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4. This project is related to the Lyapunov 2nd method, linearization technique, La Salle's invariance principle and the center manifold theory. (a) Find all the equilibra of the system *₁ =−x₁+x₁(x²+x²) *₂ = −2x₂+x²3x2 *3 = (1 + sin² x3)x3. Is the equilibrium unstable? locally asymptotically stable? (b) Prove that the subsystem *3 = −(1 + sin² £3)x3 is globally asymptotically stable at x3 = 0. (c) Use the result [b] and the stability of perturbed systems to prove that all the solution trajectories of the (x1, x2) system also converge to the origin as t goes to infinity, and hence the entire system is globally asymptotically stable.

4. This project is related to the Lyapunov 2nd method, linearization technique, La Salle's invariance principle and the center manifold theory. (a) Find all the equilibra of the system *₁ =−x₁+x₁(x²+x²) *₂ = −2x₂+x²3x2 *3 = (1 + sin² x3)x3. Is the equilibrium unstable? locally asymptotically stable? (b) Prove that the subsystem *3 = −(1 + sin² £3)x3 is globally asymptotically stable at x3 = 0. (c) Use the result [b] and the stability of perturbed systems to prove that all the solution trajectories of the (x1, x2) system also converge to the origin as t goes to infinity, and hence the entire system is globally asymptotically stable.

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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4. This project is related to the Lyapunov 2nd method, linearization technique, La Salle's invariance principle and the center manifold theory. (a) Find all the equilibra of the system *₁ *₂ *3 =−x₁ + x₁(x²+x²) = −2x2+x²3x2 = (1 + sin²x3)x3. Is the equilibrium unstable? locally asymptotically stable? (b) Prove that the subsystem *3 = (1 + sin² *3)*3 is globally asymptotically stable at x3 = 0. (c) Use the result [b] and the stability of perturbed systems to prove that all the solution trajectories of the (1, 2) system also converge to the origin as t goes to infinity, and hence the entire system is globally asymptotically stable.
Transcribed Image Text:4. This project is related to the Lyapunov 2nd method, linearization technique, La Salle's invariance principle and the center manifold theory. (a) Find all the equilibra of the system *₁ *₂ *3 =−x₁ + x₁(x²+x²) = −2x2+x²3x2 = (1 + sin²x3)x3. Is the equilibrium unstable? locally asymptotically stable? (b) Prove that the subsystem *3 = (1 + sin² *3)*3 is globally asymptotically stable at x3 = 0. (c) Use the result [b] and the stability of perturbed systems to prove that all the solution trajectories of the (1, 2) system also converge to the origin as t goes to infinity, and hence the entire system is globally asymptotically stable.
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