EBK NONLINEAR DYNAMICS AND CHAOS WITH S
EBK NONLINEAR DYNAMICS AND CHAOS WITH S
2nd Edition
ISBN: 9780429680151
Author: STROGATZ
Publisher: VST
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Chapter 7.3, Problem 9E
Interpretation Introduction

Interpretation:

  • a) Approximate the shape of the orbit r(θ) for μ1 by assuming a power series solution

    r(θ) = 1+μr1(θ)+O(μ2); substitute this series solution in a differential equation dr. By neglecting the higher order terms, derive the simple differential equation r1(θ) and solve this differential equation explicitly for r1(θ).

  • b) To find the maximum and minimum value of r on approximate orbit,- show it lies in the annulus 1-μ< r <1+μ

  • c) By using a computer, calculate r(θ) numerically for small values of μ, and plot the result on the same graph of analytical approximation for r(θ).

Concept Introduction:

  • ➢ Poincare-Bendixson Theorem: If

    • 1) R is closed, subset of the plane is bounded,

    • 2) x˙=f(x) is continuously differentiable vector field on an open set containing R

    • 3) R doesn’t have any fixed points

    There exists a trajectory C. The trajectory C is confined in R, in the sense that it starts in R and stays in R for all future time. Either C is closed orbit, or it spirals towards a closed orbit as t. In any case, R contains a closed orbit

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