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Consider the system where r² = x² + y². J = -y+x(r4-3r²+1), \ ý = x + y(r²¹ - 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)|1 < √√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)le < √² + y² < 1}, where 0 < € <<< 1.

Consider the system where r² = x² + y². J = -y+x(r4-3r²+1), \ ý = x + y(r²¹ - 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)|1 < √√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)le < √² + y² < 1}, where 0 < € <<< 1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 32E
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Consider the system where 2 = x² + y². y ý = = y + x(r² - 3r²+1), x+y(r² = 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)]1 < √√√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)<√√x² + y² <1}, where 0 < € <<< 1. (Hint: You may need the fact that the only equilibrium of system is at the origin.)
Transcribed Image Text:Consider the system where 2 = x² + y². y ý = = y + x(r² - 3r²+1), x+y(r² = 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)]1 < √√√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)<√√x² + y² <1}, where 0 < € <<< 1. (Hint: You may need the fact that the only equilibrium of system is at the origin.)
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