Chapter 6.3, Problem 6E

Interpretation Introduction

Interpretation:

Find the fixed points and classify them, sketch the neighboring trajectories

Concept Introduction:

The parametric curves traced by solutions of a differential equation are known as trajectories.

The geometrical representation of collection of trajectories in a phase plane is called as phase portrait.

The point which satisfies the condition ${\text{f(x}}^{\text{*}}\text{)=0}$ is known as fixed point, it correspond to steady state of the system (or equilibria of the system).

Closed Orbit corresponds to periodic solution of the system i.e. $\text{x(t+T)= x(t)}$ for all t.

If nearby trajectories moving away from the fixed point then the point is said to be saddle point.

If the trajectories swirling around the fixed point then it is a unstable fixed point.

If nearby trajectories moving away from the fixed point then the point is said to be unstable fixed point.

If nearby trajectories moving towards the fixed point then the point is said to be stable fixed point.

To check the stability of fixed point use Jacobian matrix

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

The point $(x^{*},y^{*})$ is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point having negative real parts and point is said to be unstable if one of its eigenvalue has positive real part. If the both eigenvalues are purely real then the fixed point is saddle point.