Formalism for chaotic behavior of the bunched beam
Using Cesaro mid of Fourier series the quasi-linear Vlasov’s equation is transformed to the integral Fredholm equation. New results on the oscillatory behavior of solution are obtained. An extension to perturbing equation is also included.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Formalism for chaotic behavior of the bunched beam / Z. Parsa, V. Zadorozhny // Вопросы атомной науки и техники. — 2001. — № 6. — С. 247-250. — Бібліогр.: 10 назв. — англ. |
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| author | Parsa, Z. Zadorozhny, V. |
| author_facet | Parsa, Z. Zadorozhny, V. |
| citation_txt | Formalism for chaotic behavior of the bunched beam / Z. Parsa, V. Zadorozhny // Вопросы атомной науки и техники. — 2001. — № 6. — С. 247-250. — Бібліогр.: 10 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Using Cesaro mid of Fourier series the quasi-linear Vlasov’s equation is transformed to the integral Fredholm equation. New results on the oscillatory behavior of solution are obtained. An extension to perturbing equation is also included.
|
| first_indexed | 2025-12-07T17:46:31Z |
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FORMALISM FOR CHAOTIC BEHAVIOR OF THE BUNCHED BEAM
Z. Parsa
Brookhaven National Laboratory, Physics Department 510 A, Upton, NY 11973
e-mail: Parsa@bnl.gov
V. Zadorozhny
Institute of Cybernetic, National Academy of Sciences of Ukraine
e-mail: Zvf@umex.istrada.net.ua
Using Cesaro mid of Fourier series the quasi-linear Vlasov’s equation is transformed to the integral Fredholm
equation. New results on the oscillatory behavior of solution are obtained. An extension to perturbing equation is
also included.
PACS: AMS 94B, 81P99, 57M20
1. INTRODUCTION
The self-consistent Vlasov equation is one of the
most frequently used equations for the time dependent
description of many-particle systems. Especially in
nuclear physics this equation has been employed to
describe multifragmentation phenomena and collective
oscillations. It is apparently not widely known that there
exists an analytical solvable model from which the
effects of self-consistency can be studied. Here such a
model is presented which shows that self-consistency
can lead to self-focused and acceleration of bunched
beam.
The kinetic equation for the beam distribution
function f has the form
0=∂+∂+
∂
∂ fFfv
t
f
vlr , (1)
where ),,( 321 xxxr = is a three-dimensional vector,
3,2,1, =ixi are Cartesian coordinates;
),,( 321 vvvv = is their velocity. In this solution the
Lorentz force ])[1( Hv
c
EqFl ×+= acting on a
nonrelativistic driving beam. Here q is the particle
charge and E is the electric field: 21 EEE += where
1E is given field and 2E is generated by a charged
bunch, H is the magnetic field and 21 HHH +=
too. The fields should satisfy the Maxwell system
∫
∫
∞
∞
=
∂
∂−+
+==><
)(
)(
,41]),[1
(
),,(
),,,(
dvfvq
ct
E
c
HrotHv
c
E
m
e
tvrf
vdtvvrfv
v
π
∫
∞
π=
)(
4 dvfqEvdi (2)
),0,01( ==
∂
∂+ Hdiv
t
H
c
Erot
where ,),,(),( dvvrtfqrt ∫=ρ
∫= dvvrtfvqrtj ),,(),( are a charge and a current
of the beam, c is the speed of light.
If we formally let 0, =∞= Hc and replace qE
by E and qρ by ρ , we get the Vlasov-Poisson
system:
0),( =∂+∂+∂ frtEfvf vxt (3)
),(4),( rtrtU π ρ−=∆ (4)
∫= dvvrtf ),,(ρ .
This system was considered by A.A. Vlasov in his
treatise on many-particle theory and plasma physics [1].
To determine the focusing and accelerating fields we
use the following auxiliary postulate.
The postulate of the existence electric and magnetic
fields realizing any motion of the bunch beam: it is
shown [2] that for any field of the velocity of charged
particle exist electric & magnetic fields that yields same
velocity field satisfying Maxwell’s equations. This
postulate makes it feasible to construct the optimal
fields using the optimal control theory [3].
2. APPROXIMATE SOLUTION OF
VLASOV’S EQUATION
Letting ticvrfvrtf ω−= ),(),,( 0 into (1) yields
0000 fifFfvfL vlr ω=∂+∂≡ . (5)
Suppose the solution Eq. (5) can be represented in the
form
∑
∞
∞−=
=
k
ikx
k ecf 0 , (6)
where the vector ),,,( 621 kkkk = , ik is an
integer, 6,,2,1 =i ; the vector vrx += , i.e. it’s sum
of the vector of position and vector of velocity of the
particle orbit, ∑=
6
1
ii xkkx ,
ii rx = , 3,2,1=i ; ii vx = , 6,5,4=i .
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 247-250. 247
Let us assume that the the vector r falls into a domain
1∆ , the vector 2∆∈r , and
,23222113121121 ∆×∆×∆×∆×∆×∆=∆×∆=∆
where ij∆ is some line segment, a sign x is the right
multiplication sign. By kc we denote Fourier’s
coefficients
∫
∆
−=≡ dxexffcc ikx
kk )(
)2(
1)( 060 π .
Thus the formula (6) is a expansion of the function
0f in the Fourier series.
Summing Eq. (6) by the method of Cesaro for any
),,2,1[ ∞∈ N we get
dyyfyxxf NN )()(1)( 060 −Φ= ∫
∆π
σ , (7)
here )(uNΦ is the Cesaro’s kernel
∏
=
=Φ
N
j
jNN uFu
1
)()( ,
2
2
sin
2
1
sin
)1(2
1)(
+
+
=
u
u
N
N
uF jN
It is easy to see
1)( =Φ∫
∆
dyuN
and
0
)(00lim →
∆
−
∞→ c
ffNN
σ ,
where ⋅ is a norm in the space of continuous
functions on )(: ∆∆ c .
Let us write down the function 0f as follows
gff n += 00 σ , where ∫
∆
=⋅σ 00 dxgfN and for
∞→N is vanishing.
Now differentiating formula (6) by the Eq. (5), we
obtain the following equation
dyyfyxgf NN )()(~1
0600 −Φ=− ∫
∆π
λ σ , (8)
where gg λ−=0 , ∆∈Φ=Φ yxL ,~
This reasoning yields Fredholm equation for the
function 0fNσ if 0g is a given function then:
∫
∆
−Φ=− dyyfyxgf NN )()(~1
0600 σ
π
σλ (9)
Define a matrix N
qrkk 1][= , as follows
∫
∆
−−Φ= dydxeeyxk iryiqx
qr )(~1
6π
It is easy to see that the matrix K is the Toeplitz
matrix which generates a vector-function { }lFvX ,=
[4].
The Eq. (9) is transformed to the linear algebraic
equation as follows:
∑
=
− ==
N
r
rqrq Nqckc
1
1 ,,1, λ , (10)
on set { } Nikxe 1=Ψ , here ],,1[ Nk i ∈ .
The Eq. (9) is an integral equation with degenerated
kernel [5].
Corollary 1. We can always find a sufficiently large
N such that there exists 0>ε , such that the
following relations are true:
εσ ω <− − ti
N efvrtf 0),,( ,
0),,(lim 0 →−
∞→
ti
NN
efvrtf ωσ ,
where f is continuous at every point ),( vr of the
domain ∆ . The function 0fNσ is a solution of Eq. (9),
and the ω is the eigenvalue of the matrix K , thus it is
frequency of a wave motion of the bunched beams.
The number ω , generally, maybe any complex
number: βαω i+= .
It can be shown in the usual way that if 0Im >ω
then 0),,( →vrtf for ∞→t , if 0Im <ω then the
solution f goes out from the domain ∆ . Finally, may
be the case such that 0=ω . These results are discussed
in more details in the next section. Under this condition
we have a stationary solution of Eq. (1).
Definition. The solution 00 =f of Eq. (1) is said
to be an asymptotically stable if for any 00 ≥t and
arbitrary 0≥ε it is possible to find such 0>δ that
implies ( ) ερδρ ≤→≤ )0),,,((, 0
00 vrtfff and
0)0),,,(( →vrtfρ as t tends ∞ . Here
ff
x ∆∈
= max)0,(ρ , ∫
∆
= dxff 22
,
),,( 00000 vrtff = .
3. CHAOTIC BEHAVIOR OF THE
BUNCHED BEAM
The motion of particles of bunched beam is evolving
in the space
{ }∞≤=ΩΩ×Ω rrrr :,ν , { }∞≤=Ω ννν : .
It is well known that the variables vr, are governed
by the following equation
vr = ,
+= ][1 vH
c
E
m
ev . (11)
Here 6Rvr ⊂Ω×Ω .
248
Thus
0][3
1
3
1
≡
∂
∂=
∂
∂ ∑∑
ks
s
v
kvH
x
v
for this reason (well known Liouville theorem) the
measure dvdr ×=∂ µ is the invariant measure for a
group tT (11), i.e. ttT µµ =0 for all ],[ ∞∞−∈t ,
that is easy to see. Consider an invariant measure on
vr Ω⋅Ω , simplify to solve linear partial differential (4)
by eigenvalue method, because we now have the
eigenvalue problem with electro-magnetic dependent
coefficients and the zero eigenvalue. We claim that the
eigenvalues will be points of the continuous spectrum
and eigenvector of (5) will be chaotic in the phase space
in the present case. It is interesting to know if it is the
case and how should one solve this kind of eigenvalue
problem when the system (11) is chaotic.
Let us consider the following operator L that is
selfadjoint extensions of the operator 0L in Hilbert
space ( )Ω2L . In accordance with the Stone theorem, the
operator ∗= LL generates a group of transformation
itL
t eU = , such that
t
U
iL t
t
ϕϕ −
=
→ 0
lim .
Let )(λke be an eigenfunction of the group tU
then
)()( λλ λ
k
ti
kt eeeU = ,
λLk dim,,1 = , where λL is a multiple of the
point )(Lσλ ∈ and σ is the spectrum of the L .
The element )(λke belongs to the space
∗
− Ω=Ω )()( 11 vv HH ([8] p. 387).
It is a direct consequence of the existence of the
invariant measure in dynamical system (11).
It is well known that )()( 1 vk He Ω∈ −λ and
)()( vk Ce Ω∉λ if it is the point of the continuous
spectrum.
In this case the first integral will be absent for
dynamical system (5) and it has become the transitive
system. In particular this reasoning yields the first
integral destruction. A. Einstein, [7] has given
conditions under which the first integral disappears.
Corollary 2. The electro-magnetic field in (11) can
generate the ergodic or chaotic motion. Suppose that
ergodic is equivalent to the chaos. This reasoning yields
an approach of the problem of deterministic chaos.
We return back to the Eq. (1) and assume that there
is the stationary solution ),,( vrgf 00 for which:
1˚ There exists a function ),,( vrgV 0 such that
0=V on the solution ),,( vrgf 00 , where 00 sf = at
0rr = , 0vv = ,˚
2˚ The function V is positive defined and founded
on an arbitrary solution ),,,( vrtsf of Eq. (1), here s
is an arbitrary function such that
),,( 000 vrtfs = , δ≤−
c
fvrtf 0000 ),,(
3˚ The derivative V of which in view of Eq. (1) is
negative.
We are going to show that in this case the solution
0f of Eq. (1) is orbital asymptotically stable.
In fact, for the function ),,( vrtV mentioned
above we have an estimate
tvrtVvrtV ),,,(),,( 12 ≤
if 12 tt < and 0lim =
∞→
V
t .
Thus the function V is decreasing and V is
representing its total time derivative, taken under the
assumption that vr, are function of t , satisfying
differential Eq. (5).
Note that a perturb have initial value, i.e. a
perturbation motion appears due to perturb of the
function s only. Now introduce into consideration a
function
∫
∆
= dtvrtfrrtV ),,()(),( η
and will show the one fulfils the conditions 1˚ – 3˚. A
function η is an arbitrary symmetric function such that
0),()( 0 =∫
∆
dvvrfrη .
Its derivative has the form
.0)(
)()(
0
00
=+
++
∂
∂=
∫
∫∫
∆
∆∆
dvfrdiv
dvfrdivdvfr
t
V
v
r
η
ηη
Indeed, in the case under consideration we get
0=
∂
∂
∫ dvf
t
η ,
=+ ∫∫ dvfdivdvfdiv vr 00 ηη
dvF
v
f
r
f
vf
r
v l∫
∂
∂
+
∂
∂
+
∂
∂ 00
0 ηη
, (12)
while
∫ =
∂
∂ 00 dvvf
r
η
, 000 =
∂
∂
+
∂
∂
lF
v
f
r
f
v .
The first integral equal to zero under the following
condition
00 →
∞→
f
v
,
i.e. the function f is a quickly decreasing with the
increasing velocity v .
Corollary 3. The function V be no positive if
0<ωeR . It is easy to verify (see above) that
249
∫ =
∂
∂ 0dvvf
r
η
.
By using this reasoning Eq. (12) yields
∫∫
∗∗ ∆∆
⋅== dvfidvvrtsfLV ωηη ),,,(
or 0≥VRe and 0≤VRe
.
Thus the particles beams under the above condition
be orbital asymptotically stable for solution 0f .
Speaking about the condition of the asymptotically
stable, we mean that the postulate in respect to the field
),( HE holds.
Thus this consideration proves that in domain
∆⊂∆ ∗ there is ),( HE such that the solution 0f
Eq. (5) is the orbital asymptotically stable. Note that if
the velocity ),,( 321 vvvv is such that the following
condition constvvv =++ 2
3
2
2
2
1 holds, then we have case
focusing and acceleration of bunched beam around 0f .
It is easy to see that we can choose any unperturbed
motion such that one is a motion of bunched beam along
arbitrary axis of rotation. This can do always, because
always, there exist electric and (or) magnetic fields
satisfying the Maxwell equation for a given arbitrary
motion, i.e. any (or) magnetic fields which satisfy the
Maxwell Eq. (2).
It follows that we can choose optimal fields.
4. CONSTRUCTION OF AN OPTIMAL
ELECTRIC FIELD
We can assume without loss of generality that the
matrix K is given in the following form
∑ −=−= ,,1 iiiii kk α ,
where iα is given number, the vector r is one-
dimension vector xr ≡ , the velocity xv = and
11 ≤≤− v , i.e. it is normalized on c (the speed of
light). Then { }π≤=∆∆×∆=∆ xx :, 121 ,
{ }1:2 ≤=∆ vv ,
+ ∞
∞+=
=
n
inx
n ex
η
ϕ
2
1)( ,
,1,0
)(ˆ
2
12
)(
=
+
=
n
nn vP
n
vP ,
nP̂ are the polynomials of Legendre.
Next we show how to choose the electrostatic field
E for the Vlasov-Poisson system
0),( =∂+∂+∂ fxtEfvf vxt (13)
),(4),( xtxtU π ρ−=∆ ,
where ),(),( xtUxtE x∂−= ,
∫
∆
= dvvxtfxt ),,(),(ρ , and E is such that the beam
of particle focused and accelerated along axis x . For
this purpose the distribution function ),,( vxtf of the
particles in phase space ∆ will be sought in the form
tievxff ω−= ),(0 . The substitution of tief ω
0 for
f yields 000 fifEfv vx ω=∂+∂ .
Next, we construct the function 0fNσ . Thus we
arrive at the following matrix N
qrkK 1][= ,
∫ ∑ ∫
∆ − ∆
−
∂
∂
=
1 2
,)()(
N
N
m
n
xrqi
qr dv
v
P
PdxxEek but
∑ ∫ ∫ ∑ ∑
− ∆ − − − −
==∂=
∂
∂N
N
N
N
N
N
mnmn
m CPPPPdv
v
P
2
1
1
0
1
1
|)(
here ,1)1( =nP ,)1()1( n
nP −=− .,..,1|| Nn =
Let the matrix K be given then the problem arises
of finding the field E under which the formulas
∫
−
−=
π
π
Nxrqi
qr dxxEeCk 1
)(
0 ])([ are fulfilled. Note
that in this situation the NN × number qrk are given
and NN × function xrqie )( − are given too, it is
necessary to find the function ).,( xtE
Let exist some number 0>L that .),( LxtE ≤
Thus we obtain the well known L - problem of
moments [3].
Now from Eq. (4) we can find ρ and U .
REFERENCES
1. A.A. Vlasov. Many-Particle Theory and its
Application to Plasma Physics. New York, 1961,
355 p.
2. V.I. Zubov. Problem of stability for the processes of
government. S.- Pet., 2001, 350 p.
3. J. Warga. Optimal control of differential and
functional equations. Academic Press, New York
and London, 1976, 622 p.
4. U. Grenander, G. Szegö. Toeplitz forms and their
applications. University of California press,
Berhelcy and Los Angeles, 1958, 305 p.
5. Z. Parsa, V. Zadorozhny. Nonlinear Dynamics on
compact and Beam Stability // Nonlinear Analysis.
2001, v. 47, p. 4897-4904.
6. F. Riesz, B. Sz.-Nagy. Leçons d’analyse
fonctionnelle. Budapest, 1953, 498 p.
7. A. Einstein. Eine Ableitung des Theorems non
Jacobi // Preuss. Akad. Wiss. 1917, pt. 2, 606-608.
8. I. Gelfand, A. Kostjuchenko. On decomposition of
differential and other operators onto eigenfunctions
// DAN SSSR. 1955, v. 103, Jfe3, p. 349-352.
9. M. Stone. On one-parameter unitary grups in Hilbert
space // Ann. Of Math. 1932, v. 33, p. 643-648.
10. Z. Parsa, V. Zadorozhny. Focusing and
Acceleration of Bunched Beams. AIP Conf. Proc.
1999, v. 530, p. 249-259.
250
FORMALISM FOR CHAOTIC BEHAVIOR OF THE BUNCHED BEAM
Z. Parsa
Brookhaven National Laboratory, Physics Department 510 A, Upton, NY 11973
e-mail: Parsa@bnl.gov
Institute of Cybernetic, National Academy of Sciences of Ukraine
2. APPROXIMATE SOLUTION OF VLASOV’S EQUATION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79899 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:46:31Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Parsa, Z. Zadorozhny, V. 2015-04-06T16:41:05Z 2015-04-06T16:41:05Z 2001 Formalism for chaotic behavior of the bunched beam / Z. Parsa, V. Zadorozhny // Вопросы атомной науки и техники. — 2001. — № 6. — С. 247-250. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: AMS 94B, 81P99, 57M20 https://nasplib.isofts.kiev.ua/handle/123456789/79899 Using Cesaro mid of Fourier series the quasi-linear Vlasov’s equation is transformed to the integral Fredholm equation. New results on the oscillatory behavior of solution are obtained. An extension to perturbing equation is also included. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Anomalous diffusion, fractals, and chaos Formalism for chaotic behavior of the bunched beam Формализм для описания хаотического поведения бунчированного пучка Article published earlier |
| spellingShingle | Formalism for chaotic behavior of the bunched beam Parsa, Z. Zadorozhny, V. Anomalous diffusion, fractals, and chaos |
| title | Formalism for chaotic behavior of the bunched beam |
| title_alt | Формализм для описания хаотического поведения бунчированного пучка |
| title_full | Formalism for chaotic behavior of the bunched beam |
| title_fullStr | Formalism for chaotic behavior of the bunched beam |
| title_full_unstemmed | Formalism for chaotic behavior of the bunched beam |
| title_short | Formalism for chaotic behavior of the bunched beam |
| title_sort | formalism for chaotic behavior of the bunched beam |
| topic | Anomalous diffusion, fractals, and chaos |
| topic_facet | Anomalous diffusion, fractals, and chaos |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79899 |
| work_keys_str_mv | AT parsaz formalismforchaoticbehaviorofthebunchedbeam AT zadorozhnyv formalismforchaoticbehaviorofthebunchedbeam AT parsaz formalizmdlâopisaniâhaotičeskogopovedeniâbunčirovannogopučka AT zadorozhnyv formalizmdlâopisaniâhaotičeskogopovedeniâbunčirovannogopučka |