Consider a nonlinear dynamical system described by 1 -1+2+2 and 2 = -1 -32. (a) Find all the equilibrium points of the system and show that one of them is at the origin. (b) For each equilibrium point, determine its type (e.g., stable/unstable node, saddle, etc.) using Jacobian linearization. (c) Show that the origin is asymptotically stable using the Lyapunov function candidate V(21, 22₂) = a² + x². (d) Is the origin globally asymptotically stable?
Consider a nonlinear dynamical system described by 1 -1+2+2 and 2 = -1 -32. (a) Find all the equilibrium points of the system and show that one of them is at the origin. (b) For each equilibrium point, determine its type (e.g., stable/unstable node, saddle, etc.) using Jacobian linearization. (c) Show that the origin is asymptotically stable using the Lyapunov function candidate V(21, 22₂) = a² + x². (d) Is the origin globally asymptotically stable?
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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Parts C and D
![Consider a nonlinear dynamical system described by = −₁+ 2x³ + x2 and 2 = -1 -22.
(a) Find all the equilibrium points of the system and show that one of them is at the origin.
(b) For each equilibrium point, determine its type (e.g., stable/unstable node, saddle, etc.) using
Jacobian linearization.
(c) Show that the origin is asymptotically stable using the Lyapunov function candidate V(1, 2);
x} + x².
(d) Is the origin globally asymptotically stable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c407875-96b5-40f7-9247-219f4b3dc200%2F1c7e6ecd-9c89-4049-972d-29ac9461d33b%2F4siwgmn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a nonlinear dynamical system described by = −₁+ 2x³ + x2 and 2 = -1 -22.
(a) Find all the equilibrium points of the system and show that one of them is at the origin.
(b) For each equilibrium point, determine its type (e.g., stable/unstable node, saddle, etc.) using
Jacobian linearization.
(c) Show that the origin is asymptotically stable using the Lyapunov function candidate V(1, 2);
x} + x².
(d) Is the origin globally asymptotically stable?
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