Chapter 6.3, Problem 10E

Interpretation Introduction

Interpretation:

To show by linearization the origin is non-isolated fixed point but in fact origin is an isolated fixed point. Classify the stability of the origin and sketch the vector field and nullclines on phase portrait. Generate a phase portrait using computer.

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 keep swirling around the fixed point, then it is a unstable fixed point.

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

If nearby trajectories are 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.

Isolated Fixed Point: If there is no any other fixed point exists in a region closed to interested fixed point, then it is called Isolated Fixed point.

Non-isolated Fixed point: If there is fixed points in a region close to interested fixed point, then it is called Non-isolated fixed point.