Chapter 6.3, Problem 1E

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 a collection of trajectories in a phase plane is called a phase portrait.

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

Closed Orbit corresponds to the 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 an unstable fixed point.

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

To check the stability of fixed-point, use the 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 eigenvalues has a positive real part. If both eigenvalues are purely real, then the fixed point is saddle point.