Chapter 6.6, Problem 8E

Interpretation Introduction

Interpretation:

To show that the given system is reversible, to verify that the system has three fixed points on the cylinder for $\frac{\text{9}}{\text{8}\sqrt{\text{2}}}\text{ > β > }\frac{\text{1}}{\sqrt{\text{2}}}$, one of which is saddle and that saddle is connected to itself by homoclinic orbit.

To show there is a band of closed orbits sandwiched between the circle $\text{x = 0}$.

To sketch the phase portrait of the system on the cylinder.

To show that as $\text{β }\to \frac{\text{1}}{\sqrt{\text{2}}}$ saddle moves toward $\text{x = 0}$ and all close orbits disappears as $\text{β =}\frac{\text{1}}{\sqrt{\text{2}}}$

To show there are two saddles for $\text{0 < β <}\frac{\text{1}}{\sqrt{\text{2}}}$ on the edge $\text{x = 0}$ and to draw the phase portrait of the system on the cylinder.

Concept Introduction:

A fixed point of a differential equation is a point where, $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$

A system for which dynamics don’t change even if signs of coordinates are interchanged is called reversible system.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.