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...

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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.

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