Chapter 9.6, Problem 3E

Interpretation Introduction

Interpretation:

• a) To plot ${\text{y(t) and y}}_{\text{r}}\text{(t)}$ on the same graph and state whether two time series are merging or not even if both are chaotic.

• b) To plot (y, z) projection of both trajectories.

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 main idea behind sending the secret message using chaos is while transmitting the message, it is masked by a louder chaos. The eavesdropping person only hears chaos, but the receiver perfectly reproduces chaos and subtracts it from the masked message and hears the original message sent by the transmitter.

• ➢ The response equations are:-

  $\begin{array}{l}\text{x(t) = x(t)}\\ {\dot{\text{y}}}_{\text{r}}\text{= rx(t)}-{\text{y}}_r-{\text{x(t)z}}_{\text{r}}\\ {\dot{\text{z}}}_{\text{r}}{\text{ = x(t)y}}_{\text{r}}-{\text{bz}}_{\text{r}}\end{array}$