1)Find the Nullclines and equilibria for the system x'(t) = r • (1 – x?), g'(t) = - and sketch the phase plane

1)Find the Nullclines and equilibria for the system x'(t) = r • (1 – x?), g'(t) = - and sketch the phase plane. 2. (5 points) Find the Nullclines and equilibria for the system x' (t) = y?, y(t) = ·(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: L(x, y) = (1 – 22)2 + + ve 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'= -VL. (This system is the "Gradient System” for L.) Show that the system in problem (2) can be written as x'(t) y(t) = -1,. (This system is the “Hamiltonian System” for L.) = Ly and These types of systems are very common in physics and engineering. Reflection 1: First Order Des Due Feb 12 at 5pm |-/20 pts ON Reflections 1.5: Ch. 1 and 2 Due Mar 8 at 5pm |-/20 pts Reflection 2: Systems Due Mar 22 at 5pm - 20 pts DN Reflection 3: Theory Due Apr 23 at 11:59pm |-/20 pts