SSvetlana vetlana SytovaSytova

Research Institute for Nuclear Research Institute for Nuclear Problems,Problems,

Belarusian State UniversityBelarusian State University

Bifurcations and Bifurcations and chaos in relativistic chaos in relativistic beams interacting beams interacting

with with three-dimensional three-dimensional

periodical structuresperiodical structures

INP

OutlineOutline

What is newWhat is new What is Volume Free Electron LaserWhat is Volume Free Electron Laser VFEL physical and mathematical models in VFEL physical and mathematical models in

Bragg geometryBragg geometry Code VOLC Code VOLC forfor VFEL simulation VFEL simulation Examples of different chaotic regimes of Examples of different chaotic regimes of

VFEL intensityVFEL intensity Sensibility to initial conditions – the Sensibility to initial conditions – the

“Butterfly” effect“Butterfly” effect The largest Lyapunov exponentThe largest Lyapunov exponent Two-parametric analysis of root to chaotic Two-parametric analysis of root to chaotic

lasing in VFELlasing in VFEL

What is new?What is new?

Investigation of chaotic behaviour of relativistic

beam in three-dimensional periodical structure

under volume (non-one-dimensional) multi-wave

distributed feedback (VDFB)

volume diffraction grating

volume diffraction grating

WhatWhat is Volume Free Electron Laser is Volume Free Electron Laser ? *? *

* Eurasian Patent no. 004665

volume “grid” resonator X-ray VFEL

Benefits of volume distributed Benefits of volume distributed feedback*feedback* frequency tuning at fixed energy of electron beam in significantly

wider range than conventional systems can provide

more effective interaction of electron beam and electromagnetic

wave allows significant reduction of threshold current of electron

beam and, as a result, miniaturization of generator

reduction of limits for available output power by the use of wide

electron beams and diffraction gratings of large volumes

simultaneous generation at several frequencies

The increment of instability for an electron beam passing through a spatially-

periodic medium in degeneration points ~1/(3+s) instead of ~1/3 for the single-wave systems (TWTA and FEL) ( s is the number of surplus waves appearing due to diffraction). The following estimation for threshold current in degeneration points is valid:

s23start LkkL1~j

*V.G.Baryshevsky, I.D.Feranchuk, Phys. Lett. 102A (1984) 141; V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, J.Phys D24 (1991) 1250

This law is universal and valid for all wavelength ranges regardless the spontaneous radiation mechanism

VFELVFEL generator generator with a "grid" volume resonator*::

Main features:

electron beam of large cross-section

electron beam energy 180-250 keV

possibility of gratings rotation

operation frequency 10 GHz

tungsten threads with diameter 100 m

* V.G. Baryshevsky et al., Nucl. Instr. Meth. B 252 (2006) 86

V.G.Baryshevsky et al., Proc. FEL06, Germany (2006), p.331

Resonant grating provides VDFB of generated radiation with electron beam

VFEL in Bragg geometryVFEL in Bragg geometry

k

0

u

L

k

0conditionsCherenkov

,02

conditionsBragg2

ku

τkτ

Equations for electron beamEquations for electron beam

]2,2[],,0[,0

,),0,(

,/),0,(

)),,,(exp(

),/(Re),,(),,(

3

232

2

pLzt

ppt

ukdz

ptd

pzti

zuztEdz

pztdk

m

e

dz

pztd

),,( pzt is an electron phase in a wave

200 )//(1 cul

System for tSystem for twowo-wave VFEL-wave VFEL::

05.05.0

,8

22

5.05.0

11

),,(),,(2

02

0

EliEiz

Ec

t

E

dpeep

j

EilEiz

Ec

t

E

pztipzti

2 2 20

2

0

, 0,1,

,

ii

k cl i

l l

- detuning from exact Cherenkov condition

0,1 are direction cosines, is an asymmetry factor

χ0, χ±1 are Fourier components of the dielectric susceptibility of the target

are system parameters,Right-hand side of this system is more complicated than usually used because it takes into account two-dimensional distributions with respect to spatial coordinate and electron phase p. So, they allow to simulate electron beam dynamics more precisely. This is very important when electron beam moves angularly to electromagnetic waves.

System of equations for BWT, System of equations for BWT, TWT etc. *TWT etc. *

System is versatile in the sense that they remain the same within some normalization for a wide range of electronic devices (FEL, BWT, TWB etc).

*N.S.Ginzburg, S.P.Kuznetsov, T.N.Fedoseeva. Izvestija VUZov - Radiophysics, 21 (1978), 1037 (in Russian).

Code Code VOLCVOLC (“ (“VOLVOLumeume CCode”),ode”),version 2.0 forversion 2.0 for VFEL simulation VFEL simulation

Some cases of bifurcation

points

Results of Results of numericalnumerical simulation (2002- simulation (2002-2007):2007):

Bragg

Laue

SASE

Laue-Laue

Bragg-Laue

Synchronism of several

modes: 1, 2, 3

VFEL

Two-wave

Three-waveAmplification

regime

j δ

βili

Bragg-BraggSASE

Generationregime with

external mirrors

Amplificationregime

Amplificationregime

Generationregime with

external mirrors

Generationregime

j L

χ

Generation threshold

Generation threshold

Generationregime

Root to chaos (parameter planes:

(j, β), (j, ), (j, L))

β

All results obtained numerically are in good agreement with analytical predictions.

Chaotic dynamics means the tendency of wide range of systems to transition into states with deterministic behavior and unpredictable behavior. We know a lot of such examples as turbulence in fluid, gas and plasma, lasers and nonlinear optical devices**, accelerators, chaos in biological and chemical systems and so on. Nonlinearity is necessary but non-sufficient condition for chaos in the system. The main origin of chaos is the exponential divergence of initially close trajectories in the nonlinear systems. This is so-called the “Butterfly effect”*** (the sensibility to initial conditions). Bifurcation is any qualitative changes of the system

when control parameter passes through the bifurcational value 0.

Dynamical systems*Dynamical systems*

*H.-G. Schuster, "Deterministic Chaos" An Introduction, Physik Verlag, (1984)** M.E.Couprie, Nucl. Instr. Meth. A507 (2003), 1 M.S.Hur, H.J. Lee, J.K.Lee., Phys. Rev. E58 (1998), 936*** E.N. Lorenz, J. Atmos. Sci. 20 (1963), 130

Example of periodic regimes of Example of periodic regimes of VFEL intensityVFEL intensity

1800 1850 1900 1950 2000 20501800

1820

1840

1860

1880

1900

1920

1940

1960

1980

2000

E(t

+d

)

E(t)

17501800

18501900

19502000

2050 17501800

18501900

19502000

2050

1800

1850

1900

1950

2000

2050

Z A

xis

E(t+d)

E(t)

0 100 200 300

0

4

8

IFT

, a .u .

0 200 400 600 800t, ns

0

500

1E+3

1.5E+3

2E+3

2.5E+3|E |

Intensity Fourier transform

Poincaré map

Attractor

VFEL Intensity

Attractor is a set of points in phase space of the dynamical system. System trajectories tend to this set. Attractors are constructed from continious time series with some delay d.

Poincaré maps or sections are a signature of periodic regimes in phase space. Poincaré map is constructed as a section of attractor by a small plan area.

Quasiperiodic oscillationsQuasiperiodic oscillations

1400 1600 1800 20001100

1200

1300

1400

1500

1600

1700

1800

1900

2000

E(t

+d

)

E(t)

0 100 200 300

0

5

10

15

IFT

, a . u .

0 100 200 300 400t, ns

0.1

1

10

1E+2

1E+3

|E |

1000120014001600

18002000

22001000 1200 1400 1600 1800 2000 2200

1200

1400

1600

1800

2000

2200

E(t

+2d

)

E(t+d)

E(t)

Quasiperiodicity is associated with the Hopf bifurcations which introduces a new frequency into the system. The ratios between the fundamental frequencies are incommensurate(Hahn, Lee, Phys. Rev.E(1993),48, 2162)

““Weak” chaotic regimeWeak” chaotic regime

2900 3000 3100 3200 3300 3400 3500 3600 37003100

3200

3300

3400

3500

3600

3700E

(t+

d)

E(t)

0 200 400 600 800t, ns

0

1E+3

2E+3

3E+3

4E +3|E |

0 200 400 600

0

2

4

6

8

10

IFT

, a .u .

3000

3200

3400

3600

3800

30003200

34003600

38003000

3200

3400

3600

3800

E(t

+2d

)

E(t+d)

E(t)

Dependence of amplitude in time seems as approximaterepetition of equitype spikes close in dimensions per approximately equal time space.

Chaotic self-oscillations Chaotic self-oscillations (hyperchaos)(hyperchaos)

0 200 400 600 800t, ns

0

5E+2

1E+3

1.5E+3

2E+3

2.5E+3

3E+3

3.5E+3

4E+3

4.5E+3

5E+3

5.5E+3|E |

0 100 200 300

0

1

2

3

4

5

IFT

, a .u.

42004300

44004500

46004700

48004900

5000 42004300

44004500

46004700

48004900

50004200

4300

4400

4500

4600

4700

4800

4900

5000

E(t

+2d

)

E(t+d)E(t)

4400 4500 4600 4700 4800 49004300

4400

4500

4600

4700

4800

E(t

+d

)

E(t)

Quasiperiodicity and intermittencyQuasiperiodicity and intermittency Intermittency is closely related to saddle-node bifurcations. This means the collision between stable and unstable points, that then disappears. (Hahn, Lee, Phys. Rev.E(1993),48, 2162)

1) 1750 А/ см2 2) 1950 А/ см2 3) 2150 А/ см2

4) 2220 А/ см2

5) 2300 А/ см2

6) 2340 А/ см2 7) 2350 А/ см2

500

1000

1500

2000

2500

3000

3500

5001000

15002000

25003000

3500

1000

1500

2000

2500

3000

3500

E(t+d)

E(t+

2d)

E(t)

Quasiperiodicity and intermittencyQuasiperiodicity and intermittency

1000

1500

2000

2500

3000

3500

5001000

15002000

25003000

3500

500

1000

1500

2000

2500

3000

3500

E(t

+2d

)

E(t+

d)

E(t)

1) 1750 А/ см2 2) 1950 А/ см2 3) 2150 А/ см2

4) 2300 А/ см2

5) 2340 А/ см2 6) 2350 А/ см2

Amplification and generation Amplification and generation regimes are first and second regimes are first and second

bifurcation pointsbifurcation points

180 200 220 240 260j, A/cm1E-9

1E-81E-71E-61E-5

0.00010.0010.010.1

110

1001000

| E |

E = 0

E =1

E

=

0.01

E = 0.0001

2

A B

L

=

20

cm

0

0

0

0

A BFirst threshold point corresponds to beginning of electron beam instability. Here regenerative amplification starts while the radiation gain of generating mode is less than radiation losses. Parameters at which radiation gain becomes equal to absorption correspond to the second threshold point after that generation progresses actively.

0 200 400 600 800t, ns

0

200

400

600

800

|E |

1

2

3

4

67

5

j is equal to:1) 350 A/ cm2, 2) 450 A/cm2, 3) 470 A/cm2, 4) 515 A/cm2, 5) 525 A/cm2,6) 528 A/cm2, 7) 550 A/cm2.

Amplification regime Amplification regime

0 200 400 600 800t, ns

0

40

80

120|E |

1

2

34

6

7

5

In simulations an important VFEL feature due to VDFB was shown. This is the initiation of quasiperiodic regimes at relatively small current near firsts threshold points.

0 200 400 600 800t, ns

0

40

80

120|E |

1

2

3

4

0 200 400 600 800t, ns

0

300

600

900|E |

1

2

3

4

j is equal to:1) 490 A/ cm2, 2) 505 A/cm2, 3) 530 A/cm2, 4) 550 A/cm2

Generation regimeGeneration regime

Initiation of quasiperiodic regimes Initiation of quasiperiodic regimes at relatively small resonator at relatively small resonator length near threshold pointlength near threshold point

10 15 20 25 30 35 40

L, cm

transmitted wave

diffracted wave

under threshold lengthperiodic regimequasiperiodic regimechaotic regime

Sensibility to initial conditionsSensibility to initial conditions for for generator regimegenerator regime

0 100 200 300 400t, ns

0.1

1

10

1E+2

1E +3

|E |

0 100 200 300 400t, ns

0.1

1

10

1E+2

1E +3

|E |

0 200 400 600t, ns

0

1E+3

2E+3

3E+3

4E +3|E |

“Weak” chaos

Quasiperiodic oscillations

j = 1950 A/cm2

j = 1960 A/cm2

0 200 400 600t, ns

0

1E+3

2E+3

3E+3

4E+3

5E+3

6E+3

7E+3

8E+3

9E+3

1E+4

1.1E+4

1.2E+4|E |

Periodic regime

2000 A/cm2

2010 A/cm2000.1 A/cm2000+10-8. A/cm

2

2

2

0 200 400 600 800t, ns

0

30

60

90|E |

Sensibility to initial conditionsSensibility to initial conditions for for amplification regimeamplification regime

0 200 400 600t, ns

0.1

1

10

1E+2

1E+3

1E +4|E |

j = 2300 А/ сm2

|Е | = 1|Е | = 1+10-15

j = 500 А/ сm2

|Е | = 1|Е | = 1.1

Periodic regime Chaotic regime

0 20 40 60 80 100t, ns

-6

-5

-4

ln (

d )

1

jThe largest Lyapunov exponentThe largest Lyapunov exponent

reconstructed with Rosenstein reconstructed with Rosenstein approach*approach*

0 20 40 60 80 100t, ns

-4

-2

0

ln(d

) j

1

**M.T.Rosenstein et al. Physica D65 (1993), 117-134

The largest Lyapunov exponent is a measurement of the stability of the underlying dynamics of time series. It specifies the mean velocity of divergence of neighboring points.

Periodic regime “Weak” chaos

Map of BWT dynamical regimes Map of BWT dynamical regimes with strong with strong reflections*reflections*

*S.P.Kuznetsov. Izvestia Vuzov “Applied Nonlinear Dynamics”, 2006, v.14, 3-35

with large-scale with large-scale and small-scale and small-scale amplitude amplitude regimesregimes

Domains with transition between Domains with transition between large-scale and small-scale large-scale and small-scale

amplitudesamplitudes

Root to chaotic lasingRoot to chaotic lasing

Larger number of principle frequencies for transmitted wave can be explained the fact that in VFEL simultaneous generation at several frequencies is available. Here electrons emit radiation namely in the direction of transmitted wave.

0 depicts a domain under beam current threshold. P – periodic regimes, Q – quasiperiodicity, I – intermittency, C – chaos.

-20 -15 -10 -5

500

1000

1500

2000

2500

j, A

/cm

2

I

C

0

P

Q

2j,

A/c

m

500

1000

1500

2000

2500

-20 -15 -10 -5

Another root to chaotic lasingAnother root to chaotic lasing

0 depicts a domain under beam current threshold. P – periodic regimes, Q –quasiperiodicity, A – domains with transition between large-scale and small-scale amplitudes, I – intermittency, C – chaos.

-20 -10 0 10 20

k L

500

1000

1500

2000

2500

3000

j, A

/cm

2

-20 -10 0 10 20

k L

500

1000

1500

2000

2500

3000

j, A

/cm

2

0

P

Q

A

I

C

ReferencesReferences

Batrakov K., Sytova S. Mathematical Modelling and Analysis, 9 (2004) 1-8.

Batrakov K., Sytova S. Computational Mathematics and Mathematical Physics 45: 4 (2005) 666–676

Batrakov K., Sytova S.Mathematical Modelling and Analysis. 10: 1 (2005) 1–8

Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems, 8 : 4(2005) 359-365

Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems, 8 : 1(2005) 42–48

Batrakov K., Sytova S. Mathematical Modelling and Analysis. 11: 1 (2006) 13–22

Batrakov K., Sytova S. Proceedings of FEL06, Germany (2006), p.41 Batrakov K., Sytova S. Proceedings of RUPAC, Novosibirsk, Russia,

(2006), p.141