ECMM718 ASSESSED CW 1, CB, MG February 10, 2017 1 DYNAMICAL SYSTEMS AND CHAOS ASSESSED CW 1 Please hand in the assessed questions by 2 PM, Friday 24 February 2017. Please hand in your work via the BART system. Note: If you hand this assignment in late, a mark of 0 will be recorded, unless you make a case for leniency which is accepted by the Mitigation Committee. Q1: (ASSESSED: 25 marks) Recall the definition of a Lipschitz function f : R ! R. Let f : R ! R; f(x) = p 3 x; compute the derivative of f and use this to argue that is not Lipschitz. Provide a direct argument (by contradiction) to show that f is not Lipschitz. Explain if this information can be used to discuss the solution of the following IVP x _ = f(x); x(0) = 0: Q2: (ASSESSED: 25 marks) Find explicitly the flow '(x0; t) for a planar dynamical system given by x _ 1 = −6x1 + 8x2; x _ 2 = −4x1 + 6x2: Using the derived '(x0; t), verify that all solutions move on curves given by ax2 1 + bx1x2 + cx2 2 = constant for some real constants a; b; c that you should find. Justify that we can take a = 1. Given this, verify that b2 > 4c; this means that the curves are hyperbolas. What does this imply about the Liapunov stability and asymptotic stability of the origin for this system? Justify why the equilibrium will persist under sufficiently small perturbations of the system. Q3: (ASSESSED: 25 marks) Consider a vector field f : R2 ! R2: f x y = x( yy− −x2) : Indicate the stationary points and their type (saddle, sink, source, center) of the differential equation: x y __ = f x y : Show that there is exactly one invariant horizontal line L and Compute the solution on L. Let !(z) be the omega-limit set of the trajectory starting at z. For which z = (x; y) 2 R2, does !(z) = (0; 0) hold? Q4: (ASSESSED: 25 marks) Let A; B be n × n matrices. Write down the exponential function eA. Does eA+B = eAeB hold? Justify your answer. Let N = λ 0 λ 1 :ECMM718 ASSESSED CW 1, CB, MG February 10, 2017 2 Compute etN. Consider the differential equation (x y _ _ = = − y x − 2y: (1) Is the origin stable/asyptotically stable? Justify your answer. Compute the explicit solution for (1) with initial values x(0) = y(0) = 1. You could start by writing the derivative in matrix form using the normal Jordan form and the formula for etN obtained above.