1. KAM Theorem

2. Energetic Pendulum  As the energy of a double pendulum increases the invariant tori change. Some tori break up into discrete segments Some new sets of tori emerge replacing old tori.  JJ   l l m m

3. Rational Winding  Smooth curves on the Poincare section are perturbed by the non-linear effects.  An operator T can represent one iteration of the section For finite values there are resonancesFor finite values there are resonances Appears as broken toriAppears as broken tori Islands represent rational winding numbersIslands represent rational winding numbers

4. Elliptic and Hyperbolic  As the tori break up new fixed points emerge. Some are not strictly fixed, but are cyclic.  The points near the fixed points have different properties. Elliptic points are stable with near points staying near Hyperbolic points are unstable  JJ elliptic hyperbolic

5. Poincare-Birkhoff  Consider three nearby tori Rest frame of middle tori Map T s is fixed Perturbation is zero  Turn on perturbation Curves are distorted Angular coordinates unchanged under map Radial may be changed except for even number of fixed points C+C+ C-C- C0C0 CC C+C+ C-C- TsCTsC elliptic hyperbolic

6. Kolmogorov, Arnol’d, and Moser  The KAM theorem relates the strength of the perturbation to the breaking tori. Most tori remain for a small perturbation.Most tori remain for a small perturbation. The set of remaining tori occupy a finite area in the Poincare section.The set of remaining tori occupy a finite area in the Poincare section. The tori that break have rational winding numbersThe tori that break have rational winding numbers The last to break are the most irrationalThe last to break are the most irrational

7. Fraction Expansion  For any irrational number , there is a rational approximation. Improves as s increasesImproves as s increases  Expansions are better as continued fractions. The expansion is uniqueThe expansion is unique r, s integers

8. Irrationality  Truncating the expression after n steps gives a rational fraction.  The slowest convergence is if all a n = 1. This is  the golden mean.This is  the golden mean. It is the “most” irrational number.It is the “most” irrational number. r, s integers

9. KAM Tori  The tori destroyed obey a a relationship with their winding number. K is the measure of all possible winding numbersK is the measure of all possible winding numbers The one valued  is the lastThe one valued  is the last next  JJ