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.grURL: http://www.math.upatras.gr/~crans

OutlineDefinition of the SALI CriterionBehavior of the SALI CriterionOrdered orbitsChaotic orbitsApplications to:2D maps4D maps2D Hamiltonians (Hnon-Heiles system)3D Hamiltonians4D maps of accelerator dynamicsConclusions

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)=(x1(0), x2(0),,xk(0))

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) =(x1(0), x2 (0),, xk(0))

We shall follow the evolution in time of two different initial deviation vectors (e.g. V1(0), V2(0)), and define SALI (Skokos Ch., 2001, J. Phys. A, 34, 10029) as:

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: (Skokos & Bountis, 2003, Prog. Th. Phys. Supp., 150, 439)A general deviation vector can then be written as a linear combination:

We have found that all deviation vectors, V(t), tend to the tangent space of the torus, since a1,a2 oscillate, while |a3|, |a4| ~ t-10 as t.which for B=3A and any E has a second integral: F=(xpy y px)2as, for example, in the Hamiltonian:

Behavior of the SALI for chaotic motion The evolution of a deviation vector can be approximated by : where 1>2> are the Lyapunov exponents. Thus, we derive a leading order estimate for v1(t) :and an analogous expression for v2(t): So we get:+(Skokos and Bountis, 2004, J. Phys. A, 37, 6269)

We have tested the validity of this important approximation

SALI ~ exp[-(1-2)t]

on many examples, like the 3D Hamiltonian:with 1=1, 2=1.4142, 3=1.7321, =0.09Slope= (1-2)1=0.037=Lmax2 =0.011

Applications 2D mapFor =0.5 we consider the orbits:ordered orbit A with initial conditions x1=2, x2=0.chaotic orbit B with initial conditions x1=3, x2=0.

Applications 4D mapFor =0.5, =0.1, =0.1 we consider the orbits:ordered orbit C with initial conditions x1=0.5, x2=0, x3=0.5, x4=0. chaotic orbit D with initial conditions x1=3, x2=0, x3=0.5, x4=0.CDDCAfter N~8000 iterationsSALI~10-8

Applications Hnon-Heiles systemFor E=1/8 we consider the orbits with initial conditions:Ordered orbit, x=0, y=0.55, px=0.2417, py=0Chaotic orbit, x=0, y=-0.016, px=0.49974, py=0Chaotic orbit, x=0, y=-0.01344, px=0.49982, py=0

Applications Hnon-Heiles systemInitial conditions taken on the y axis (x=0, py=0)

Applications Hnon-Heiles system

We consider the 3D HamiltonianApplications 3D Hamiltonianwith A=0.9, B=0.4, C=0.225, =0.56, =0.2, H=0.00765.

Applications 4D Accelerator mapWe consider the 4D symplectic mapdescribing 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 x1 and x3 are the particles horizontal and vertical deflections from the ideal circular orbit, x2 and x4 are the associated momenta and 1 , 2 are related to the accelerators tunes qx , qy by1 =2qx , 2 =2qyOur problem is to estimate the region of stability of the particles motion (the so-called dynamical aperture of the beam)

qx =0.61903 qy =0.4152

SALI quickly detects the thin chaotic layer around the 8-tori resonance

A more global study of the dynamics:We compute orbits with x1 varying from 0 to 0.9, while x2, x3, x4 are the same as in the orbit of the 8-tori resonance Plotting SALI after N=105 iterations for each point on this line we find:

2. We consider 1,922,833 orbits by varying all x1, x2, x3, x4 within spherical shells of width 0.01 in a hypersphere of radius 1. (qx=0.61803 qy=0.4152)

ConclusionsThe Small Alignment Index (SALI) is an efficient criterion for distiguishing ordered from chaotic motion in Hamiltonian flows and symplectic maps of any dimensionality.It has significant advantages over the usual computation of Lyapunov exponents. It can be used to characterize individual orbits as well as chart chaotic and regular domains of different scales and sizes in phase space.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.We are now generalizing our results by adding to our models space charge effects and synchrotron oscillations.

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.