Math 134 Spring 2021: Mini-Exam 7 Please upload your answers by 1:30pm on Tuesday, 5/11 • You must show complete work to earn credit

Math 134 Spring 2021: Mini-Exam 7 Please upload your answers by 1:30pm on Tuesday, 5/11 • You must show complete work to earn credit. Open book, Open note. • You can use the internet, but LIST which resources you used. 1. (5 points) Find the Nullclines and equilibria for the system x0 (t) = x · (1 − x2 ), y 0 (t) = −y 3 and sketch the phase plane. 2. (5 points) Find the Nullclines and equilibria for the system x0 (t) = y 3 , y 0 (t) = x · (1 − x2 ) and sketch the phase plane. 3. (4 points) For BOTH of the systems above, identify the equilibrium points, calculate the Jacobian, and explain what the linearization theorem tells us about the stability of each equilbrium point. Explain your results in your own words. 4. (4 points) Verify that the following function is a Lyapunov function for both systems and check if it is a strict Lypaunov function: 1 1 L(x, y) = (1 − x2 )2 + y 4 4 4 Based on this Lyapunov function, classify the equilibria you found in the previous part as ‘Asymptotically Stable’, ‘Stable’, or ‘Inconclusive’. 5. (2 points) Show that the system in problem (1) can be written as X 0 = −∇L. (This system is the “Gradient System” for L.) Show that the system in problem (2) can be written as x0 (t) = Ly and y 0 (t) = −Lx . (This system is the “Hamiltonian System” for L.) These types of systems are very common in physics and engineering.