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MATH 557 Homework Set #10 Fall 2015 38. Use Lyapunov theorems to discuss the stability of the zero solution of y 00 + y 0 + y 3 = 0 Suggestions: Rewrite the equation as a first order system x0 = f (x) and look for a n Lyapunov function in the form V (x) = αxm 1 + βx2 for some α, β > 0 and positive even integers m, n. You should be able to show that 0 is globally asymptotically stable. 39. Let ψ(t) be a bounded solution of x0 = f (x) for t ≥ t0 . Show that ω(ψ) is compact. Suggestions: Let C + (ξ) = {φ(t, 0, ξ) : t ≥ 0} be the semiorbit starting at ξ, and show \ ω(ψ) = C + (ψ(τ )) τ ≥0 40. Let x0 = f (x) be a gradient system with f (0) = 0. Show that all of the eigenvalues of ∂f (0) are real. What does this say about the behavior of solutions of the system near ∂x the origin? 41. Show that the linear constant coefficient system x0 = Ax (A real) is of gradient type if and only if A is symmetric. (Suggestion: consider V (x) = xT Ax.) 42. In a domain U ⊂ R3 show that if x0 = f (x) is a gradient system then ∇ × f = 0 in U . The converse is almost, but not quite exactly correct. What is a correct converse statement? (You may just quote from a textbook here, if you like.) Note that these results can be applied for U ⊂ R2 also, just by specializing to vector fields with a zero third component. 43. Let k1 , k2 be constants and consider the system x01 = k1 x1 − x1 (x21 + x22 ) − x1 x22 x02 = k2 x2 − x2 (x21 + x22 ) − x2 x21 a) Show that the disk B(0, R) is a positive invariant set for sufficiently large R. b) Is this a gradient system? c) Find all stable equilibrium points.