Consider a continuously differentiable function f : R^2 → R^2 and a differential equation x' = f(x) with an equilibrium x^∗ = 0, and suppose that there exists a homoclinic orbit O(x) connecting to this equilibrium (in both forward and backward time). Which of the following three statements is true? (a) The equilibrium x^∗ = 0 is stable. (b) The equilibrium x^∗ = 0 is unstable. (c) It depends on the right hand side f whether the equilibrium x^∗ = 0 is stable or unstable.
Consider a continuously differentiable function f : R^2 → R^2 and a differential equation x' = f(x) with an equilibrium x^∗ = 0, and suppose that there exists a homoclinic orbit O(x) connecting to this equilibrium (in both forward and backward time). Which of the following three statements is true? (a) The equilibrium x^∗ = 0 is stable. (b) The equilibrium x^∗ = 0 is unstable. (c) It depends on the right hand side f whether the equilibrium x^∗ = 0 is stable or unstable.
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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Consider a continuously differentiable function f : R^2 → R^2 and a differential equation x' = f(x) with an equilibrium x^∗ = 0, and suppose that there exists a homoclinic orbit O(x) connecting to this equilibrium (in both forward and backward time). Which of the following three statements is true?
(a) The equilibrium x^∗ = 0 is stable.
(b) The equilibrium x^∗ = 0 is unstable.
(c) It depends on the right hand side f whether the equilibrium x^∗ = 0 is stable or unstable.
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