Chapter 12.2, Problem 2E

Interpretation Introduction

Interpretation:

To show transformations $\text{T' and T"}$ are area-preserving and $\text{T'}$ is not.

Concept Introduction:

• Henon map is given by

  ${\text{x}}_{\text{n+1}}{\text{= y}}_{\text{n}}+1-{\text{ax}}_{\text{n}}^{\text{2}},{\text{ y}}_{\text{n+1}}{\text{= bx}}_{\text{n}}$

  Here, a and b are adjustable parameters.

• The first transformation is

  $\text{T': x' = x, y' =1+y}-{\text{ax}}^{\text{2}}$

  The second transformation is

  $\text{T'': x'' = bx', y'' = y'}$

  The third transformation is

  $\text{T''': x''' = y'', y''' = x''}$

  The composite transformation $\text{T = T''' T'' T'}$ gives the Henon mapping using notation ${\text{(x}}_{\text{n}}{\text{,y}}_{\text{n}}\text{) for (x,y)}$ and ${\text{(x}}_{\text{n+1}}{\text{,y}}_{\text{n+1}}\text{) for (x''',y''')}$

• The properties of Henon map are

  1. It is invertible

  2. It is dissipative

  3. For certain parameter values, it has a trapping region.

  4. Some trajectories of it may escape to infinity.

• The Jacobian matrix is given as

  $J=\left(\begin{array}{cc}\frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\\ \frac{\partial g}{\partial x}& \frac{\partial g}{\partial y}\end{array}\right)$

  If the determinant of the Jacobian matrix is $1$, then the transformation is said to be area preserving otherwise not.