Phys. 4267/6268 Final Exam Due 04/30/09 (12:00 noon)
Problem 1
Consider a two dimensional system:
˙x=x(3 −2x−y),
˙y=y(2 −x−y).
1. Find the fixed points and determine their stability.
2. Draw the nullclines and sketch the phase portrait.
3. Sketch the basin(s) of attraction of the stable fixed point(s).
Problem 2
The dynamics of a system is determined by the following ODE
¨x+ω2x+x3+ ˙x3= 0,
with ω=O(1).
1. Find the fixed point of this system.
2. Use perturbation theory to find (approximately) the solution near the fixed point. Hint: You may want to use
rescaling in order to recast the equation in the form suitable for applying the perturbation theory.
3. Determine the stability of the fixed point using the solution you have found.
4. Find the Lyapunov function for this system and use it to verify the answer you got for the stability.
Problem 3
Consider the following system:
˙x=x³a−px2+y2´−y,
˙y=y³a−px2+y2´+x,
˙z=−z,
where ais a real parameter.
1. Define an appropriate Poincar´e section and compute the corresponding Poincar´e map.
2. Show that the system has a periodic orbit for some aand determine its stability.
3. Find al l the Floquet multipliers for the periodic orbit.
4. What is the largest Lyapunov exponent of this system (for all a)?
5. If the system undergoes a bifurcation as avaries, determine the critical value of a. Is the bifurcation of one of
the standard types? If not, why?