 # 1.) Let x˙ = y 2 − x − 1 y˙ = y 2 − 2x a.)...

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1.) Let x˙ = y 2 − x − 1 y˙ = y 2 − 2x a.) Find all fixed points and classify them. b.) Sketch the nullclines and determine the motion on them. c.) Draw a reasonable phase portrait. 2.) Prove that if x˙ = 3x + y 3 − xy 2 y˙ = 6y + x 3 − x 2 y has a periodic solution then it must cross a circle of radius 3 centered at the origin. 2 3.) a.) Prove that all solutions to are bounded. x¨ + 4x + 3x 5 = 0 b.) If x(0) = 2 and x ′ (0) = 1, give the least upper bound for |x ′ (t)| in the solution to x¨ + 4x + 3x 5 = 0 (that is, give the maximum possible value of |x ′ (t)|) 4.) Prove that x˙ = −y y˙ = x − y(1 − x 2 − 2y 2 ) hasaperiodic solution. 5.) Let x˙ = x(e 2−x− y − 1) y˙ = y(e 3− x−y −1) where x(t),y(t) ≥ 0 represent the population of two species. a.) Find the four fixed points of the system. b.) Classify each of the fixed points from(a). c.) Sketch the nullclines and draw a reasonable phase portrait. 6.) x˙= y˙= y y −x + 1 + βy3 (1 + (x − 1)2 ) Here β > 0 a.) Find all fixed points of the system. Why does the linear classification fail? b.) Find a Lyapunov function of the form V (x, y) = (x − 1)2 + ay2 and use it to classify the fixedpoints. 7.) Suppose u is an eigenvector for the matrix A. Show that u is also an eigen- 3 vector for the matrix exponential e A and find the corresponding eigenvalue. 4 t 3... x 8.) Consider thefollowingsystemofdifferential equations x˙ = −x − x 2 y y˙ = µy − 2z − y(y 2 + z 2 ) z˙ = 2y + µz − z(y 2 + z 2 ) a.) Let µ = −1. Let S denote the set of initial conditions close to the origin whose solution goes to (0,0,0) as t → ∞. Determine the dimension of S. b.) Describe the qualitative change that occurs as µ varies from −1 to 1. 9.) In this problem we will focus on the differential equation x + at 2 x¨ + btx˙ + cx = 0 where a, b, c are real constants. Consider the substitution y = tx˙ and z = ty˙ a.) Show that t 2 x¨ = z −y b.) Show that tz˙ = t 3 ... + 3t 2 x¨ + tx˙ c.) Use (a), (b), and the original differential equation to show that tz˙ = (3 − a)(z − y) + (1 − b)y − cx tx˙ x x d.) Show that the differential equation is equivalent to ty˙ tz˙ = A y z Give an explict expression for the 3×3 matrix A. 5 10.) Let x˙ = −y + x(x 2 + y 2 )* sin(pi/sqrt(x^2+y^2)) y˙ = x + y(x 2 + y 2 )* sin(pi/sqrt(x^2+y^ 2)) away from the origin (define the right-hand side of each to be 0 at the origin). a.) Show that there are infinitely many periodic solutions. b.) Classify the stability of each periodic solution. c.) Explain why there is no meaningful classification to assign for the fixed point (0,0), even in the nonlinear picture.

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