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Consider a nonlinear dynamical system described by 1 -1 + 2x³ + x2 and 2 = -1 -₂. (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 Lyapunoy function candidate V(#₁, 2) = x² + x². (d) Is the origin globally asymptotically stable?

Consider a nonlinear dynamical system described by 1 -1 + 2x³ + x2 and 2 = -1 -₂. (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 Lyapunoy function candidate V(#₁, 2) = x² + x². (d) Is the origin 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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parts B and C

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?
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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