support an interpretation of irregular motion, but it must be checked against trajectories, transients, step-size sensitivity, bifurcation structure, and convergence-aware 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.
Displayed components: compare the x, y, and z curves because they can emphasize different frequencies for the same trajectory.
Spectral mode: a Welch PSD measures power per unit frequency, whereas an amplitude spectrum measures Fourier-component magnitude.
Frequency range: zooming too far out can compress useful structure near low frequencies.