Problem List MATH 5173 Spring, 2014

Transcription

1 Problem List MATH 5173 Spring, 2014 The notation p/n means the problem with number n on page p of Perko. 1. 5/3 [Due Wednesday, January 15] 2. 6/5 and describe the relationship of the phase portraits [Due Wednesday, January 15] 3. 6/6 [Due Wednesday, January 15] 4. 9/1 5. 9/3a 6. 10/5 7. Suppose ẋ = λx and ẏ = µy for λµ 0. Obtain a formula of the form x = h(y), y = h(x), or h(x, y) = 0 for the solution curve through (x 0, y 0 ) (0, 0), for all such pairs (x 0, y 0 ). Draw all qualitatively distinct phase portraits (there are three of them) /6 Hint. 1. Without loss of generality you may assume that the invariant subspace is E = {x 1,..., x p, 0,..., 0}. Invariance of E shows up in A as a block of zeros. 2. If matrices are partitioned into blocks then the matrices can be multiplied as if the blocks were numbers, as long as each product of the relevant blocks is compatible: ( ) ( ) ( ) A B R S AR + BT AS + BU = C D T U CR + DT CS + DU ( ) ( ) λ 1 e λt te 9. For J = show that exp(tj) = λt 0 λ 0 e λt by writing ( ) ( ) ( ) λt t λt 0 0 t = + := S + N 0 λt 0 λt 0 0 and computing exp(n) directly from the definition. 10. For x(t) = x 0 e λt + y 0 te λt and y(t) = y 0 e λt, the solution to the linear system of the previous problem, prove that if λy 0 0 then x = x0 y 0 y + 1 λ log y y 0 y. 1

2 2 11. Show that in polar coordinates the system ẋ = P (x, y), ẏ = Q(x, y) (for any functions P and Q, not necessarily linear) becomes r 2 θ = [xq(x, y) yp (x, y)] (x=r cos θ,y=r sin θ) rṙ = [xp (x, y) + yq(x, y)] (x=r cos θ,y=r sin θ). Hint. Exercise 27/9 of Perko. ( ) a b 12. For the system ẋ = Jx, J =, b 0, change to polar coordinates to b a obtain an uncoupled system θ = f(θ), ṙ = g(r). Solve the system explicitly (find θ(t) and r(t)) and use the solution to rigorously derive the phase portraits possible (with attention to the signs of a and b 0) :1 Note that you are not to solve the system :5 15. Show that if η(t) solves ẋ = f(x), x(t 0 ) = x 0, then there exists a constant c such that µ(t) := η(t + c) solves ẋ = f(x), x(0) = x 0. Find c explicitly. 16. Suppose f : I E R R n R n is continuous and consider the initial value problem and the integral equation ẏ = f(t, y), y(t 0 ) = y 0 (1) η(t) = y 0 + t t 0 f(s, η(s)) ds. (2) a. Show that if η : J I E is continuous and satisfies (2) then η is differentiable and satisfies (1). b. Show that if η : J I E is differentiable and satisfies (1) then the integral in (2) exists and η satisfies (2). 17. In the proof of Lemma 1 in Section 2.4, we assumed in a proof by contradiction that an endpoint t 0 of I lay in the interval I. Prove that if this is so then there exists a neighborhood N of t 0 in I on which both u 1 and u 2 are defined and continuous, that both lim t t0 u 1 (t) and lim t t0 u 1 (t) exist and have a common value u 0, and that u 0 E : :4

3 3 20. Suppose f : E R n R n is at least C 1 and that ϕ : (α, β) R n is a solution of the initial value problem ẋ = f(x), x(0) = x 0 E. Let g(x) = kf(x), k R\{0}. Express the solution η(t) of the initial value problem ẋ = g(x), x(0) = x 0, in terms of ϕ(t), being careful about its domain. Discuss the relationship of the phase portraits of the two differential equations on E. (Compare with Problem 2 on this list.) 21. Same problem as Problem 24 but for g(x) = k(x)f(x), where k : R n R \ {0} is at least C Prove that if f : R n R n, f is at least C 1, and there exists M R such that f(x) M for all x R n, then for any x 0 R n the unique solution to the initial value problem ẋ = f(x), x(0) = x 0, exists for all t R. 23. Prove that if f : R n R n and is at least C 1, and if we are studying geometric properties of the phase portrait of the system ẋ = f(x) of ordinary differential equations on R n, then we may assume without loss of generality that solutions exist on all of R. Hint. You may use the results of the previous two problems :2 [Due Wednesday, March 19] :7 [Due Wednesday, March 19] 26. Duffing s equation is ẍ x + x 3 = 0 a. Write an equivalent system of first order equations (*) ẋ = P (x, y), ẏ = Q(x, y). b. Show that H : R 2 R : (x, y) 1 4 x4 1 2 x y2 is constant on trajectories, so that every level curve of H is an invariant set for (*). c. Using a symbolic manipulator like Maple or Mathematica, or by hand, find H 1 (h) for any value of h for which it contains an equilibrium of (*), and for nearby h. Using (*), decompose these level curves into trajectories and add arrows. [Due Wednesday, March 19] 27. Rework 2.5:7 with ẋ = x 2 on R \ {0} replaced by ẋ = 1 + x 2 on R. Hint. Review a calculus text if need be to integrate (1 + x 2 ) 1 and to recall the properties of the function arctan : R ( π/2, π/2) and its graph. It is always true (i.e., is true for all x R) that arctan(tan x) is defined and is x but the opposite order is tricky because we have selected a branch of the tangent function in order to define an inverse for it. [Due Wednesday, March 26] 28. Rework Problem 26 with the second order differential equation there replaced by ẍ + x x 2 = 0 and H replaced by H(x, y) = 3x 2 + 3y 2 2x 3. [Due Wednesday, March 26]

4 4 29. In the proof of the Flowbox Theorem, f(0) = (k, 0,..., 0) T and g was defined by g : [ α, α] N R R n 1 R n R n : (t, (x 2,..., x n )) η(t, (0, x 2,..., x n )). a. Compute dg(0, 0). The answer is an n n matrix of specific constants. Hint. The first column is η. For the remaining columns you may set t = 0 before computing the first partial derivatives. b. Apply the Inverse Function Theorem to conclude that there exists a neighborhood V of (0, 0) in [ α, α] N and a C r mapping h : W := g(v ) V such that (i) g h = id W and (ii) h g = id V. [Due Wednesday, March 26] 30. Prove the Integral Equation Lemma: If u(t) is differentiable and solves the initial value problem ẋ = A x + f(x), x(0) = x 0 (3) on an interval I then u(t) solves the integral equation x(t) = exp(ta) x 0 + t 0 exp((t s)a) f(x(s)) ds (4) on I. Conversely, if u(t) is continuous and solves the integral equation (4) on I then it is differentiable and solves the initial value problem (3) on I. Hint. For one direction use the calculus fact d dt [ ] b(t) f(t, s) ds = f(t, b(t))b (t) f(t, a(t))a (t) + a(t) b(t) a(t) f (t, s) ds. t For the other direction define a new variable z by z = exp( ta) x. Equate two expressions for ẋ to obtain an ordinary differential equation that z solves. Integrate it from 0 to t and revert to the original variable x. [Due Wednesday, April 2] 31. Let L be a linear mapping on R n and let λ(x) be a C 1 mapping on R n that is bounded by some constant ν > 0 (i.e., λ(x) ν for all x R n ). Show that if ν is sufficiently small then for every x 0 R n the unique solution η(t, x 0 ) to ẋ = L x + λ(x), x(0) = x 0, exists for all t 0. At the end address the question of whether or not ν really needs to be small, or if it can be any positive real number. Hint. For any t > 0 for which η(t, x)) exists bound η(t, x) x 0 independently of t using the triangle inequality involving x, η(t, x), and ϕ(t, x) = exp(tl)x, the global flow of ẋ = L x, using the fact that η(t, x) satisfies η(t, x 0 ) = exp(ta) x 0 + [Due Wednesday, April 2] t 0 exp((t s)a) λ(η(s, x)) ds 32. Use the Hartman-Grobman Theorem to classify the singularity specified, or state why the theorem does not apply. a. ẋ = x(x y), ẏ = y(2x y) at (0, 0). b. ẋ = x x 3, ẏ = 2y + y 2 at (0, 0). c. ẋ = x y, ẏ = 1 e x at (0, 0).

5 5 d. ẋ = x + e y, ẏ = y at ( 1, 0). e. ẋ = y + x x 3, ẏ = y at each singularity. [Due Wednesday, April 2] 33. Classify the singularity at the origin of the system ẋ = 2x 2 y 2, ẏ = 3xy as either stable, asymptotically stable, or unstable. (The x-axis is invariant and each of the half-planes it determines is a global elliptic sector.) 34. Any two parts of 2.9:2 35. Any two parts of 2.9: :7 and 2.9:8. Hint. Recall Example 4 at the end of the section and ignore the author s hint. 37. For the planar phase portrait shown in Figure 2 of Section 3.3 describe the alpha and omega limit sets of every point in the plane. To most easily present your answer divide the plane into points, curves, and open regions on which every point has the same limit sets, which you can describe using a drawing. For example, one region could be the set of points to the right of the curve y 2 x x3 = 0 (the fish ), and you could say that for x in this region, α(x) = ω(x) =. (Incidentally, the system of differential equations is given in Example 3 of that section, but that is not important.) [Due Wednesday, April 16] 38. The same problem as Problem 37 for the phase portrait of the system in Problem 33. [Due Wednesday, April 16] 39. Retry just 2.9:7 (the first part of Problem 36 above) with the following hints: (a) (Not really a hint, just a comment: the problem as stated is missing the hypotheses that f be continuous and g be differentiable. Assume this.) (b) Instead of the usual auxiliary variable y = ẋ that we have used before introduce y = ẋ + F (x). This will get you the system described. (c) Argue (geometrically if you have to) that the fact that G(0) = 0 (explain why) and the hypothesis G(x) > 0 if 0 < x < ɛ (some positive ɛ) force g (0) 0 and that g changes sign from negative to positive at zero. (d) Now the hypothesis that g(x)f (x) > 0 if 0 < x < ɛ implies that f(0) 0. (e) Assume the inequalities in hints (c) and (d) are strict in order to finish. [Due Wednesday, April 16] 40. Prove as a corollary to the Poincaré-Bendixson Theorem the Poincaré Annular Region Theorem: Suppose A is the diffeomorphic image of an annulus (loosely, it is a ring-shaped region with smooth inner and outer boundary; a typical example is the closed region between two concentric circles). Suppose f is a smooth vector field (i.e., at least C 1 ) defined on a neighborhood of A with the following properties: (a) f points into A (respectively, out of A ) at every point of the topological boundary A of A ; (b) f has no critical point in A A.

6 6 Then f has a cycle that lies wholly within A. (In fact the cycle has to enclose the hole in A in its interior.) 41. Prove Dulac s Criterion: Suppose E R 2 is a simply connected open set, f : E R 2 and B : E R are C r, r 1, and div(bf) is not identically zero on E and is of one sign on E. Then no closed orbit of the system ẋ = f(x) lies wholly within E. (The function B is termed a Dulac function. ) Hint. Resist the urge to apply the Product Rule. 42. Show that the planar system ẋ = 2xy ẏ = 2xy x 2 + y has no closed orbits. Hint. Figure out why no closed orbit can intersect the y-axis. Show that a Dulac function is B(x, y) = 2/x Show that the planar system ẋ = x(1 + x 2 2y 2 ) ẏ = y(1 4x 2 + 3y 2 has no closed orbits by showing that there exist positive constants r and s such that B(x, y) = x r y s is a Dulac function. 44. It can be shown (by constructing a suitable Poincaré Annular Region) that the van der Pol oscillator ẋ = y ẏ = x + λ(1 x 2 )y (studied by Balthasar van der Pol in the 1920 s) has at least one limit cycle for all λ > 0. Use the Dulac function B(x, y) = (x 2 + y 2 1) 1 2 (discovered by Leonid Cherkas in the late 20th century) to show that the cycle is unique. Hint. The function B fails to exist on the unit circle.