(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the most general two-dimensional quadratic map with a constant Jacobian is the Henon map:

x_{n+1}=y_{n}+1-ax^{2}_{n}

y_{n+1}=bx_{n},

where a,b are positive constants.

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2. Relevant equations

From the general quadratic map,

x_{n+1}=f_{1}+a_{1}x_{n}+b_{1}y_{n}+c_{1}x^{2}_{n}+d_{1}x_{n}y_{n}+e_{1}y^{2}_{n}

y_{n+1}=f_{2}+a_{2}x_{n}+b_{2}y_{n}+c_{2}x^{2}_{n}+d_{2}x_{n}y_{n}+e_{2}y^{2}_{n}

Linear transformations (deformations, shifts, and rotations) can be used to transform this into the Henon map, or an equivalently simple canonical form - that is, when the Jacobian is required to be constant.

3. The attempt at a solution

Requiring the Jacobian determinant of the most general quadratic map to be constant, we get five equations that relate the 12 coefficients - the coefficients of the five different terms {x,y,x^{2}, xy, y^{2}} in the determinant must be zero so that the determinant is always constant. The five equations are essentially:

a_{1}d_{2}+2c_{1}b_{2}=2b_{1}c_{2}+d_{1}a_{2}

2a_{1}e_{2}+d_{1}b_{2}=b_{1}d_{2}=b_{1}d_{2}+2e_{1}a_{2}

c_{1}e_{2}=e_{1}c_{2}

d_{1}c_{2}=c_{1}d_{2}

e_{1}d_{2}=d_{1}e_{2}

Apparently, using shifts, linear transformations, and rotations, together with these relations, one can reduce the general quadratic map to the the Henon map. But brute forcing the transformation in all its generality leaves me with a large mess of intractable algebra. Does anyone have any ideas on how to do this problem, i.e. how to look for sensible transformations that actually work? Homework's due at the end of Friday and I've been stuck on this for awhile, so I'd really appreciate any advice.

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# How to show the Henon map is the most general quad. map with a constant Jacobian?

Can you offer guidance or do you also need help?

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