Chapter 6.4, Problem 10E

Interpretation Introduction

Interpretation:

To show that, for $\text{n=2}$ the system reduced to ${\dot{\text{x}}}_{\text{1}}{\text{ = x}}_{\text{1}}\left({\text{x}}_2{\text{- 2x}}_{\text{2}}{\text{x}}_{\text{1}}\right)$, ${\dot{\text{x}}}_{\text{2}}{\text{ = x}}_{\text{2}}\left({\text{x}}_{\text{1}}{\text{- 2x}}_{\text{1}}{\text{x}}_2\right)$. Find the all fixed points. Prove that $\text{u(t)}\to \text{1 as t }\to \infty$ and $\text{v(t)}\to 0\text{ as t }\to \infty$, also prove that $\left(\text{x(t),y(t)}\right)\to \left(\frac{1}{2},\frac{1}{2}\right)$ Draw the phase portrait for this system.

Concept Introduction:

Fixed points of the system are the points where $\dot{\text{x}}\text{ = 0}$, $\dot{\text{y}}\text{ = 0}\text{.}$

$\text{x -}$ nullclines are a set of points in the phase plane where $\frac{\text{dx}}{\text{dt}}\text{ = 0}$. At these points where the vectors are either upward or downward.

$\text{y -}$ nullclines is a set of points where $\frac{\text{dy}}{\text{dt}}\text{ = 0}$. These are the points where vectors are in horizontal direction.