Chapter 6.5, Problem 2E

Interpretation Introduction

Interpretation:

To find and classify the equilibrium points of the system ${\ddot{\text{x}}\text{ = x - x}}^{\text{2}}$. To sketch the phase portrait of the given system. To find an equation for the homoclinic orbit that separates closed and nonclosed trajectories.

Concept Introduction:

Fixed points occur where $\dot{\text{x}}\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{A = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$

The Eigenvalue $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$

The trajectories that start and end at the same fixed point are called homoclinic orbits. The equation for the homoclinic orbit that separates closed and nonclosed trajectories can be calculated for $\text{E = 0}$.

The total energy is given by $\text{E = }\frac{\text{1}}{\text{2}}{\text{m}\dot{\text{x}}}^{\text{2}}\text{ + V(x)}$