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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Published in:Вопросы атомной науки и техники
Date:2001
Main Authors: Parsa, Z., Zadorozhny, V.
Format: Article
Language:English
Published: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2001
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Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/79899
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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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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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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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fulltext 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
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