Question

1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems (Here r and θ are polar coordinates in the between the nullclines, and on the nullclines themselves equilibrium point plane.) a. b. r )0-2) 2-2 3. Consider the system ay( -1). a. Find and classify all the equilibrium points b. Show that the system has a circular limit cycle e. Determine the stability of the cycle. d. Give an argument showing that the system has no closed trajectories besides the limit cycle you found in part b. (Hint: for parts b, e, and d t change the system into polar coordinates.) 4. Consider the system y(+2) a. Find all the eqibrium points, and linearize the system about each equilibrium point to find the type of the equilibrium pont b. Show that the system is a gradient system, and conclude that has no periodic solutions. c. Sketch the phase portrait. Explain how you determined what the phase portrait looks like d. (For students in Math 5103 only.) Prove that the phase portrait is symmetric about the lie 1/2

Answer 1

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