Chapter 12.5, Problem 1E

Interpretation Introduction

Interpretation:

Sketch the basins for the weakly damped double-well oscillator for the unforced case, $\text{F = 0}$. Explain how the shape of the basins depends on the size of the damping parameter and what happens when the damping coefficient tends to become zero. Derive the conclusion from these results for predictability of the unforced system.

Concept Introduction:

• The expression for the Forced Double-Well oscillator is,

  ${\ddot{\text{x}}\text{ + δ}\dot{\text{x}}\text{ - βx + αx}}^{\text{3}}\text{ = Fcos}\left(\text{ωt}\right)$

  Here the parameters shows,

  $\text{δ}$ is the damping coefficient,

  $\text{ω}$ is the driving frequency,

  $\text{F}$ is the amplitude of the driving force,

  $\text{α}$ and $\text{β}$ are the positive constant which is used to determine the potential barrier separating the wells.