1. Review of dynamic aperture studies Yiton Yan SLAC

2. Dynamic aperture (DA) studies tracking Default method: symplectic element by element tracking – however, attention will be more on nonlinear maps. Taylor-map tracking – accurate, but not exactly symplectic Generating-type mixed-variable map tracking – implicit procedure, accurate, symplectic, but with singularities. Irwin factorization Integrable polynomial Resonance basis map and nPB tracking Optimization of dynamic aperture – parameterized Lie map for Resonance basis map vs. direct DA optimization (which uses GA).

3. SSC collider ring Largest ring for DA study. Highly nonlinear small DA small interest region fast convergence Accurate one-turn Taylor map

4. Direct 12 th -order and 11 th -order Taylor map tracking Compare the 11 th -order with the 12 th –order Taylor map tracking Not exactly symplectic, seems OK with this much turn?

5. Directly 12 th -order Taylor map tracking Compared with element-by- element tracking 5 cm dipole

6. SSC 4-cm dipole case 1 million turn 11 th -order Taylor map Seems OK.

7. But 10 th -order Taylor map tracking shows biased artificial diffusion due to not syplectic. But is it accurate?

8. Lie transformation Dragt-Finn factorization Can be trasnferred back to Taylor map with one-order (or more orders) higher

9. The 10 th -order Taylor map is re-expanded to the 11 th -order Taylor map and works! Degree of accuracy symplecticity

10. Mixed variable map and Generating function It is a mixed variable Vps obtained through, implicitly, generating function, but no need to get the generating function. It retains the same accuracy compared to the same-order Taylor It is symplectic at any truncated order Tracking time is about twice of the same-order Taylor map tracking, but of course much faster compared to several orders higher Taylor map. But with singularities - OK.

11. Conversion

12. Implicit mixed-variable power-series tracking Direct iteration will do.

13. The same SSC 4-cm diameter dipole injection lattice with mixed-variable map tracking

14. Irwin (Kick) Factorization A map is completely reconstructed into a minimum number of kick maps such with an order by order accuracy while push the errors to higher order. Very elegant idea. Very fast. However, Spurious terms present unknown accuracy concern

15. Explicit integrable polynomial where Symplectic integrator is then used for separating the integrable polynomials. Lower homogeneous order terms uses higher order symplectic integrator while higher homogenous order terms uses lower order symplectic integrator or even just separate them. Computation time is slower than minimized-term Irwin factorization. However, it is more naturally close to the original Lie map and therefore generally less concerns about accuracy.

16. Use of symplectic integrator for separation of homgeneous integrable polynomial

17. Resonance basis map and nPB tracking for PEP-II HER and LER dynamics aperture stuides Taylor map can be trasnformed into a Deprit-type Lie map and then further transformed into the following resonance-basis map This is a goal-moving soccer game

18. nPB tracking nPB converge very fast. Not exactly symplectic. OK for PEP-II HER and LER tracking 1000 turns are needed for PEP-II due to damping

19. PEP II Swamp plot See resonances Easier to choose an working tune

20. Side issue – tune shift with amplitude

21. Normalized tune shift and resonance driving terms They are analytical Can this be used for indirect dynamic aperture optimization otherwise the direct dynamic aperture optimization is not analytical and so must use GA which usually take longer time. All done. Thanks.