Problem 3. Consider the differential equation i(t) = y – x³ ý(t) = -x³ – y3 a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to conclude the same for the original nonlinear system? Why or why not? b) Show that the "energy function" E(x,y) = + is non-increasing along the solution, i.e., using the multivariable chain rule verify that E(x(t), y(t)) < 0 c) Use part b) to show that the origin is stable, i.e., suppose that you are given R> 0, then provide a sufficient condition on r > 0 such that, if x²(0) + y²(0) < r², then we can conclude that the solution x(t), y(t) satisifes x²(t) + y²(t) < R² for all t > 0. [Argue as done in class for the simple harmonic oscillator system]
Problem 3. Consider the differential equation i(t) = y – x³ ý(t) = -x³ – y3 a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to conclude the same for the original nonlinear system? Why or why not? b) Show that the "energy function" E(x,y) = + is non-increasing along the solution, i.e., using the multivariable chain rule verify that E(x(t), y(t)) < 0 c) Use part b) to show that the origin is stable, i.e., suppose that you are given R> 0, then provide a sufficient condition on r > 0 such that, if x²(0) + y²(0) < r², then we can conclude that the solution x(t), y(t) satisifes x²(t) + y²(t) < R² for all t > 0. [Argue as done in class for the simple harmonic oscillator system]
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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