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x' = (4 + x) (1 - x + y), y' = (y - 1) (1 + x + ay); a > 0.

x' = (4 + x) (1 - x + y), y' = (y - 1) (1 + x + ay); a > 0.

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 (a) Find the critical points. ○ ( − 1, 2), (-4, 1), (-4, 2), (−¹+ ○ (2,1), (–4, 1), (4, 1), (–4, ³/1) ○ (2,1), (4, 1), (4, 2), (²) (2,1), (-4,1), 3 -1 -2 1). (-4, ²- ). ( -¹ +68², 1-7-²a) 1 a + α > 1+α 1+α -1+α ○ (²) 1+α 1+α (2, − 1), (−4, − 1), (–4, ¾/), (- - eTextbook and Media The bifurcation point α = (a) coincide. (b) Determine the value of a > 0, denoted by ao, (bifurcation point) where two critical points coincide. x' (4 + x) (1 − x + y), y' = (y − 1) (1 + x + ay); a > 0. For a 3,the critical point (x2, y₂) is For a <3,the critical point (x2, y2) is (c) By finding the approximating linear systems and their eigenvalues, determine how the stability properties of these two critical points change as a passes through the bifurcation point ao. For a > 3,the critical point (x3, Y3) is For a <3,the critical point (x3, y3) is where the second ✓and fourth a saddle ✓which is a spiral a node ✓which is v which is ✓which is asymptotically stable. V unstable. critical points listed in part stable.
Transcribed Image Text:Consider the system (a) Find the critical points. ○ ( − 1, 2), (-4, 1), (-4, 2), (−¹+ ○ (2,1), (–4, 1), (4, 1), (–4, ³/1) ○ (2,1), (4, 1), (4, 2), (²) (2,1), (-4,1), 3 -1 -2 1). (-4, ²- ). ( -¹ +68², 1-7-²a) 1 a + α > 1+α 1+α -1+α ○ (²) 1+α 1+α (2, − 1), (−4, − 1), (–4, ¾/), (- - eTextbook and Media The bifurcation point α = (a) coincide. (b) Determine the value of a > 0, denoted by ao, (bifurcation point) where two critical points coincide. x' (4 + x) (1 − x + y), y' = (y − 1) (1 + x + ay); a > 0. For a 3,the critical point (x2, y₂) is For a <3,the critical point (x2, y2) is (c) By finding the approximating linear systems and their eigenvalues, determine how the stability properties of these two critical points change as a passes through the bifurcation point ao. For a > 3,the critical point (x3, Y3) is For a <3,the critical point (x3, y3) is where the second ✓and fourth a saddle ✓which is a spiral a node ✓which is v which is ✓which is asymptotically stable. V unstable. critical points listed in part stable.
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