Chapter 8.7, Problem 11E

Interpretation Introduction

Interpretation:

Using reversibility argument the in-phase periodic state of ${\dot{\varphi }}_i=\Omega +a\sin {\varphi }_i+\frac{1}{N}{\displaystyle \sum _{j=1}^N\sin {\varphi }_j}$ is not attracting, even if the nonlinear terms are kept attracting is to be proved.

Concept Introduction:

A nonlinear system of the form $\dot{\text{x}}\text{ = F}\left(\text{x}\right)$ is reversible if it is invariant under the transformation $\text{t}\to \text{-t and x}\to \text{R(x)}$ where ${\text{R}}^2\text{(x) = x}$

According to the theorem $\mathrm{6.6.1}$ if the system is reversible, then the origin ${\text{x}}^{\text{*}}\text{ = 0}$ is a nonlinear center.