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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.

Advanced Engineering Mathematics
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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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.
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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