INTRODUCTION TO QUANTUM CHAOS
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     NBIfAFG Fysik 711B			         Fall semester 1999 
     Henrik Bruus, Predrag Cvitanovic', Andy Jackson
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Problem session 9:  10:30 - 12:15 tue  9 nov 99, Aud. C - NBI Blegdamsvej 

To go beyond where Gutzwiller was in 1971 we have to 
find periodic orbits and determine their stabilities.

Exercises:
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Numerical computation of linearized stability
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         sect 5.1  implement the integration of the linear stability
		Jacobian matrix for the Roessler flow. Compute its
		eigenvalues for a few trajectories with evalution time
		roughly comparable to one or a few returns to a 
		Poincare section.

         sect 2.2.2 (recommended, but optional:) implement Ronnie's
		(originally Henon's) clever method for fast and
		accurate construction of Poincare sections.

Qualitative dynamics
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	exer 2.1  (easy:) determine all possible itineraries for prime cycles 
		up to length 4 for the one-dimensional bimodal repeller 
		by doodling.
	exer 4.16 (cute, but optional:) count prime binary cycles by 
		thinking

Numerical computation of periodic orbits
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         exer 11.2 (a) and (b) only; find all prime cycles up to length 4
		for the one-dimensional repeller either by inverse
		iteration, or by the Newton method. Tabulate also the
		the cycle stabilities and the cycle Lyapunovs 
		\lambda = ln |\Lambda_p|/n_p.
	
         sect  6.2 Implement the Newton routine for finding periodic points.
		Find the  Roessler Poincare section fixed point (the one
		inside the strange attractor) and compute the eigenvalues
		of the linearized stability matrix.  If one eigenvalue is
		not 1 with high accuracy, you are not there yet.

	 sect 5.1.3, 5.3.1 (optional:) try plotting the stable and 
		unstable manifolds for this Roessler Poincare section
		fixed point. If successful, try constructing a return
		map from the unstable manifold, with the length measured
		along the manifold. Return map should look sort of like
		a parabola.

         sect 2.1.2 (optional:) try using your newly won skills on collinear
                helium. Do not be surprised that all trajectories run away
                from you! - however, periodic ones shoudl be very
		robust.

If we manage to generate the colinear Helium periodic orbits in next
two weeks, we are on the way to compute helium spectrum. I'll be
proud of you.


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