Chapter 8.7, Problem 12E

Interpretation Introduction

Interpretation:

To show that all the characteristic multipliers equal $+1$, by calculating the linearized Poincare map, for the system of N identical oscillators, ${\dot{\text{θ}}}_{\text{i}}{\text{=f(θ}}_{\text{i}}\text{)+}\frac{\text{K}}{\text{N}}{\displaystyle \sum _{\text{j=1}}^{\text{N}}{\text{f(θ}}_{\text{j}}\text{)}},\text{for i=1,}\mathrm{......}\text{, N}$, where $\text{K>0}$, $\text{f(θ)>0}$, and $\text{f(θ)}$ is smooth and $\text{2π}$- periodic for all $\text{θ}$ so that the in-phase solution is periodic.

Concept Introduction:

• ➢ If a system of the form ${\text{x}}^{\text{*}}\text{=f(x)}$ has an infinitesimal perturbation such that ${\text{x}}^{\text{*}}{\text{+v}}_{\text{0}}$ is in surface of section $\text{S}$, after the first return to $\text{S}$,

  ${\text{x}}^{\text{*}}{\text{+v}}_{\text{1}}{\text{=P(x}}^{\text{*}}{\text{+v}}_{\text{0}}{\text{)=P(x}}^{\text{*}}{\text{)[DP(x}}^{\text{*}}{\text{)]v}}_{\text{0}}{\text{+O(||v}}_{\text{0}}{\text{||}}^{\text{2}}\text{)}$

  where an $\text{(n-1)×(n-1)}$ matrix ${\text{[DP(x}}^{\text{*}}\text{)]}$ is known as a linearized Poincare map.

• ➢ The in-phase solution is given by ${\text{θ}}_1{\text{(t)=θ}}_2\text{(t)=}\mathrm{......}{\text{θ}}_N{\text{(t)=θ}}^{*}\text{(t)}$, where ${\text{θ}}^{*}\text{(t)}$ is a common waveform.

• ➢ If all perturbation remains the same after one cycle, then the corresponding system has all characteristic multipliers equal $+1$.