# One Dimensional Maps - Diagnostics

## Timeseries

Shows graphically the iteration of the map. The iterations can be stepped through one at a time by hitting "Step" or iterated dynamically hitting "Start". The iteration speed (roughly in iterations per second) is set by the "Speed" slider.

The iteration of the nth order composition of the map function is displayed by setting "Compose" to n.

To study the sensitivity to small changes in initial conditions, set Delta-x to a small value. Two traces will be seen, corresponding to the two orbits starting from the nearby initial point.

## Spectrum

Plots the power spectrum of the variable defined by y-ax. The power spectrum is generated by successively accumulating Points number of points, performing a fast Fourier transform on these points, and calculating the magnitude squared to give the power spectrum of this segment of data. This is then averaged in with the power spectrum of previous segments. The maximum frequency in the power spectrum is the Nyquist frequency, which is in this case. The x-axis is scaled to this maximum frequency. The resolution on the frequency axis is then 2/(Points).

Segmenting the data introduces apparent discontinuities in the data to be transformed, which leads to broadening of the transform even for periodic signals and to high frequency tails. These effects are reduced by multiplying each segment by a window function that goes continuously to zero at the beginning and end of each data segment. The windowing function used is set by the parameter WinNum:

 WinNum Window Description Singularity in: 0 None Top Hat Function 1 Bartlett Tent First derivative 2 Welch Parabolic First derivative 3 Hanning Sinusoidal Second derivative

For more details see "Numerical Recipes" by W.H. Press et al. Note that there is no difficulty if the number of points over which the data is accumulated is a multiple of the period of a periodic orbit: in this case no window is the best choice.

## Histogram

Plots the histogram of the distribution of x values, which gives an approximation to the "invariant measure". The histogram is calculated as a running average of the fraction of points falling into each of Bins equal size divisions of the unit interval. Particular ranges of x can be enlarged by dragging the mouse over the stationary plot (the y-coordinates of the drag rectangle have no significance in this case.)

## Bifurcation

Displays the "bifurcation plot" showing the values of x visited in the dynamics (after the transient time set by "Transient") for each value of a. The a range is repeatedly scanned, with values slightly offset, to fill in the plot range.

## Lyapunov

Plots the Lyapunov exponent for each a.
[Instructions] [Equations]