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In his famous paper A two-dimensional mapping with strange attractor, Hénon gives an explicit trapping region for the Hénon map with $a=1.3$ and $b=0.4$.

In the figure below, the quadrilateral $Q=ABCD$ is mapped to the curvilinear quadrilateral $A_1B_1C_1D_1$ which is totally contained in $Q$. Therefore, all orbits of the Hénon map starting in $Q$ remain in $Q$. It is in this sense that $Q$ is a trapping region for the map. Hénon gives explicit coordinates for the points $A,B,C,D$.

How did Hénon find the quadrilateral $Q$ explicitly?

How does one find an explicit trapping region for other values of the map parameters $a$ and $b$?

Are there any other large quadrilaterals or triangles that are trapping regions for the Hénon map?

enter image description here

(Picture from the book "Chaotic evolution and strange attractors" by Ruelle)

lhf
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  • Related to https://math.stackexchange.com/questions/2319302/what-is-a-trapping-region and https://math.stackexchange.com/questions/2962375/strategy-for-finding-trapping-region-of-a-discrete-dynamical-system-hénon-map – lhf Mar 15 '21 at 21:37
  • In the paper Characteristic exponents and strange attractors, Feit seems to describe a more complicated trapping region in the general case – lhf Mar 15 '21 at 21:44
  • There are other trapping regions. When you numerically compute the basin of attraction of the attractor and the attractor itself you can get very good guesses for trapping regions. – SillyMathematician Mar 16 '21 at 13:26
  • @SillyMathematician, any references for explicit trapping regions? Thanks. – lhf Mar 16 '21 at 18:48
  • @lhf Can any one help me : The group of bijection $G$ of the set $X={1, 2,3,..,n}$ where $n\ge 3$ acts on $S$ via $i\to f(i)$ . Then find the number of orbits of the natural action of $G$ on $X×X×X$ defined by $f•(i, j, k) =(f(I), f(j), f(k)) $ – SoG Mar 29 '23 at 03:56

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