Chapter 9.2, Problem 2E

Interpretation Introduction

Interpretation:

To show that there is a certain ellipsoidal region E of the form ${\text{rx}}^{\text{2}}{\text{ + σy}}^{\text{2}}\text{ + σ}{\left(\text{z - 2r}\right)}^{\text{2}}\le \text{C}$ ${\text{rx}}^{\text{2}}{\text{ + σy}}^{\text{2}}\text{ + σ}{\left(\text{z - 2r}\right)}^{\text{2}}\le \text{C}$ such that all trajectories of the Lorenz equations eventually enter E and stay in there forever. To obtain the smallest possible value of $\text{C}$.

Concept Introduction:

• ➢ The Lorenz equations are given as

  $\dot{\text{x}}=\text{σ}\left(\text{y - x}\right)$,

  $\dot{\text{y}}=\text{rx - xz - y}$,

  $\dot{\text{z}}=\text{xy - bz}$

  Here, $\text{σ, r, b > 0}$.