Critical points and phase-plane analysis. Consider the system of coupled ODES: x' = y - x², y = y² - xy-2y. (i) Find the critical points (ro, yo) E R² of this system. Hint: One critical point is (0,0), and there are two more critical points. (ii) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. (iii) Find the eigenvalues of each of the linear systems found in part (ii). Also, for the real eigenvalues only, construct the corresponding eigenvectors. y 5 4 3 2 1 of -2 -1 0 (1) X 2 3 (iv) The figure shows 5 actual trajectories on the phase plane (x, y), of the solutions to system (1). Copy this figure by hand, to your answer book. Finish your plot by marking all 3 critical points and adding arrowheads on each of the 5 trajectories. REMARK: Be as clear as possible.
Critical points and phase-plane analysis. Consider the system of coupled ODES: x' = y - x², y = y² - xy-2y. (i) Find the critical points (ro, yo) E R² of this system. Hint: One critical point is (0,0), and there are two more critical points. (ii) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. (iii) Find the eigenvalues of each of the linear systems found in part (ii). Also, for the real eigenvalues only, construct the corresponding eigenvectors. y 5 4 3 2 1 of -2 -1 0 (1) X 2 3 (iv) The figure shows 5 actual trajectories on the phase plane (x, y), of the solutions to system (1). Copy this figure by hand, to your answer book. Finish your plot by marking all 3 critical points and adding arrowheads on each of the 5 trajectories. REMARK: Be as clear as possible.
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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PLEASE ANSWER ALL PARTS OF QUESTION CLEARLY
Transcribed Image Text:6. Critical points and phase-plane analysis. Consider the system of coupled ODES:
x' = y - x², y = y² - xy - 2y.
(i) Find the critical points (xo, yo) € R² of this system.
Hint: One critical point is (0,0), and there are two more critical points.
(ii) For each critical point, find the approximate linear ODE system that is valid in a
small neighbourhood of it.
(iii) Find the eigenvalues of each of the linear systems found in part (ii). Also, for the
real eigenvalues only, construct the corresponding eigenvectors.
5
4
3
2
1
0
2
-2 -1 0
X
(1)
2
3 4
(iv) The figure shows 5 actual trajectories on the phase plane (x, y), of the solutions to
system (1). Copy this figure by hand, to your answer book. Finish your plot by
marking all 3 critical points and adding arrowheads on each of the 5 trajectories.
REMARK: Be as clear as possible.
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