Recurrence Plot of Mathematical Functions and Constants

size s

tolerance ϵ

value/function

π

view

point

density

mesh

A recurrence plot illustrates the recurrence of states in a phase space where all the possible states of a system can be seen. Recurrence plots can be used to view and study mathematical functions such as sine and sinc or constants like

π

,

e,

and so on. In the case of a function

f

, the values used are the finite sequence

σ={f(-s/2),f(s/2+1),…,f(s/2-1),f(s/2)}

, where

s

is the size. In the case of a number, the values used are the digits of its decimal expansion taken to

s

places.

The expression plotted is

R(x,y)=H(ϵ-σ(x)-σ(y))

, where

H

is the Heaviside step function,

σ

is the sequence, and

ϵ

is a kind of tolerance.

The point view is a graphical representation of the matrix

R

, which is binary because of the unit step function. In the density view, the points are grouped in clusters to give a smoother representation of the

R

matrix, and the matrix rows are rotated (vertical shift).

The mesh draws lines that highlight the white spaces for the point view and gives reference rulers for the density view.