Question 6 Consider the following system ax-bxy, y -cy + day, where a, b, c and d are strictly positive constants. (2.1) Show that the equilibrium points of the system are (0,0) and (,).
Question 6 Consider the following system ax-bxy, y -cy + day, where a, b, c and d are strictly positive constants. (2.1) Show that the equilibrium points of the system are (0,0) and (,).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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plz solve question 6(a) it within 30-40 mins I'll give you multiple upvote
![Question 6
Consider the following system
ax - bay,
-cy + day,
where a, b, c and d are strictly positive constants.
(2.1) Show that the equilibrium points of the system are (0,0) and (,).
(2.2) Given L(x, y)=da - clna + by - alny + K, where K is a constant. Verify that
L=0 along the solutions of the dynamical system.
(2.3) Determine the value for K such that L()-0.
(2.4) Show that L(x, y) is a Lyapunov function for the equilibrium point (,), using the
value of K determined above.
[6]
(2.5) Given that L(x, y) is a Lyapunov function for (2, 2), what can you conclude about
the nonlinear stability of the equilibrium? What can you conclude from the prop-
erties of L(x, y) about the solutions of the system close to this equilibrium?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09a8a06e-3bed-47f3-ae1c-fada2f53743e%2Fcc762a32-6221-40b3-b45e-a0876aaf4e79%2Fb5ipck_processed.png&w=3840&q=75)
Transcribed Image Text:Question 6
Consider the following system
ax - bay,
-cy + day,
where a, b, c and d are strictly positive constants.
(2.1) Show that the equilibrium points of the system are (0,0) and (,).
(2.2) Given L(x, y)=da - clna + by - alny + K, where K is a constant. Verify that
L=0 along the solutions of the dynamical system.
(2.3) Determine the value for K such that L()-0.
(2.4) Show that L(x, y) is a Lyapunov function for the equilibrium point (,), using the
value of K determined above.
[6]
(2.5) Given that L(x, y) is a Lyapunov function for (2, 2), what can you conclude about
the nonlinear stability of the equilibrium? What can you conclude from the prop-
erties of L(x, y) about the solutions of the system close to this equilibrium?
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