Fix-lines and stability G. Franchetti and F. Schmidt GSI, CERN AOC-Workshop - CERN 6/2/2015 G. Franchetti and F. Schmidt1

Introduction: close to 3Q x = N 6/2/2015 G. Franchetti and F. Schmidt2

Stability Domain 6/2/2015 G. Franchetti and F. Schmidt Very useful close to the resonances Example, slow extraction near q=1/3 Analytic theory do not exist far from the continuum limit, but possible near resonances q = It is possible to give an analytic estimate of the border of stability 3

6/2/2015 G. Franchetti and F. Schmidt4 Good for 1 dimensional resonances

In proximity of 6/2/2015 G. Franchetti and F. Schmidt Unstable fix points stable orbits are “bounded” by the unstable fix points 5

every particle inside the circle is “stable” Only at special angles particles out of the circle are “stable” θ Properties of the stability domain 6/2/2015 G. Franchetti and F. Schmidt6

Representation of stability 6/2/2015 G. Franchetti and F. Schmidt7 Θ J 2π Unstable fix point Complete stability Partial stability

Close to Q x + 2Q y = N 6/2/2015 G. Franchetti and F. Schmidt8

6/2/2015 G. Franchetti and F. Schmidt9 Dynamics near near of 3 rd order resonance is of interest

Relevance 6/2/2015 G. Franchetti and F. Schmidt10 normal sext skew sext normal quad skew quad skew sext normal quad One error seed Periodic resonance crossing of a coupled 3 rd order resonance induced by space charge Issues of beam halo prediction and of resonance compensation

Experimental signature of something strange 6/2/2015 G. Franchetti and F. Schmidt11 Resonance:q x +2 q y = 19 G. Franchetti, S. Gilardoni, A. Huschauer, F. Schmidt, R. Wasef

Unfortunately coupled resonances require a 4D treatment of the dynamics 6/2/2015 G. Franchetti and F. Schmidt structure of resonances is controlled by the equations 12

Do we find fix points in proximity of 6/2/2015 G. Franchetti and F. Schmidt ? Wrong concept! These are closed lines in the 4D phase space of which we can only see the projections 13 Forthere are fix-lines F. Schmidt PhD

x x’ y y’ Close to the resonance 6/2/2015 G. Franchetti and F. Schmidt14 4D phase spaceOne turn x x’ y y’ 4D phase space

x x’ y y’ Close to the resonance 6/2/2015 G. Franchetti and F. Schmidt15 4D phase spaceOne turn x x’ y y’ 4D phase space

x x’ y y’ 6/2/2015 G. Franchetti and F. Schmidt16 4D phase spaceOne turn x x’ y y’ 4D phase space After one turn each point on the fix-line is mapped into the fix-line

Fix-line projections 6/2/2015 G. Franchetti and F. Schmidt17 Frank Schmidt PhD

Fundamental Questions 6/2/2015 G. Franchetti and F. Schmidt18 1)Can we characterize these objects with a mathematic expression? 2)Can we predict their extension as functions of lattice nonlinear errors? 3)How many fix-lines do we have ? 4)Can a fix-line be stable or unstable? What does it mean ? 5)Do we have “secondary” tunes? What does it mean ? 6)Do we have a concept of “island”? 7)How are fix-lines related to the stability domain ? (Where Do We Come From? What Are We? Where Are We Going?)

1) Analytic form 6/2/2015 G. Franchetti and F. Schmidt projection of a fix-line close to the 3 rd order coupled resonance a x, a y are the invariant of the fix line, they depends on the “distance” from the resonance, and they are CONSTANT t y parameter that specify the canonical transformation in which makes a x, a y time independent. t parameter that parametrize the fix-line 19 M is an integer that depends on the sign of the “distance” of the resonance

6/2/2015 G. Franchetti and F. Schmidt20 The shape of the line is set by the order of the resonance. The amplitude of the fix-line is determined by a x, a y Therefore ignoring the position of a particle in a fix-line, the fix-line can be identified by a proper pair of a x, a y

2) Fix-line extension function of lattice 6/2/2015 G. Franchetti and F. Schmidt21 with K 2j are the normal integrated sextupolar strength of all errors and correctors Fix-line is given by

3) How many fix-lines ? 6/2/2015 G. Franchetti and F. Schmidt Near 3Q x = N there are 3 unstable fix points Near 2Q y + Q x = N there are infinite fix-lines !! What is the meaning of this ? 22

All the infinite fix-lines 6/2/ axax ayay G. Franchetti and F. Schmidt Collection of all (a x,a y ) satisfying the fix-line equation

4) Fix-lines are stable or unstable? 6/2/ axax ayay unstable fix-lines stable fix-lines G. Franchetti and F. Schmidt

5) The secondary tune: the dynamics off the fix-line 6/2/2015 G. Franchetti and F. Schmidt25 Particle coordinates parameterization In a linear lattice a x, a y are constant Near the resonance becomes time dependent Only on a fix-lineare constant

The problem of the secondary tune 6/2/2015 G. Franchetti and F. Schmidt26 In a 1D resonance stable points means that (x, p x ) The angular velocity  secondary tunes

The problem of the secondary tune 6/2/2015 G. Franchetti and F. Schmidt27 What does it means if we have infinite fix-lines ? (a x, a y )fix-lines in around which fix-line is this particle oscillating ?

The problem of the secondary tune 6/2/2015 G. Franchetti and F. Schmidt28 What does it means if we have infinite fix-lines ? (a x, a y )fix-lines in There is only one special direction of oscillation, which identify a unique fix-line

A very strange stability… 6/2/2015 G. Franchetti and F. Schmidt29 it oscillates around another fix-line - simulations - theory Δa x turns original fix-line = secondary frequency

All the fix-lines 6/2/ axax ayay unstable fix-lines stable fix-lines secondary frequency  0 G. Franchetti and F. Schmidt

6) Islands or not Islands… that is the problem 6/2/2015 G. Franchetti and F. Schmidt31 fix- line “island” ?

7) Stability & fix-lines 6/2/ G. Franchetti and F. Schmidt32 particle coordinates parameterization An interesting, and strange coordinate Two invariants of motion a strange invariant

Consequences of the first invariant 6/2/2015 G. Franchetti and F. Schmidt axax ayay unstable fix-lines stable fix-lines C 33 a x,a y moves along this line

The second invariant: level lines 6/2/2015 G. Franchetti and F. Schmidt34 C is constant

Meaning 6/2/2015 G. Franchetti and F. Schmidt35

Meaning 6/2/2015 G. Franchetti and F. Schmidt36 stable unstable

6/2/2015 G. Franchetti and F. Schmidt37 This point does not move  a x is constant a x, a y = 2 a x + C is a fix-line

Comparison with simulations 6/2/2015 G. Franchetti and F. Schmidt Initial condition of particles that are stable very nice ! 38

Measuring the stable area 6/2/2015 G. Franchetti and F. Schmidt Particle with initial a x, a y fix C, then the allowed ΔΩ is this 39 C = const. axax

Measuring the stable area 6/2/2015 G. Franchetti and F. Schmidt40 C = const. ax 2 pi 0

Stability domain 6/2/2015 G. Franchetti and F. Schmidt axax ayay unstable fix-lines stable fix-lines C 2 pi 0 We can characterize the stability domain with 41 the fix-line is the edge of stability

Stability domain 6/2/2015 G. Franchetti and F. Schmidt42 Here the coordinates are beyond the stable fix-line

Comparison with simulations 6/2/2015 G. Franchetti and F. Schmidt Really good! also in situation off of the “single harmonics limit” From trackingFrom theory 43

Advantages 6/2/2015 G. Franchetti and F. Schmidt fine exploration of the stability domain with tracking 100 processors 2 hours 30x30x100 initial conditions tracking 1000 turns CPU time = 7x10 5 sec. fine exploration of the stability domain with analytic theory CPU time = 3 seconds gain > 2x10 5 stability during infinite turns ! 44

Conclusion/Outlook 6/2/2015 G. Franchetti and F. Schmidt45 Now we know what is the dynamics here Now we know what is the dynamics here Fundamental ingredient to create the asymmetric halo (hence to predict halo extension and maybe density), for high intensity bunched beams Fundamental ingredient to create the asymmetric halo (hence to predict halo extension and maybe density), for high intensity bunched beams S. Aumon, F. Kesting, R. Singh, G. Franchetti

6/2/2015 G. Franchetti and F. Schmidt46 This is the amplitude of an “island”

Stability in a 1D system 6/2/2015 G. Franchetti and F. Schmidt Resonances 47 A. Bazzani et al. Yellow Report