1.3 STABILITY OF EQUILIBRIUM POINTS

In this section you will learn that the phase plane diagram of a system of equations can be deduced from the eigenvalues of the associated matrix. You will learn

- That a simple system is unstable if one of the eigenvalues are real and positive or the eigenvalues are imaginary and have positive real part, otherwise the equilibrium point is stable.

- To classify the various types of equilibrium points

- To classify the equilibrium points using only the trace, determinant and discriminant of the matrix.

1.3.1 Systems with matrices having real eigenvalues

Investigation 1

Two real eigenvalues of the same sign.

You are going to investigate the phase portrait for systems with two positive and two negative eigenvalues. Click on investigation 1 to open the Maple window.

Investigation 2

Two real eigenvalues of opposite sign

You are going to investigate the phase portrait for a system with two real eigenvalues of opposite signs. Click on investigation 2 to open the Maple window..

Investigation 3

Two coincident real eigenvalues

You are going to plot phase portraits of systems with matrices having two coincident eigenvalues. You will consider both diagonal and non diagonal matrices. Click on investigation 3 to open the maple window.

1.3.2 Systems with matrices with complex eigenvalues.

Investigation 4

You are going to draw the phase portraits for systems with complex eigenvalues with real parts positive or negative. Click on investigation 4 to open the Maple window.

1.3.3 Systems with imaginary eigenvalues

Investigation 5

The eigenvalues will be purely imaginary if the real part is zero.You are going to draw the phase portrait for such a system. Click on investigation 5 to open the maple window.

1.3.4 Trace-Determinant diagram

Since the nature of the equilibrium point depends on the eigenvalues which in turn depend on

it follows that the nature of the equilibrium point can be determined from these three values.

The tr- det diagram divides the plane into regions corresponding to different types of equilibrium point.

- Worked example 1

Click on worked example 1 to open the maple window for this work sheet.