Chapter 12.2, Problem 6E

Interpretation Introduction

Interpretation:

To determine the stability of the fixed points of the Hénon map as the functions of $\text{a}$ and $\text{b}$.

To show that one fixed point is always unstable, while the other is stable for $\text{a}$, which is slightly larger than ${\text{a}}_{\text{0}}$.

To show that this fixed point loses its stability in a flip bifurcation $\left(\text{λ = -1}\right)$ at ${\text{a}}_{\text{1}}\text{ = }\frac{\text{3}}{\text{4}}{\text{ (1 - b)}}^{\text{2}}$.

Concept Introduction:

• The Hénon map is given as:

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

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

• The roots of the quadratic equation are given as:

  $\text{x = }\frac{\text{- b ± }\sqrt{{\text{b}}^{\text{2}}\text{ - 4ac}}}{\text{2a}}$

• The Jacobian matrix is given as:

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

• The Eigen value $\text{λ}$ of the Jacobian matrix can be calculated as:

  $\left|\text{J - λI}\right|\text{ = 0}$