Chapter 12.4, Problem 4E

Interpretation Introduction

Interpretation:

Numerically integrate the Lorenz equations for standard parameters $\text{r = 28, b =}\frac{\text{8}}{\text{3}}\text{, σ =10}$ and obtain the long time series for $\text{x(t)}$ and then by using attractor reconstruction method for various values of the delay and plot $\text{(x(t), x(t + τ))}$ also find the value of $\text{τ}$ for which the reconstructed attractor is similar to actual Lorenz attractor.

Concept Introduction:

• The Lorenz equations are

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

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

• Rössler system is

  $\begin{array}{l}\dot{\text{x}}=-\text{y}-\text{z}\\ \dot{\text{y}}=\text{x}+\text{ay}\\ \dot{\text{z}}=\text{b}+\text{z(x}-\text{c})\end{array}$

• In attractor reconstruction method the time delay $\text{τ}$ is introduced, and by observing graphs, for which value of $\text{τ}$ the phase space look similar to actual Lorenz attractor.