dx dt = u2+y− 2 ( + ), dy dt =-2 tuy y ( +), where ER is a parameter. (a) Show that the origin is the only critical point of this system. (b) Find the linear system that approximates the system in the vicinity of the origin. Determine the type and stability of the critical point at the origin for different values of μ. (c) Express the system in terms of polar coordinates. (d) Show that when μ> 0, there is a periodic solution r = √√√. By solving the system found in part (c), conclude that this periodic solution attracts all other non-zero solutions.
dx dt = u2+y− 2 ( + ), dy dt =-2 tuy y ( +), where ER is a parameter. (a) Show that the origin is the only critical point of this system. (b) Find the linear system that approximates the system in the vicinity of the origin. Determine the type and stability of the critical point at the origin for different values of μ. (c) Express the system in terms of polar coordinates. (d) Show that when μ> 0, there is a periodic solution r = √√√. By solving the system found in part (c), conclude that this periodic solution attracts all other non-zero solutions.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter1: Systems Of Linear Equations
Section1.1: Introduction To Systems Of Linear Equations
Problem 71E: Find a system of two equations in two variables, x1 and x2, that has the solution set given by the...
Question
Consider the autonomous system
![dx
dt
= u2+y− 2 ( + ),
dy
dt
=-2 tuy y ( +),
where ER is a parameter.
(a) Show that the origin is the only critical point of this system.
(b) Find the linear system that approximates the system in the vicinity of the origin. Determine the type and
stability of the critical point at the origin for different values of μ.
(c) Express the system in terms of polar coordinates.
(d) Show that when μ> 0, there is a periodic solution r = √√√. By solving the system found in part (c), conclude
that this periodic solution attracts all other non-zero solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e303283-8045-4d81-bcb2-cabd4c10c125%2Fb74381de-b14c-481b-bf91-6251d2009eb6%2Fff4p9ae_processed.png&w=3840&q=75)
Transcribed Image Text:dx
dt
= u2+y− 2 ( + ),
dy
dt
=-2 tuy y ( +),
where ER is a parameter.
(a) Show that the origin is the only critical point of this system.
(b) Find the linear system that approximates the system in the vicinity of the origin. Determine the type and
stability of the critical point at the origin for different values of μ.
(c) Express the system in terms of polar coordinates.
(d) Show that when μ> 0, there is a periodic solution r = √√√. By solving the system found in part (c), conclude
that this periodic solution attracts all other non-zero solutions.
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