Chapter 10.4, Problem 8E

Interpretation Introduction

Interpretation:

To solve the Lorenz equation numerically for $\text{σ = 10, β = }\frac{\text{8}}{\text{3}}$, and r near to $\text{166}$.

1. To show if $\text{r = 166}$ all trajectories are attracted to a stable limit cycle and plot both the xz projection of the cycle, and the time series $\text{x(t)}$.

2. To show that if $\text{r = 166}\text{.2}$ the trajectory looks like the old limit cycle for much of the time but occasionally it is interrupted by chaotic bursts.

3. To show that as r increases the burst become more frequent and last longer.

Concept Introduction:

• The three-different system of ordinary differential equations are,

  $\begin{array}{l}\frac{\text{dx}\left(\text{t}\right)}{\text{dt}}\text{ = σ}\left(\text{y-x}\right)\\ \frac{\text{dy}\left(\text{t}\right)}{\text{dt}}\text{ = x}\left(\text{r-z}\right)\text{-y}\\ \frac{\text{dz}\left(\text{t}\right)}{\text{dt}}\text{ = xy-bz}\end{array}$

  Here the variable x, y and z is the system at time t and $\text{σ, ρ, β}$ are the positive parameter of the Lorenz equation.

• The Lorenz system has able to reduce the system into three $\text{-}$ dimension state space and determine the place of first chaotic system. And intermittency is the rate of a signal that rotates randomly between randomly, points and moderately shorts asymmetrical bursts.