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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

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

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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))

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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) =(x 1 (0), x 2 (0),, x k (0))

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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

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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:

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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:

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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 1 > 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)

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9 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, =0.09 Slope= ( 1 - 2 ) 1 =0.037=L max 2 =0.011

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10 Applications 2D map For =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.

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11 Applications 4D map For =0.5, =0.1, =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

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12 Applications Hnon-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

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

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14 Applications Hnon-Heiles system y pypy E=1/8 t=1000

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15 We consider the 3D Hamiltonian Applications 3D Hamiltonian with A=0.9, B=0.4, C=0.225, =0.56, =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

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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 particles horizontal and vertical deflections from the ideal circular orbit, x 2 and x 4 are the associated momenta and 1, 2 are related to the accelerators tunes q x, q y by 1 =2q x, 2 =2q y Our problem is to estimate the region of stability of the particles motion (the so-called dynamical aperture of the beam)

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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

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

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