What Are the Key Characteristics of Henon Attractors?

[MathAmateur]

1. Homework Statement

I am studying Henon Attractors. The Henon map is recursively defined as follows:

$$x_{t+1} = a - x^2_{t} + by_{t}<br /> y_{t+1} = x_{t}$$

I am supposed to find the fixed point (may be unstable) that is contained with the chaotic behavior

The Attempt at a Solution

It is clear that to find the fixed point would be when $$x_{t+1}, x_{t}, y_{t}$$
are all equal (Lets call them all $$x_{b}$$). It seemed obvious to just plug in
$$x_{b}$$ and solve the quadratic, but the book had an extra term in the answer and I do not know where it came from:
$$-x^2_{b} + (b-1) x_{b} +a = 0$$. Where did the -1 in the b-1 term come from?

 

[praharmitra]

If you just plug in x_b for x_t, y_t, and x_{t+1}, then the equation you get is exactly what the book says. Do your math again.

 

[MathAmateur]

Oh, yes, the x_{b} on the left side is subtracted from the right to make it equal to 0. Duh!

Could you now how I insert a new line in the Latex equation? I can't seem to keep those two first equations from running together.

 

[praharmitra]

Oh, yes, the x_{b} on the left side is subtracted from the right to make it equal to 0. Duh!

Could you now how I insert a new line in the Latex equation? I can't seem to keep those two first equations from running together.

The latex symbol for new line is \\

 

[MathAmateur]

I did the following and got the following result:

x_{t+1} = a - x^2_{t} + by_{t} \\
y_{t+1} = x_{t}

$$x_{t+1} = a - x^2_{t} + by_{t}\\<br /> y_{t+1} = x_{t}$$