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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) = 5 + is non-increasing along the solution, i.e., using the multivariable chain rule verify that

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) = 5 + is non-increasing along the solution, i.e., using the multivariable chain rule verify that

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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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) = 5 i.e., using the multivariable chain rule verify that x4 + 4 is non-increasing along the solution, d E(x(t), y(t)) < 0 dt 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]
Transcribed Image Text: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) = 5 i.e., using the multivariable chain rule verify that x4 + 4 is non-increasing along the solution, d E(x(t), y(t)) < 0 dt 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]
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