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Periodic orbits and their linear stability

According to Poincaré Birkhoff's theorem, there exists for each pair (p,q) with p;SPMgt;1 and tex2html_wrap_inline595 and 0;SPMlt;q/p;SPMlt;1 a periodic orbit of period p which winds around the table q times. These periodic orbits are called Birkhoff periodic orbits. In general, there exist many more orbits of period p. It is an open question whether the set of periodic orbits can form a set of positive Lebesgue measure in the phase space. There exist some results for small periods [20, 19]. We cannot decide this question for our case.
Even if the set of periodic orbits forms a set of zero Lebesgue measure, they still generally have a significant influence on the global phase space picture. A stable periodic orbit keeps a whole neighborhood nearby stable, which prevents ergodicity. A first step in understanding the stability of the orbits is finding out whether they are linear stable. The stability of periodic orbits of small order can be decided by hand. We will treat orbits of period 2 and 4.

propo103

Proof. Look at an orbit of period 2 hitting the table at points tex2html_wrap_inline635 with curvatures tex2html_wrap_inline637 and tex2html_wrap_inline639 and let l be the distance between the two points. A computation of the trace tex2html_wrap_inline643 of the Jacobean of tex2html_wrap_inline645 (see Proposition 4 of [22] or Proposition 3 of [23]) shows that the orbit is elliptic if and only if tex2html_wrap_inline647 or tex2html_wrap_inline649 .

Consider a 2-orbit of length tex2html_wrap_inline651 , that meets the billiard table at points with radii of curvature tex2html_wrap_inline653 and tex2html_wrap_inline655 , with tex2html_wrap_inline657 . If one of the inequalities tex2html_wrap_inline659 or tex2html_wrap_inline661 is satisfied, then the orbit is hyperbolic, while if either tex2html_wrap_inline663 or tex2html_wrap_inline665 is satisfied, the orbit is elliptic. For our 2-orbits, tex2html_wrap_inline667 , so we get elliptic orbits when tex2html_wrap_inline669 and hyperbolic orbits when tex2html_wrap_inline671 .

We look first at the horizontal 2-orbit tex2html_wrap_inline673 . The vertical 2-orbit is obviously identical to the horizontal one. The horizontal orbit has length 2. The curvature at the endpoints is tex2html_wrap_inline323 , 1, and 0 for tex2html_wrap_inline677 , respectively. This gives radii of curvature of 0, 1, and tex2html_wrap_inline323 for tex2html_wrap_inline677 , respectively. For tex2html_wrap_inline683 and the orbit is hyperbolic. And for tex2html_wrap_inline685 , and the orbit is elliptic. So the horizontal orbit is hyperbolic if p;SPMlt;2 and elliptic if p;SPMgt;2. The diagonal orbit has tex2html_wrap_inline691 . The curvature at the endpoints is tex2html_wrap_inline693 . The radius of curvature is then tex2html_wrap_inline695 . The orbit is elliptic if tex2html_wrap_inline669 , or

displaymath699

Multiplying both sides by tex2html_wrap_inline701 , we get tex2html_wrap_inline703 , which means p;SPMlt;2. So the diagonal 2-orbit is elliptic if p;SPMlt;2. Similarly, the diagonal 2-orbit is hyperbolic if p;SPMgt;2.
End of the proof.

propo137

Proof. We calculate the Hessian for tex2html_wrap_inline733 . Let tex2html_wrap_inline735 be the central angle for the tex2html_wrap_inline737 point of the orbit, tex2html_wrap_inline739 be the angle between the path of the particle leaving the tex2html_wrap_inline741 point and the directed tangent to the p-curve at that point. tex2html_wrap_inline743 gives the distance of the straight line connecting the points given by tex2html_wrap_inline735 and tex2html_wrap_inline747 , and tex2html_wrap_inline749 is the curvature at the tex2html_wrap_inline737 point. The Hessian is then

displaymath753

with

displaymath755

In our case we have by symmetry tex2html_wrap_inline757 and all of the tex2html_wrap_inline759 . The determinant of the Hessian then simplifies to tex2html_wrap_inline761 . Define tex2html_wrap_inline763 . The orbit is hyperbolic if D;SPMgt;2 and and elliptic if D;SPMlt; 2 (see [22]). For the symmetric 4-orbits we have

displaymath769

First look at the tex2html_wrap_inline733 -orbit. We get with tex2html_wrap_inline773 , and tex2html_wrap_inline775 , so that

displaymath777

We have already calculated tex2html_wrap_inline779 at these points, so we get

displaymath781

displaymath783

So we get that the tex2html_wrap_inline733 -orbit is hyperbolic for p;SPMlt;2 and parabolic for tex2html_wrap_inline789 . (Numerical experimentation suggests that the tex2html_wrap_inline733 -orbit is stable for p;SPMgt;2).

For the orbit tex2html_wrap_inline795 , all four points have tex2html_wrap_inline797 . We then calculate

displaymath799

and

displaymath801

The tex2html_wrap_inline803 -orbit is elliptic if tex2html_wrap_inline805 and tex2html_wrap_inline807 . So the orbit is elliptic if 1;SPMlt;p;SPMlt;2 and hyperbolic for p;SPMgt;2.
End of the proof.

Next: Invariant curves and topological Up: Billiards in the unit Previous: Numerical computation of orbits

Oliver Knill, Jul 10 1998