Chapter 9.2, Problem 5E

Interpretation Introduction

Interpretation:

To give a partial sketch of the stability diagram, using the analytical results obtained about bifurcations in the Lorenz equations. To plot the pitchfork and Hopf bifurcation curves in the $\left(\text{σ, r}\right)$ parameter plane, by assuming $\text{b = 1}$, and $\text{σ, r}\ge \text{0}$.

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, $\text{σ, r, b > 0}$.

• ➢ The Jacobian matrix is given by

  $\text{A = }\left(\begin{array}{ccc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}& \frac{\partial \dot{\text{x}}}{\partial \text{z}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}& \frac{\partial \dot{\text{y}}}{\partial \text{z}}\\ \frac{\partial \dot{\text{z}}}{\partial \text{x}}& \frac{\partial \dot{\text{z}}}{\partial \text{y}}& \frac{\partial \dot{\text{z}}}{\partial \text{z}}\end{array}\right)$

• ➢ The Eigen value $\text{λ}$ can be calculated using the characteristic equation

  $\left|\left(\text{A - λI}\right)\right|\text{ = 0}$