On The Duffing Type Behaviors Of Gursey Instantons

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3 (1), 2009, 120 - 126

©BEYKENT UNIVERSITY

ON THE DUFFING TYPE BEHAVIORS OF

GURSEY INSTANTONS

Fatma AYDOĞMUŞ, K.Gediz AKDENİZ, Serap SAĞALTICI

Istanbul University,

Faculty of Science, Department of Physics, Istanbul, Turkey

Cem ÖNEM

Trakya University,

Department of Physics, Edirne, Turkey

Received: 10.07.2008 Revised: 26.07.2008 Accepted: 08.01.2009

ABSTRACT

Recently the new different quantum sense of 1956 Gursey non-linear conformally invariant pure spinor wave equation (GSW) in four dimensions has been considered and nonlinear dynamical structures of the Gursey spinor solitons are investigated.

In this work we investigate the Duffing type behaviours of the Gursey Instanton in phase space.

Keywords: Nonlinear systems, Bifurcation, Chaos

GÜRSEY İNSTANTONLARININ DUFFİNG TİPİ

DAVRANIŞLARI

ÖZET

Çok yakınlarda Gürsey tarafından 1956 yılında önerilen dört boyutlu konformal invariant non-lineer saf spinör etkileşmeli dalga denkleminin yeni farklı kuantum anlamı ele alınmış ve Gürsey spinor solitonlarının non-lineer dinamik yapıları incelenmiştir.

Bu çalışmada biz Gürsey spinor instantonlarının faz uzayındaki Duffing tipi davranışlarını inceliyoruz.

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I. INTRODUCTION

In 1956; Gursey proposed a conformally invariant pure spinor wave (GSW) equation in four dimensions, with a non-linear self-coupled spinor term which contains no derivatives higher than the first [1], and Kortel found a class of exact wave solutions to the GSW using the Heisenberg ansatz [2]. Much later; the special case of the Kortel solutions was shown to be instantonic which reflected the spontaneous symmetry breaking of the conformal invariance [3]. It has also been shown that the GSW equation may lead a consistent quantum field theory of composite particles made out of fundamental fermions in the symmetric phase, with an asymptotically free coupling strength [4]. More recently; new attempts to make different quantum sense of the GSW equation has been considered [5], and disorder structure of the solitons with mass [6] and instanton-like spinor wave solutions of the GSW equation has been investigated in phase space [7].

In this work we want to discuss the Duffing type chaotic behaviours of the instanton-like Gürsey spinors in phase space using our recent work [7].

II. MATERIALS AND METHOD

We start with 1956 Gursey non-linear spinor wave equation in four dimensions

4 - 1

( Y )

= -

3 Y

(1)

Here Y is a spinor field and g is a dimensionless coupling constant. When the Heisenberg ansatz

Y = \ixMrMx( s) + <( s)]« (2)

is inserted into Eq.(1), one can obtain the following nonlinear differential equation system

1

2 sx + 4x-a<p{ sx2 + < )3 = 0 ( 3 a )

1 2< +ax(sx

2

+ < )

3

= 0

(3b) 1

Here a is an arbitrary spinor constant (a = g(aa)3 ), x ( s ) and <p(s) are real functions and

,2 2

^

= t - r

.

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2 f (z) + 2 f (z) - g(z)(f

2

(z) + g

2

(z))

3

= 0 (4a)

2 g ( z ) - 2 g ( z ) + f ( z ) ( f 2(z) + g2(z))3 = 0 (4b) where f(z) and g(z) are dimensionless .

III. INSTANTONIC ATTRACTORS IN PHASE SPACE

Using the Runge-Kutta numerical method with MAPLE, we obtain the following phase diagrams of equation system (4):

The attractor in Figure 1 shows that g(z)-f(z) phase is closed system. i.e. the space-time extension of the instantonic behaviours in the GSW is autonomous and stable.

Figure 1: The phase diagram of equation system in Eq.(4)

Secondly, Figures 2a,2b and 2c show us the evaluation of the space-time extension of the instantonic attractors in 3-dimension. They inform us that the stability is preserved during the space-time evolution. In this case the Runge-Kutta numerical solutions are obtained by fortran and plotted with W-plot graphic programme.

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Figüre 2a: g(z)-f(z) space in 3-dimension a 2 1 D 1 2 --3

\

/ A - . 0 0 5 0 0 5 ^ o - . 0 1 - 3 - 1 1 2 3

Figüre 2b: g(z)-f(z) space in 3-dimension

3 2 - | I o -1 -2 -3 ,oı 4 o / / - . 0 0 5 ııırp t«ji rp?jp»iTf ı f -,01 3 - 1 1 2 3

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f(z)-f'(z) and g(z)-g'(z), the attractors in Figure 2a and 2b behave as the solution of "weakly" nonlinear Duffing differential equation [8]. But in Figure 3, f'(z)-g'(z) attractor moves as the solution of "strongly" nonlinear Duffing differential equation. Both cases also confirm that the chaotic structure of the Gursey instanton in space-time extension is stable.

-3 -2 -i o 1 a Î

y(D

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.1.5 , , r , ,

- 1 4 4 » t t ) tf!)

Figure 4: g(z)-g'(z) phase space

16

-1.5

• 15 -1 -6 0 .5 1 15 f(D

Figure 5: f'(z)-g'(z) phase space

IV. CONCLUSION

The main conclusion of this work can be summarized as: Within the conformal symmetry breaking technique [3], the phase-space investigation with Heisenberg ansatz of GSW gives us more information about the dynamical properties of Gursey instanton. This work also points to the Duffing type chaotic behaviours of instanton, which can play more fruitful role in vacuum transitions of quarks [9].

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REFERENCES

[1] F. GURSEY: Nuovo Cimento 3, 988 (1956). [2] F. KORTEL: Nuovo Cimento 4, 210 (1956).

[3] K.G. AKDENIZ: Lettere al Nuovo Cimento 33, 40 (1982).

[4] K.G. AKDENIZ, M. ARIK, M. DURGUT, M. HORTACSU, S. KAPTANOGLU and N.K. PAK: Phys. Lett.,B 116B, 34 and 41 (1982); K.G. AKDENIZ, M. ARIK, M. HORTACSU and N.K. PAK: Phys. Lett.,B 124B, 79 (1983).

[5] M. HORTACSU, B. C. LUTFUOGLU and F. TASKIN: Modern Physics Letters A 22: 2521 (2007).

[6] K.G. AKDENIZ, F. OZOK, S. SAGALTICI and C. ONEM: "On Nonlinear Dynamics of Spinor Type Solitons in Gursey Equation with Mass Term", Istanbul University Department of Physics preprint, Unpublished (2005).

[7] K.G.AKDENlZ,F.AYDOGMUS,S.SAGALTICI and C.ONEM, "Chaotic Dynamics of Gursey Instantons in Phase Space" Paper presented in Nonlinear Dynamical Analysis-2007, Petersburg, Russia (2007).

[8] W. SZEMPLINSKA-STUPNICKA, G. IOOSS, F.C. MOON, "ChaoticMotions in Nonlinear Dynamical Systems ", Spinger-Verlag Wien-New York (1988).