Chapter 9.4, Problem 2E

Interpretation Introduction

Interpretation:

• a) To state why it is called as the “tent map”.

• b) To find all fixed points and classify their stability.

• c) To show map has period $\text{-2}$ orbit and classify the stability of the orbit.

• d) To find period $\text{-3}$ and period $\text{-4}$ points and classify their stability.

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

• ➢ Strange attractor: It is not same as a fixed point, limit cycle, a point, a curve or surface. It is a fractal with fractional dimension between $\text{2 and 3}$.

• ➢ Chaos: It is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

• ➢ The plot of function ${\text{z}}_{\text{n}+1}=\text{f}\left({\text{z}}_{\text{n}}\right)$ is called a Lorenz map. It gives information about dynamics on the attractor.

• ➢ If $\left|f^{\prime }\left(z\right)\right|>1$ then the limit cycle exists and it is unstable.