The Periodic Transport
205
Eqs. (8.4) and (8.5) are very useful for the design process to determine
where the chromaticities are generated and where the best locations are to
place the sextupoles for correction. Yet computing them together with the
higher order terms become almost trivial with the Differential Algebraic (DA)
technique. Recall that the tunes are given by
ν x,y =
1
2π
arccos
tr ˆ
M x,y
2
.
The δ dependent tunes are simply
ν x,y (δ) =
1
2π
arccos
tr ˆ
M x,y (δ)
2
,
which contain the tunes and the chromaticities to arbitrary order. For example, ν x,y are the constant part and ξ x,y are the linear coefficients.
8.3 A Glimpse at Nonlinear Effects
Linear motion around a fixed point is completely classified by the two cases
we discussed previously, namely the stable or unstable case. This situation
is fundamentally different in the nonlinear case; it is in fact much
more complicated and interesting, and represents one example of the modern
research field dealing with just such questions.
While this is not at all the place to try to develop a complete understanding
of the nonlinear effects that may appear, let us spend some time to stake the
territory and make some general observations. First we may expect that as
long as the motion is close enough to the fixed point, it is dominated by
linear effects, and depending on whether we have stability or not, we see
either stable elliptic motion or unstable hyperbolic motion. While we may
expect that linearly unstable motion will in most cases also stay unstable if
we consider the nonlinear effects, linear stable motion will not usually
stay nonlinearly stable. In fact, if the amplitudes of the motion become
large, the effects of nonlinearity will become noticeable over-proportionally,
and eventually they will become dominating, in most cases leading to instability for large amplitudes.
One can then try to heuristically separate the phase space into a region that
appears stable for a reasonable number of turns, and a region that appears
unstable. According to the previous arguments, in most cases the stable
region will be near the fixed point, and the unstable region will be away from
the fixed point. The region of transition between the apparently stable and
205
Eqs. (8.4) and (8.5) are very useful for the design process to determine
where the chromaticities are generated and where the best locations are to
place the sextupoles for correction. Yet computing them together with the
higher order terms become almost trivial with the Differential Algebraic (DA)
technique. Recall that the tunes are given by
ν x,y =
1
2π
arccos
tr ˆ
M x,y
2
.
The δ dependent tunes are simply
ν x,y (δ) =
1
2π
arccos
tr ˆ
M x,y (δ)
2
,
which contain the tunes and the chromaticities to arbitrary order. For example, ν x,y are the constant part and ξ x,y are the linear coefficients.
8.3 A Glimpse at Nonlinear Effects
Linear motion around a fixed point is completely classified by the two cases
we discussed previously, namely the stable or unstable case. This situation
is fundamentally different in the nonlinear case; it is in fact much
more complicated and interesting, and represents one example of the modern
research field dealing with just such questions.
While this is not at all the place to try to develop a complete understanding
of the nonlinear effects that may appear, let us spend some time to stake the
territory and make some general observations. First we may expect that as
long as the motion is close enough to the fixed point, it is dominated by
linear effects, and depending on whether we have stability or not, we see
either stable elliptic motion or unstable hyperbolic motion. While we may
expect that linearly unstable motion will in most cases also stay unstable if
we consider the nonlinear effects, linear stable motion will not usually
stay nonlinearly stable. In fact, if the amplitudes of the motion become
large, the effects of nonlinearity will become noticeable over-proportionally,
and eventually they will become dominating, in most cases leading to instability for large amplitudes.
One can then try to heuristically separate the phase space into a region that
appears stable for a reasonable number of turns, and a region that appears
unstable. According to the previous arguments, in most cases the stable
region will be near the fixed point, and the unstable region will be away from
the fixed point. The region of transition between the apparently stable and