Chapter 12.5, Problem 2E

Interpretation Introduction

Interpretation:

Consider the double-well oscillator with parameters $\text{δ = 0}\text{.15}$, $\text{F = 0}\text{.3}$ and $\text{ω = 1}$.

$\begin{array}{l}\dot{\text{x}}\text{ = y}\\ {\dot{\text{y}}}^{\text{ = -δy+x-x}}\text{+Fcos}\left(\text{ωt}\right)\end{array}$

To show that the system has at least two coexisting attractors: a large limit cycle and a smaller attractor. Also, plot both in a Poincare section.

Concept Introduction:

• According to the Poincare $\text{-}$ Bendix son Theorem,

  1. There is a fixed point.

  2. There is a periodic orbit.

  3. There is anassociated set composed of a finite number of fixed points with homo clinic and hetero clinic orbits connecting together.

• The expression for the Forced Double $\text{-}$ Well oscillator is,

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

  Here, $\text{δ}$ is the damping coefficient, $\text{ω}$ is the driving frequency, F is the amplitude of the driving force, and $\text{α}$ and $\text{β}$ are the positive constant, which is used to determine the potential barrier separating the wells.

• Attractor: An attractor is the smallest unit at which the point cannot decompose further into any more attractors with distinct basins of attraction.