Chapter 9.5, Problem 5E

Interpretation Introduction

Interpretation:

• a) Find the change in variables as $\epsilon \to 0$.

• b) To find the conserved quantities for the new system.

• c) To show that the new system is volume preserving.

• d) To explain that the Lorenz equations might show conservative features for $\text{r}\to \infty$.

• e) To solve the system obtained in part (a) numerically and state its long term behavior for the large value of r.

Concept Introduction:

• ➢ Lorenz equations:-

  $\begin{array}{l}\dot{\text{x}}=\sigma (\text{y}-\text{x})\\ \dot{\text{y}}=\text{rx}-\text{y}-\text{xz}\\ \dot{\text{z}}=\text{xy}-\text{bz}\\ \text{Here σ, r, b > 0}\end{array}$.

  The solution of the Lorenz equations oscillates irregularly for a wide range of parameters, never exactly repeating, but always remaining in the bounded region of the phase space.

• ➢ The conserved quantity of a dynamical system is defined as the function of the dependent variable whose value remains constant along each trajectory of the system.

• ➢ The Rayleigh number is $\frac{{\text{πgrq}}_{\text{1}}}{{\text{K}}^{\text{2}}\text{v}}$, which is dimensionless. It is a measure of how hard the system is driven relative to dissipation. Here, g is the gravity, ${\text{q}}_{\text{1}}$ is the inflow, K is the leakage, and v is damping. The steady rotation of the waterwheel is possible only if the Rayleigh number is of a large value.