1. Applications of the SALI Method for Detecting Chaos and Order in Accelerator Mappings Tassos Bountis Department of Mathematics and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, Patras, Greece Work in collaboration with Haris Skokos and Christos Antonopoulos E-mail: tassos50@otenet.gr URL: http://www.math.upatras.gr/~crans

2. 2 Outline Definition of the SALI Criterion Behavior of the SALI Criterion Ordered orbits Chaotic orbits Applications to: 2D maps 4D maps 2D Hamiltonians (Hénon-Heiles system) 3D Hamiltonians 4D maps of accelerator dynamics Conclusions

3. 3 Definition of the Smaller Alignment Index (SALI) Consider the k-dimensional phase space of a conservative dynamical system (e.g. a Hamiltonian flow or a symplectic map), which may be continuous in time (k =2N) or discrete: (k=2M). Suppose we wish to study the behavior of an orbit in that space X(t), or X(n), with initial condition : X(0)=(x 1 (0), x 2 (0),…,x k (0))

4. 4 To study the dynamics of X(t) - or X(n) – one usually follows the evolution of one deviation vector V(t) or V(n), by solving: Either the linear (variational) ODEs (for flows) Or the linearized (tangent) map (for discrete dynamics) where DF is the Jacobian matrix evaluated at the points of the orbit under study. Consider a nearby orbit X’ with deviation vector V(t), or V(n), and initial condition V(0) =X’(0) – X(0) =(Δx 1 (0), Δx 2 (0),…, Δx k (0))

5. 5 We shall follow the evolution in time of two different initial deviation vectors (e.g. V 1 (0), V 2 (0)), and define SALI (Skokos Ch., 2001, J. Phys. A, 34, 10029) as: Two different behaviors are distinguished: 1)The two vectors V 1, V 2 tend to coincide or become opposite along the most unstable direction SALI(t) →0 and the orbit X(t) is chaotic 2)There is no unstable direction, the deviation vectors tend to become tangent to a torus and SALI(t) ~ constant and the orbit X(t) is regular

6. 6 Behavior of SALI for ordered motion Suppose we have an integrable 2D Hamiltonian system with integrals H and F. Form a vector basis of the 4D phase space: Choosing 2 vectors along the 2 independent directions: f H =( H px, H py, -H x, -H y ), f F =( F px, F py, -F x, -F y ) and 2 vectors perpendicular to these directions: H=(H x, H y, H px, H py ), F=(F x, F y, F px, F py ) (Skokos & Bountis, 2003, Prog. Th. Phys. Supp., 150, 439) A general deviation vector can then be written as a linear combination:

7. 7 We have found that all deviation vectors, V(t), tend to the tangent space of the torus, since a 1,a 2 oscillate, while |a 3 |, |a 4 | ~ t -1 →0 as t→∞. which for B=3A and any E has a second integral: F=(xp y – y p x ) 2 as, for example, in the Hamiltonian:

8. 8 Behavior of the SALI for chaotic motion Behavior of the SALI for chaotic motion The evolution of a deviation vector can be approximated by : where σ 1 >σ 2 >… are the Lyapunov exponents. Thus, we derive a leading order estimate for v 1 (t) : and an analogous expression for v 2 (t): So we get: +… (Skokos and Bountis, 2004, J. Phys. A, 37, 6269)

9. 9 We have tested the validity of this important approximation SALI ~ exp[-(σ 1 -σ 2 )t] on many examples, like the 3D Hamiltonian: with ω 1 =1, ω 2 =1.4142, ω 3 =1.7321, Η=0.09 Slope= –(σ 1 -σ 2 ) σ 1 =0.037=L max σ 2 =0.011

10. 10 Applications – 2D map For ν=0.5 we consider the orbits: ordered orbit A with initial conditions x 1 =2, x 2 =0. chaotic orbit B with initial conditions x 1 =3, x 2 =0.

11. 11 Applications – 4D map For ν=0.5, κ=0.1, μ=0.1 we consider the orbits: ordered orbit C with initial conditions x 1 =0.5, x 2 =0, x 3 =0.5, x 4 =0. chaotic orbit D with initial conditions x 1 =3, x 2 =0, x 3 =0.5, x 4 =0. C D D C After N~8000 iterations SALI~10 -8

12. 12 Applications – Hénon-Heiles system For E=1/8 we consider the orbits with initial conditions: Ordered orbit, x=0, y=0.55, p x =0.2417, p y =0 Chaotic orbit, x=0, y=-0.016, p x =0.49974, p y =0 Chaotic orbit, x=0, y=-0.01344, p x =0.49982, p y =0

13. 13 Applications – Hénon-Heiles system t=1000 t=4000 Initial conditions taken on the y – axis (x=0, p y =0)

14. 14 Applications – Hénon-Heiles system y pypy E=1/8 t=1000

15. 15 We consider the 3D Hamiltonian Applications – 3D Hamiltonian with A=0.9, B=0.4, C=0.225, ε=0.56, η=0.2, H=0.00765. Behavior of the SALI for ordered and chaotic orbits Color plot of the subspace x, p x with initial conditions y=z=p y =0, p z >0

16. 16 Applications – 4D Accelerator map We consider the 4D symplectic map describing the instantaneous “kicks” experienced by a proton beam as it passes through magnetic focusing elements of the FODO cell type (see Turchetti & Scandale 1991, Vrahatis & Bountis, IJBC, 1996, 1997). Here x 1 and x 3 are the particle’s horizontal and vertical deflections from the ideal circular orbit, x 2 and x 4 are the associated momenta and ω 1, ω 2 are related to the accelerator’s tunes q x, q y by ω 1 =2πq x, ω 2 =2πq y Our problem is to estimate the region of stability of the particle’s motion (the so-called dynamical aperture of the beam)

17. 17 In Vrahatis and Bountis, IJBC, 1996, 1997, we computed near the boundary of escape stable periodic orbits of very long period, e.g. 13237 (!). Note that SALI converges long before L max becomes zero. q x =0.61903 q y =0.4152

18. 18 Making a small perturbation in the x - tune Δq x =-0.001, i.e. with q x =0.61803 q y =0.4152, we detect ordered motion near an 8 – tori resonance

19. 19 SALI quickly detects the thin chaotic layer around the 8- tori resonance points with |x 3 |<0.04

20. 20 Near the beam’s dynamical aperture, SALI quickly detects a chaotic orbit, which eventually escapes after about 82000 iterations (q x =0.628615 q y =0.4152)

21. 21 A more “global” study of the dynamics: 1.We compute orbits with x 1 varying from 0 to 0.9, while x 2, x 3, x 4 are the same as in the orbit of the 8-tori resonance Plotting SALI after N=10 5 iterations for each point on this line we find:

22. 22 2. We consider 1,922,833 orbits by varying all x 1, x 2, x 3, x 4 within spherical shells of width 0.01 in a hypersphere of radius 1. (q x =0.61803 q y =0.4152)

23. 23 Conclusions 1.The Small Alignment Index (SALI) is an efficient criterion for distiguishing ordered from chaotic motion in Hamiltonian flows and symplectic maps of any dimensionality. 2.It has significant advantages over the usual computation of Lyapunov exponents. 3.It can be used to characterize individual orbits as well as “chart” chaotic and regular domains of different scales and sizes in phase space. 4.It has been successfully applied to “trace out” the dynamical aperture of proton beams in 2 space dimensions, passing repeatedly through FODO cell magnetic focusing elements. 5.We are now generalizing our results by adding to our models space charge effects and synchrotron oscillations.

24. 24 References Vrahatis M., Bountis, T. et al. (1996) IJBC, vol. 6 (8), 1425. Vrahatis M., Bountis, T. et al. (1997), IJBC, vol. 7 (12), 2707. Skokos Ch. (2001) J. Phys. A, 34, 10029. Skokos Ch., Antonopoulos Ch., Bountis T. C. & Vrahatis M. (2003) Prog. Theor. Phys. Supp., 150, 439. Skokos Ch., Antonopoulos Ch., Bountis T. C. & Vrahatis M. (2004) J. Phys. A, 37, 6269.