Chaos Theory: Hénon map (Part 2)

Continuing on from our introduction of the Hénon map, we will look at its dependence on initial conditions and take a look at the bifurcation diagram which is created by varying our variable a, such that

$x_{n+1}=y_n+1-ax_n^2, \qquad \qquad y_{n+1}=bx_n$.

Dependence on initial conditions

For the 1-D maps we have explored in the previous chapters, we have shown that they are sensitive to initial conditions, we can show this is true for the Hénon map as well.

Hénon map: Sensitivity to initial conditions

Using the classical Hénon map parameter values and comparing two different initial conditions $(0.1,0.1)$ and $(0.10001,0.10001)$ represented by the coloured lines blue and red respectively then we can observe that the map is sensitive to initial conditions (Figure above). Note that the orbits overlap or are close for the first 26 iterations then appear to become unidentical to one another.

Bifurcation diagram

In this section we’ll be looking at the Hénon map for different values of parameter $a$, with a fixed $b=0.3$. Like the logistic map, there’s a wide range of different behaviours dependent upon our choice of $a$. We can show this in a bifurcation diagram. Note that since the Hénon map is a 2-D map then we have bifurcation diagrams for our $x$ and $y$ values.

Bifurcation diagrams for the classical Hénon map for $0 \leq a \leq 1.5$

The diagrams show that the Hénon map shares the same route to chaos as the Logistic map, i.e. period doubling route to chaos. The first bifurcation occurs approximately at $a=0.36$, the second at $a=0.91$, the bifurcations keep occurring results in orbits of period 4,8,16,…,$\infty$. It can be shown that the rate at which these bifurcations occur converge to the Feigenbaum constant (see previous posts). It’s interesting to observe that despite the diagrams having different y-axis ranges, the diagram is identical. In addition, its interesting to note that the bifurcation diagram has periodic windows, with a large period-7 orbit appearing at $a \approx 1.22$ amongst the chaos, only to go through more period doublings and transition back into chaos.