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W. Herr and E. Forest
The same techniques can be employed to maximise the dynamic aperture, in
the ideal case beyond the limits of the physical aperture. Usually one can define
tolerances for the allowed multipole components of the magnets or the optimized
parameters for colliding beams when the dominant non-linear effect comes from
beam-beam interactions.
3.8.4.1 Long Term Stability and Chaotic Behaviour
In accelerators such as particle colliders, the beams have to remain stable for many
hours and we may be asked to answer the question about stability for as many as 10 9
turns in the machine. This important question cannot be answered by perturbative
techniques. In the discussion of Poincare surface-of-section we have tasted the
complexity of the phase space topology and the final question is whether particles
eventually reach the entire region of the available phase space.
It was proven by Kolmogorov, Arnol’d and Moser (KAM theorem) that for
weakly perturbed systems invariant surfaces exist in the neighbourhood of integrable
ones. Poincare gave a first hint that stochastic behaviour may be generated in nonlinear systems. In fact, higher order resonances change the topology of the phase
space and lead to the formation of island chains on an increasingly fine scale.
Satisfactory insight to the fine structure of the phase space can only be gained with
numerical computation. Although the motion near resonances may be stochastic,
the trajectories are constrained by nearby KAM surfaces (at least in one degree of
freedom) and the motion remains confined.
3.8.4.2 Practical Implications
In numerical simulations where particles are tracked for millions of turns we would
like to determine the region of stability, i.e. dynamic aperture. Since we cannot
track ad infinitum, we have to specify criteria whether a particle is stable or not.
A straightforward method is to test the particle amplitudes against well defined
apertures and declare a particle lost when the aperture is reached. A sufficient
number of turns, usually determined by careful testing, is required with this method.
Usually this means to find the particle survival time as a function of the initial
amplitude. In general the survival time decreases as the amplitude increases and
should reach an asymptotic value at some amplitude. The latter can be identified as
the dynamic aperture.
Other methods rely on the assumption that a particle that is unstable in the long
term, exhibits features such as a certain amount of chaotic motion.
Typical methods to detect and quantify chaotic motion are:
• Frequency Map Analysis [24, 25].
• Lyapunov exponent [26].
• Chirikov criterion [27].
W. Herr and E. Forest
The same techniques can be employed to maximise the dynamic aperture, in
the ideal case beyond the limits of the physical aperture. Usually one can define
tolerances for the allowed multipole components of the magnets or the optimized
parameters for colliding beams when the dominant non-linear effect comes from
beam-beam interactions.
3.8.4.1 Long Term Stability and Chaotic Behaviour
In accelerators such as particle colliders, the beams have to remain stable for many
hours and we may be asked to answer the question about stability for as many as 10 9
turns in the machine. This important question cannot be answered by perturbative
techniques. In the discussion of Poincare surface-of-section we have tasted the
complexity of the phase space topology and the final question is whether particles
eventually reach the entire region of the available phase space.
It was proven by Kolmogorov, Arnol’d and Moser (KAM theorem) that for
weakly perturbed systems invariant surfaces exist in the neighbourhood of integrable
ones. Poincare gave a first hint that stochastic behaviour may be generated in nonlinear systems. In fact, higher order resonances change the topology of the phase
space and lead to the formation of island chains on an increasingly fine scale.
Satisfactory insight to the fine structure of the phase space can only be gained with
numerical computation. Although the motion near resonances may be stochastic,
the trajectories are constrained by nearby KAM surfaces (at least in one degree of
freedom) and the motion remains confined.
3.8.4.2 Practical Implications
In numerical simulations where particles are tracked for millions of turns we would
like to determine the region of stability, i.e. dynamic aperture. Since we cannot
track ad infinitum, we have to specify criteria whether a particle is stable or not.
A straightforward method is to test the particle amplitudes against well defined
apertures and declare a particle lost when the aperture is reached. A sufficient
number of turns, usually determined by careful testing, is required with this method.
Usually this means to find the particle survival time as a function of the initial
amplitude. In general the survival time decreases as the amplitude increases and
should reach an asymptotic value at some amplitude. The latter can be identified as
the dynamic aperture.
Other methods rely on the assumption that a particle that is unstable in the long
term, exhibits features such as a certain amount of chaotic motion.
Typical methods to detect and quantify chaotic motion are:
• Frequency Map Analysis [24, 25].
• Lyapunov exponent [26].
• Chirikov criterion [27].