Chapter 12.2, Problem 9E

Interpretation Introduction

Interpretation:

For the Henon map T, taking standard values of the parameter $\text{a =1}\text{.4, b = 0}\text{.3}$:

1. To plot Q and its image T(Q). Here, Q is the quadrilateral with vertices $\left(-1.33,0.42\right),\left(1.32,0.133\right),\left(1.245,-0.14\right),\left(-1.06,-0.5\right)$

2. To prove that T(Q) is contained in Q.

Concept Introduction:

• The equations for the Henon map are as follows:-

  $\begin{array}{l}{\text{x}}_{\text{n+1}}{\text{= y}}_{\text{n}}+1-{\text{ax}}_{\text{n}}^{\text{2}}\\ {\text{y}}_{\text{n+1}}={\text{bx}}_{\text{n}}\end{array}$

  Here a and b are adjustable parameters.

• 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 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 ${\text{(x}}_{\text{n+1}}{\text{,y}}_{\text{n+1}}\text{) for (x''',y''')}$.