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:

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

Bifurcation

Lyapunov