Chapter 10.2, Problem 8E

Interpretation Introduction

Interpretation:

Plot orbit diagram for map equation ${\text{x}}_{\text{n+1}}{\text{= r x}}_{\text{n}}{\text{-x}}_{\text{n}}^{\text{3}}$. Also, try different initial conditions

Concept Introduction:

• ➢ Standard Periodic-doubling: It is a discreet dynamic system equation in which a small change in a parameter value in the system equation changes to the new behavior with double the period of the original system. The appearance of numerous period-doubling cascades is among the most prominent features of parameterized maps, that is, smooth one-parameter families of maps $\text{F: R ×M}\to \text{M}$ Here, $\text{M}$ is a smooth locally compact manifold without boundary, typically ${\text{R}}^{\text{N}}$, each cascade has infinitely many period-doubling bifurcations, and it is typical to observe that, whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic $\text{F}$.

• ➢ Logistic map: A logistic map is defined as the polynomial map of the degree two and chaotic behavior can arise from a non-linear dynamic equation.