Chapter 12.1, Problem 8E

Interpretation Introduction

Interpretation:

For the given map equations

$\begin{array}{l}{\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}\text{cos}\alpha -\left(y_n-x_n^2\right)\text{sin}\alpha \\ {\text{y}}_{\text{n+1}}{\text{= x}}_{\text{n}}\text{sin}\alpha \text{+ }\left(y_n-x_n^2\right)\text{cos}\alpha \end{array}$

1. To verify the map is area preserving

2. To find inverse mapping.

3. To explore the map using the computer for various values of $\alpha$

Concept Introduction:

• ➢ Jacobian matrix is used for checking the area behavior of the system

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

  If the determinant of the Jacobian matrix is 1, then the area is preserved; if the determinant is less than 1, then the area is contracted; if the determinant is greater than 1, then the area expands.

• ➢ Hénon map is given by

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

  ${\text{y}}_{\text{n + 1}}{\text{ = bx}}_{\text{n}}$

  where $\text{a,b}$ are adjustable parameters.

• ➢ Properties of Hénon map:

  1. It is invertible.

  2. It is dissipative.

  3. It has a trapping region for certain parameter values.

  4. Some orbits of the Hénon map escape to infinity.