Chapter 2.2, Problem 1E

Interpretation Introduction

Interpretation:

To analyze the nonlinear system graphically and to sketch the vector field graphically on the real line. All fixed points are to be determined; their stability should be classified, and graph of $\text{ x(t)}$ is to be sketched for different initial conditions. Also, if possible, the analytic solution for $\text{ x(t)}$ is to be obtained.

Concept Introduction:

The given equation is ${\dot{\text{x}}\text{=4x}}^{\text{2}}\text{-16}$. Initially, we want to find the fixed point of the given equation. Fixed point of the given equation are values of x at which $\dot{\text{x}}\text{=0}$.

The point at which the velocity is zero can be obtained by graphing the function $\dot{\text{x}}$ vs $\text{x}$(position).

Stable points are points at which the local flow is toward them. They represent stable equilibria at which small disturbances damp out in time away from it.

Unstable points are points at which the local flow is away from them. They represent unstable equilibria.