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Consider the system in R² −y + x(r² − 3r² + 1) == Ꮎ = x + y(r¹ — 3r²+1), - r = √√√x² + y². Use Poincare-Bendixon theorem to answer the following questions: (a) Prove that the origin is an unstable equilibrium. (b) Show that the set {0 < || < 1} contains a periodic orbit. (c) Show that the set {1 < || < 2} contains a periodic orbit. (d) Find and determine the stability type of all limit cycles and equilibria.

Consider the system in R² −y + x(r² − 3r² + 1) == Ꮎ = x + y(r¹ — 3r²+1), - r = √√√x² + y². Use Poincare-Bendixon theorem to answer the following questions: (a) Prove that the origin is an unstable equilibrium. (b) Show that the set {0 < || < 1} contains a periodic orbit. (c) Show that the set {1 < || < 2} contains a periodic orbit. (d) Find and determine the stability type of all limit cycles and equilibria.

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 system in R² x = − y + x(r² − 3r² + 1) Ö= x + y(r¹ — 3r² + 1), r = x2 + y². Use Poincare-Bendixon theorem to answer the following questions: (a) Prove that the origin is an unstable equilibrium. (b) Show that the set {0 < |☎| < 1} contains a periodic orbit. (c) Show that the set {1 < || < 2} contains a periodic orbit. (d) Find and determine the stability type of all limit cycles and equilibria.
Transcribed Image Text:Consider the system in R² x = − y + x(r² − 3r² + 1) Ö= x + y(r¹ — 3r² + 1), r = x2 + y². Use Poincare-Bendixon theorem to answer the following questions: (a) Prove that the origin is an unstable equilibrium. (b) Show that the set {0 < |☎| < 1} contains a periodic orbit. (c) Show that the set {1 < || < 2} contains a periodic orbit. (d) Find and determine the stability type of all limit cycles and equilibria.
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