Consider the nonlinear system of differential equations dy dt dx dt = x - e², xy² - - xy. (a) Find all the critical points for the nonlinear system. (b) For the critical point where both x 0 and y = 0: (i) Find the linearisation of the system with the critical point translated to (0,0). (ii) Using eigenvalues and eigenvectors, find the general solution of the linearised system in part (i). (iii) For the linearised system in part (i): find all straight line orbits, determine the behaviour of the orbits as t→∞ and t→→∞, • determine the slopes at which orbits meet the coordinate axes. Hence sketch (by hand) a phase portrait for the linearised system around (0, 0), showing all straight line orbits and at least four other orbits, and identify the type and stability of the critical point. (iv) Determine whether the linear system in part (i) can be used to approximate the be- haviour of the non-linear system near the critical point? Explain your answer. (c) Use PPLANE to sketch a global phase portrait for the nonlinear system in the region -2 ≤ x ≤ 4 and -2 ≤ y ≤ 4, showing the behaviour of the orbits near each critical point. (d) Based on the global phase portrait, discuss what happens to x(t) and y(t) as t → ∞ if y (0) is positive.
Consider the nonlinear system of differential equations dy dt dx dt = x - e², xy² - - xy. (a) Find all the critical points for the nonlinear system. (b) For the critical point where both x 0 and y = 0: (i) Find the linearisation of the system with the critical point translated to (0,0). (ii) Using eigenvalues and eigenvectors, find the general solution of the linearised system in part (i). (iii) For the linearised system in part (i): find all straight line orbits, determine the behaviour of the orbits as t→∞ and t→→∞, • determine the slopes at which orbits meet the coordinate axes. Hence sketch (by hand) a phase portrait for the linearised system around (0, 0), showing all straight line orbits and at least four other orbits, and identify the type and stability of the critical point. (iv) Determine whether the linear system in part (i) can be used to approximate the be- haviour of the non-linear system near the critical point? Explain your answer. (c) Use PPLANE to sketch a global phase portrait for the nonlinear system in the region -2 ≤ x ≤ 4 and -2 ≤ y ≤ 4, showing the behaviour of the orbits near each critical point. (d) Based on the global phase portrait, discuss what happens to x(t) and y(t) as t → ∞ if y (0) is positive.
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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Transcribed Image Text:Consider the nonlinear system of differential equations
dy
dt
dx
dt
= x - e²,
= xy² — xy.
=
(a) Find all the critical points for the nonlinear system.
(b) For the critical point where both x = 0 and y ‡ 0:
(i) Find the linearisation of the system with the critical point translated to (0,0).
(ii) Using eigenvalues and eigenvectors, find the general solution of the linearised system in
part (i).
(iii) For the linearised system in part (i):
find all straight line orbits,
determine the behaviour of the orbits as t→∞ and t→→∞,
• determine the slopes at which orbits meet the coordinate axes.
Hence sketch (by hand) a phase portrait for the linearised system around (0, 0), showing
all straight line orbits and at least four other orbits, and identify the type and stability
of the critical point.
(iv) Determine whether the linear system in part (i) can be used to approximate the be-
haviour of the non-linear system near the critical point? Explain your answer.
(c) Use PPLANE to sketch a global phase portrait for the nonlinear system in the region
-2 ≤ x ≤ 4 and -2 ≤ y ≤ 4, showing the behaviour of the orbits near each critical point.
(d) Based on the global phase portrait, discuss what happens to x(t) and y(t) as t→ ∞ if
y (0) is positive.
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