Consider the system * - ( ) - x' X. a. Solve the system for a = 0.5. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin. b. Solve the system for a = 2. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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ODE question

Consider the system
x'
=
-1
X.
-α
-1
a. Solve the system for a = 0.5. What are the eigenvalues of the coefficient matrix? Classify
the equilibrium point at the origin.
b. Solve the system for a = 2. What are the eigenvalues of the coefficient matrix? Classify
the equilibrium point at the origin.
c. In parts a. and b., solutions of the system exhibit two quite different types of behavior.
Find the eigenvalues of the coefficient matrix in terms of a, and determine the value of a
between 0.5 and 2 where the transition from one type of behavior to the other occurs. This
value of a is called a bifurcation value for this problem.
Transcribed Image Text:Consider the system x' = -1 X. -α -1 a. Solve the system for a = 0.5. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin. b. Solve the system for a = 2. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin. c. In parts a. and b., solutions of the system exhibit two quite different types of behavior. Find the eigenvalues of the coefficient matrix in terms of a, and determine the value of a between 0.5 and 2 where the transition from one type of behavior to the other occurs. This value of a is called a bifurcation value for this problem.
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