Cycles of the Hénon map

Introduction

The Hénon map is a simple two-dimensional extension of the logistic map. It is defined as:
xk+1 = 1 + yk − a xk2,
yk+1 = b xk.              
We however shall transform the variables as xk → xk / a. So the new map is
xk+1 = a + yk − xk2,
yk+1 = b xk,            
which is reduced to the logistic map when b = 0.

Cycle polynomials

Since the map has two parameters, the onset (bifurcation) points, correspond to a polynomial equation of a and b, instead of a single value. Below we list the polynomials of A = 4 a and b at the onset and bifurcation points of the n-cycles below.
nOnsetBifurcation
1 eq. eq.
2 eq. eq.
3 eq. eq.
4 eq. eq.
5 eq. eq.
6 eq. eq.
7 eq. eq.
8 eq. eq.

Programs

Program Description
hen.ma A self-contained Mathematica script that computes the onset and bifurcation polynomials of the parameters a and b for the n-cycles.
For the onset polynomial of the 6-cycles, type
math < hen.ma 6 a
The output would be ab6a.txt.
For the bifurcation polynomial of the 5-cycles, type
math < hen.ma 5 b
The output would be ab5b.txt.
mkinterp.py A Python script that generates a Mathematica script, which computes the polynomial from interpolating the list of values. This is used in parallel runs, see here for details.
Click here to see how to use the Mathematica scripts .ma files.

References

Primary reference

General information

Hénon map

Last updated on April 2nd, 2014.