Posted online: 2024-03-18 19:41:03Z by SciLag Admin
Cite as: G-240318.1
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In April 2023, a workshop titled "Exploring Hénon Maps: Real, Complex, and Beyond" took place at BIRS, Banff, with the goal of uniting global specialists focusing on various aspects of Hénon maps. See [1] for a detail description and all problems announced during the meeting.
The Hénon map, also referred to as the Hénon–Pomeau attractor/map, is a discrete-time dynamical system that is extensively studied due to its chaotic behavior. The map transforms a point $(x_n, y_n)$ in the plane to a new point defined by the equations:
$$ \begin{cases} x_{n+1}=1-ax_{n}^{2}+y_{n}\\ y_{n+1}=bx_{n} \end{cases} $$
The behavior of the map depends on two parameters, $a$ and $b$, with classical values of $a = 1.4$ and $b = 0.3$ resulting in chaotic dynamics. However, for different parameter values, the map may exhibit chaotic, intermittent, or periodic behavior. The Hénon map was initially introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, initial points in the plane tend to either approach a set of points known as the Hénon strange attractor or diverge to infinity. This attractor is a fractal, exhibiting smoothness in one direction and a Cantor set structure in another. Numerical estimates suggest a correlation dimension of $1.21 \pm 0.01$ or $1.25 \pm 0.02$ (depending on the dimension of the embedding space) and a Box Counting dimension of $1.261 \pm 0.003$ for the attractor of the classical map.
This group of problem contains several problems from this event, that appeared in [1].
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