Chapter 12.1, Problem 9E

Interpretation Introduction

Interpretation:

For standard map equations ${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}{\text{+ y}}_{\text{n+1}}{\text{, y}}_{\text{n+1}}{\text{= y}}_{\text{n}}{\text{+ k sin(x}}_{\text{n}})$:

1. To show that the map is an area preserving all values of k.

2. To plot the various orbits for $\text{k = 0}$.

3. To plot the phase portrait for $\text{k = 0}\text{.5}$ using computer.

4. To show that the phase portrait contains both islands and chaos for $\text{k =1}$.

5. To show that in the chaotic sea, all the islands are almost engulfed for $\text{k = 2}$.

Concept Introduction:

• The standard map equations are ${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}{\text{+ y}}_{\text{n+1}}{\text{, y}}_{\text{n+1}}{\text{= y}}_{\text{n}}{\text{+ k sin(x}}_{\text{n}})$, which are also written as:

  ${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}{\text{+ y}}_{\text{n}}{\text{+ k sin(x}}_{\text{n}}){\text{, y}}_{\text{n+1}}{\text{= y}}_{\text{n}}{\text{+ k sin(x}}_{\text{n}})$

• 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.