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4- Which of the following is a Hamiltonian system? (a) x = y² - x, y = x, (c) i = y +3x, j=y, (b)x=x²-1, y = -y, (d) x = x + y², y = -y

4- Which of the following is a Hamiltonian system? (a) x = y² - x, y = x, (c) i = y +3x, j=y, (b)x=x²-1, y = -y, (d) x = x + y², y = -y

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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1- We can dropping the higher order term in only one of the following system to derived the linearized system (a) x = x(1 - y²), y = -xy, (b) x = y² + 2x + 4, y = y + 3 (c) x = y, y = -x- y³, (d) x = x² - y, y = -x + 5 2- If L(x, y) = ax² + by² is a Lyapunov function at the origin for the system * = y + x³, y = -x + y3, a = b = 1. Then compute the expression: L(x, y) - 2 (x² + y) = ...... 3- If L(x, y) = ax² + by² is a Lyapunov function at the origin for the system x = 2x + xy², y = y(1-x²), a = b = 1, then this system is: (b) unstable (c) Asymptotic stable (d) stable if x < 0, y > 0. 4- Which of the following is a Hamiltonian system? (a) x = y² - x, y = x, (b)x=x²-1, y = -y, (c) i = y +3x, j=y, (d) x = x + y², y = -y 5- Let the dynamical system is given by x=ay-bxy, y = −dx + cxy. The fixed point computed as: (a) (1,1) and (0,0) (b) (0,0) and (d), (c) (0,0) and (2), (d) (0,0) and (2)
Transcribed Image Text:1- We can dropping the higher order term in only one of the following system to derived the linearized system (a) x = x(1 - y²), y = -xy, (b) x = y² + 2x + 4, y = y + 3 (c) x = y, y = -x- y³, (d) x = x² - y, y = -x + 5 2- If L(x, y) = ax² + by² is a Lyapunov function at the origin for the system * = y + x³, y = -x + y3, a = b = 1. Then compute the expression: L(x, y) - 2 (x² + y) = ...... 3- If L(x, y) = ax² + by² is a Lyapunov function at the origin for the system x = 2x + xy², y = y(1-x²), a = b = 1, then this system is: (b) unstable (c) Asymptotic stable (d) stable if x < 0, y > 0. 4- Which of the following is a Hamiltonian system? (a) x = y² - x, y = x, (b)x=x²-1, y = -y, (c) i = y +3x, j=y, (d) x = x + y², y = -y 5- Let the dynamical system is given by x=ay-bxy, y = −dx + cxy. The fixed point computed as: (a) (1,1) and (0,0) (b) (0,0) and (d), (c) (0,0) and (2), (d) (0,0) and (2)
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