Math 4210 / MAE 5790 Final Exam, Spring 2015 (due in Malott 533 by 12 noon on Monday, May 11) This exam is open book and open notes. You may refer to other books if you wish. Also,

feel free to use a computer (it will help for some parts of the test). The only thing that is

not allowed is discussion of the test with other people.

All the problems are deliberately open-ended. I want to test whether youve

learned to ask the right questions about dynamical systems thats almost as

important as being able to give the right answers.

Please explain your reasoning and show the steps in your arguments. Where

relevant, consider such topics as fixed points, closed orbits, limit cycles, stability,

nullclines, trapping regions, symmetry, Liapunov functions, heteroclinic orbits,

quasiperiodicity, chaos, cobweb diagrams, conserved quantities, bifurcations, bifurcation

diagrams, stability diagrams, and so on. Use geometrical, analytical, and numerical

arguments, as appropriate.

Grades will be available online by 4 pm on May 15. Good luck!

1. (20 pts) Draw the phase portrait for the system

x = x (x2 + y2 2) y = y(x2 + y2 3x +1).

(Dont just solve it on a computer. Please analyze the system using methods we developed in the course.) 2. (20 pts) Analyze the three-dimensional system

&x = yz&y = z&z = xy.

3. (30 pts) Let a > 0 be an arbitrary positive real number, and consider the following sequence:

x1 = a, x 2 = aa , x 3 = a

aa( ) , K

where the general term is

xn+1 = a(xn ) .

Analyze the long-term behavior of the sequence xn{ } as n , and discuss how that long-term behavior depends on a . For instance, show that for certain values of a , the terms xn tend to some limiting value. How does that limit depend on a ? For which values of a is the long-term behavior more complicated? What happens then? 4. (30 pts) Analyze the system

x = x b x y1+ x

,

y = y x1+ x

ay

where x, y 0 and a,b > 0 are parameters. Hint: To get started, see problem 8.2.9. But try to go even farther. For example, you should investigate (both analytically and numerically) the systems saddle-node bifurcations, and other bifurcations, as you vary the parameters a and b . I dont expect you to solve this problem completelydo what you can, while spending a reasonable amount of time on it.