Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system $x\text{'}=f\left(x\right)$ with critical points at $A\left(-1,0\right),B\left(0,0\right),C\left(0,1\right)$ are shown in the figure below:

(b)

To determine

$\underset{t\to \infty }{\lim }\varphi \left(t\right)$ for each of the following set of initial conditions

i) $x_0={\left(0,0.1\right)}^T$

ii) $x_0={\left(0,-0.1\right)}^T$

iii) $x_0={\left(2,-1.5\right)}^T$

iv) $x_0={\left(-2,1.5\right)}^T$

Where the solution of the initial value problem $x\text{'}=f\left(x\right)$, $x\left(0\right)=x_0$ is denoted by $\varphi \left(t\right)$

with critical points at $A\left(-1,0\right),B\left(0,0\right),C\left(0,1\right)$ and the direction field and the phase portrait are shown in the figure below: