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Math 2280 Quiz 9 1. Given the system dx dt dy dt = εx + y − x(x2 + y 2 ) , = −x + εy − y(x2 + y 2 ) find the linearization about the fixed point at the origin and classify its stability and type if (a) ε > 0 and (b) ε < 0. The Jacobian for the system is J(x, y) = ε − 3x2 − y 2 −1 − 2xy 1 − 2xy ε − x2 − 3y 2 . We are interested in its eigenvalues at (0, 0), so we must have ε−λ 1 = (ε − λ)2 + 1 ⇒ (ε − λ)2 = −1 ⇒ ε − λ = ±i ⇒ λ = ε ± i. 0 = |J(0, 0) − λI| = −1 ε − λ So if ε > 0 the fixed point at (0, 0) is an unstable spiral. If ε < 0, then the fixed point at (0, 0) is an asymptotically stable spiral. 2. Linearize the system dx dt dy dt = 60x − 4x2 − 3xy = 42y − 2y 2 − 3xy about the fixed point at (0, 21) and classify the fixed point’s stability and type. The Jacobian for this system is J(x, y) = 60 − 8x − 3y −3y so J(0, 21) = −3x 42 − 4y − 3x −3 0 −63 −42 , , which has eigenvalues λ1 = −3 and λ2 = −42. This tells us that the fixed point at (0, 21) is an asymptotically stable node. 1