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MAT332 Dynamical Systems Yury G. Kudryashov 1. Given a vector field with a critical point at the origin and a function L, determine if L is a first integral and check whether L is a Lyapunov function near the origin. (a) x = y + x2, y = -x+ y', L(x,y) =22+y2; (b) ic = 2x - 4x y3, y = -3y + 6xy4, L(x, y) = 23yz; (c) x = siny, y = - sin x, L(x, y) = cos x + cosy; (d) x = siny + 0.1 sinx, y = - sin x + 0.2 siny, L(x, y) = cosx + cosy; (e) x = siny + 0.1 sinx, y = - sin x - 0.1 siny, L(x, y) = cosx + cosy. Sketch phase portraits for the last three systems near the origin and explain what qualitative properties of the pictures relate to different answers in these parts. 2. Find a first integral of the following second-order ODE: x = cosx, then check whether it is a Lyapunov function for x = cos x-0.1x near the point (x, x) = (7,0). 3. Verify that L(x, y) = x2 + y2 is a Lyapunov function for the equation x = y + x3(1 - 2 cosy), y = -x - y'(1 - cos x) near the origin. Use this function to find a specific domain that is surely included in the attracting basin of the origin, i.e. all solutions originating from this domain tend to zero. The domain does not need to be the largest possible. 4. Rewrite the following differential equations in polar coordinates. Use the result to describe the qualitative behaviour of the solutions of these ODEs. Sketch the phase portraits of these ODEs and compare them to the predicted qualitative behaviour. (a) x = 2y, y = -2x; (b) ic = 0.1x -y - x(x2 + y?)2, y = x + 0.ly -y(x2 + y2)2. 5. Find two vector fields (x, y) = (Pi(x, y), Q1(x, y)) and (x, y) = (P2(x, y), Q2(x, y)) and two functions Fi(x, y), F2(x, y) such that each function is a first integral of the corresponding vector field but their sum is not the first integral of the vector field (x, y) = (Pi(x, y) + P2(x, y), Q1(x, y) + Q2(x, y))