Recurrence plots, bifurcation diagrams, and fractals

Recurrence plots

Recurrence plot of over the ‐plane.

Bifurcation-related plots

Nestings of the map starting with as a function of . The number of iterations increases from red (0) to blue (80). The left graphic shows the real parts and the right graphic shows the imaginary parts.

Nestings of the map starting with as a function of . The initial iterations are ignored and the real parts are shown.

Iterations and powers in the complex plane

The real part and the imaginary part of the scaled value of in the . The left graphic shows the real part and the right graphic shows the imaginary part.

The real part and the imaginary part of the scaled value of in the . The left graphic shows the real part and the right graphic shows the imaginary part.

Lattice sums

The real part and the imaginary part of the lattice sum in the . The left graphic shows the real part and the right graphic shows the imaginary part. The steep vertical walls are caused by branch cuts of the individual summands.

Fractals and Julia sets

Mandelbrot set of the map over the . The coloring indicates the number of iterations before .

Julia sets of the map over the .