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, (c) x = y, y = -x- y³, (b) x = y² + 2x +4, y = y + 3 (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: i(x,y) - 2(x² + y¹) = ...... 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: (a) Stable (b) unstable (c) Asymptotic stable (d) stable if x < 0,y > 0.
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, (c) x = y, y = -x- y³, (b) x = y² + 2x +4, y = y + 3 (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: i(x,y) - 2(x² + y¹) = ...... 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: (a) Stable (b) unstable (c) Asymptotic stable (d) stable if x < 0,y > 0.
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: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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