Real Eigenvalues

Consider the linear homogeneous system

displaymath92

In order to find the eigenvalues, consider the characteristic polynomial

displaymath94.

In this section we will consider the case of the quadratic equation above when it has two distinct real roots (that is, if tex2html_wrap_inline96 ). The roots (eigenvalues) are

displaymath98,

and

displaymath100.

Here we know that the differential system has two linearly independent straight-line solutions

displaymath102,

where tex2html_wrap_inline104 (respectively tex2html_wrap_inline106 ) is an eigenvector associated to the eigenvalue tex2html_wrap_inline108 (respectively tex2html_wrap_inline110 ). We also know that the general solution (which describes all of the solutions) to the system has the form

displaymath112.

Keep in mind that tex2html_wrap_inline104 and tex2html_wrap_inline106 are two constant vectors.

Let us discuss the behavior of the solutions when tex2html_wrap_inline118 (meaning the future) and when tex2html_wrap_inline120 (meaning the past). Since the eigenvalues are distinct, one is bigger than the other one. Assume that we have

displaymath122.

It is easy to see that we have

displaymath124

Behavior when tex2html_wrap_inline118

In this case we will consider the equation

displaymath128.

Since

displaymath130,

(because tex2html_wrap_inline132 ) then it is clear that when tex2html_wrap_inline118 , we have

displaymath136.

Behavior when tex2html_wrap_inline118

In this case we will consider the equation

displaymath142

Since

displaymath144

(because tex2html_wrap_inline146 ) then it is clear that when tex2html_wrap_inline120 , we have

displaymath150

Remark:Since the two eigenvalues are real numbers, we have three cases to consider depending on their signs:

Case 1: Both are positive

displaymath158.

In this case we have

displaymath160,

meaning that the solutions emanate from the origin (if you go to the past, you will die at the origin). When tex2html_wrap_inline118 , Y(t) explodes.

In this case the origin plays the role of a source. Clearly, the origin is the only equilibrium point.

Case 2: Both eigenvalues are negative

displaymath168.

In this case we have

displaymath170,

meaning that in the future the solutions die at the origin. When tex2html_wrap_inline120 , Y(t) explodes.

In this case, the origin plays the role of a sink. Clearly, the origin is the only equilibrium point.

Case 3: The eigenvalues have different signs

displaymath178.

In this case, the origin behaves like a saddle.

Remark: It is clear from the above discussions that one may decide about the signs of the eigenvalues just by looking at some solutions on the phase plane (depending whether we have a saddle, a sink or a source).

Example: Consider the three phase planes and decide about the sign-distribution of the associated eigenvalues.

Phase Plane I
Phase Plane II
Phase Plane III

Answer:

For the phase-plane I, the origin is a source. Therefore, the two eigenvalues are both positive.
For the phase-plane II, the origin is a saddle. Hence, the two eigenvalues are opposite signs.
For the phase-plane III, the origin is a sink. Hence, the two eigenvalues are negative.

Example: Consider the harmonic oscillator equation

displaymath180.

Discuss the behavior of the spring-mass.

Answer: First, translate this equation to the system

displaymath182,

where

displaymath184

The characteristic polynomial of this system is

displaymath186.

The eigenvalues are

displaymath188.

It is clear that both of them are negative. Hence, the origin is a sink. Meaning that, regardless of the initial condition, the mass will always tend to its equilibrium, or rest, position.
Note that if V is an eigenvector associated to the biggest eigenvalue tex2html_wrap_inline192 , then all the solutions tend to the origin tangent to that vector V. In this case we have

displaymath196.

Remark: The case when one of the two eigenvalues is zero will be discussed in another section separately.

If you would like more practice, click on Example.

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour