Chapter 12.2, Problem 15E

Interpretation Introduction

Interpretation:

To show the Lozi map contracts areas if $-1<\text{b}<1$.

Concept Introduction:

• The expressions for the Lozi map are:-

  ${\text{x}}_{\text{n+1}}{\text{=1+y}}_{\text{n}}-\text{a}\left|{\text{x}}_{\text{n}}\right|,{\text{ y}}_{\text{n+1}}{\text{= bx}}_{\text{n}}$

  Here, a and b are real parameters: $-1<b<1$.

• The 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 $\text{1}$, then the area is preserved; if the determinant is less than $\text{1}$, then the area is contracted. If the determinant is greater than $\text{1}$, then the area expands.