Lyapunov Exponents of Thirring Instantons

Year 2022, , 529 - 536, 30.06.2022

Beyrul Canbaz

https://doi.org/10.17798/bitlisfen.1051969

Cited By: 1

Abstract

Recently, nonlinear differential equations corresponding to pure spinor instanton solutions were obtained by using Heisenberg anzat in the 2D Thirring Model, which has been used as a toy model in Quantum field theory. In addition, the evolution of spinor type instanton solutions in phase space was investigated according to the change in the constant parameter β. Spinor instanton dynamics is a special case in which nonlinear terms play an important role. Chaos describes certain nonlinear dynamical systems that depend very precisely on initial conditions. Lyapunov exponents are an important method for measuring stability and deterministic chaos in dynamical systems. Lyapunov exponents characterize and quantify the dynamics of small perturbations of a state or orbit in state space. In this study, Lyapunov spectrum of spinor type instanton solutions was investigated by examining the largest local and global Lyapunov exponents. As a result of the Lyapunov Spectrum, it was determined that the spinor type instanton solutions exhibit chaotic behavior at parameter value β = 2. Periodic and quasi periodic behaviors were detected when the parameter values were β < 2. In cases of β > 2, weak chaotic behaviors were observed. This study demonstrates that Thirring Instantons, which are spinor type instanton solutions, exhibit chaotic properties.

Keywords

Instanton, spinor, Lyapunov exponent, chaos theory, nonlinear systems

Thanks

I would like to thank K. Gediz Akdeniz for his support during the preparation of this study.

Thirring İnstantonlarının Lyapunov Üstelleri

Abstract

Son zamanlarda, Quantum alan teorisinde denek modeli olarak kullanılan 2 boyutlu Thirring Model de Heisenberg anzatı kullanılarak saf spinör instanton çözümlerine karşılık gelen lineer olmayan diferansiyel denklemleri elde edildi. Ayrıca, spinör tipi instanton çözümlerinin, β sabit parametresinin değişime göre faz uzayında ki evrimi incelendi. Spinör instanton dinamikleri, doğrusal olmayan terimlerin önemli bir rol oynadığı özel bir durumdur. Kaos, başlangıç koşullarına çok hassas bir şekilde bağlı olan belirli doğrusal olmayan dinamik sistemlerini tanımlar. Lyapunov üstelleri, dinamik sistemlerde kararlılığı ve deterministik kaosu ölçmek için önemli bir yöntemdir. Lyapunov üstelleri, durum uzayında bir durum veya yörüngenin küçük düzensizliklerinin dinamiklerini karakterize eder ve nicelendirir. Bu çalışmada, spinör tipi instanton çözümlerinin Lyapunov üstellerinin spektrumu incelendi. Yerel ve global en büyük Lyapunov üstelleri incelenerek spinör tipi instantonların kaotik davranışları araştırıldı. Lyapunov Spektrumu sonucuna göre spinör tipi instanton çözümlerinin parametre değeri = 2 olduğunda kaotik durum sergilediği tespit edildi. Parametre değerleri < 2 olduğu durumlarda periyodik ve kuazi periyodik davranışlar tespit edildi. > 2 durumlarında ise zayıf kaotik durumlar olduğu görüldü. Sonuç olarak bu çalışma, spinör tipi instanton çözümleri olarak Thirring İnstantonların kaotik özellikler sergilediğini göstermektedir.

Keywords

İnstanton, spinör, Lyapunov üstelleri, kaos teori, lineer olmayan sistemler

References

• [1] M. Dunajski, Solitons, Instantons, and Twistors, Oxford University Press, New York, 2010.
• [2] W. E. Thirring, “A Soluble. Relativistic Field Theory,” Anal. Phys., vol. 3, pp. 91, 1958.
• [3] K. G. Akdeniz and A. Smailagić, “Classical solutions for fermionic models,” Il Nuovo Cimento A, vol. 51, pp. 345– 357, 1979.
• [4] K. G. Akdenizand M. Hortacsu, “Functional determinant for the Thirring model with instanton,” II Nuovo Cimento A, vol. 59, pp. 181-188, 1980.
• [5] B. Canbaz, C. Onem, F. Aydogmus and K. G. Akdeniz, “From Heisenberg ansatz to attractor of Thirring Instanton,” Chaos, Solitons & Fractals, vol. 45, no. 2, pp. 188–191, 2012.
• [6] N. Yılmaz, B. Canbaz, M. Akıllı and C. Onem, 2018. “Study of the stability of the fermionic instanton solutions by the scale index method,” Physics Letters A, vol. 382, no. 32, pp. 2118-2121, 2012.
• [7] B. Canbaz, “Genel Hizalama İndeksi Yöntemiyle 2 Boyutlu Saf Fermiyonik Modelde Kaosun İncelenmesi,” Avrupa Bilim ve Teknoloji Dergisi. vol. 33, pp. 161-166, 2022.
• [8] J. Greick, Chaos: making a new science, Oxford Sciences Publications, 19-26. Oxford, England, 1987.
• [9] T. P. Shimizu, K. A. Takeuchi, “Measuring Lyapunov exponents of large chaotic systems with global coupling by time series analysis,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 28, no. 12, pp. 121103, 2018.
• [10] R. L. Devaney, An Introduction to Chaotic Dynamical Systems. 3rd Edition, CRC Press, 2021.
• [11] D. Feldman, Chaos and Dynamical Systems (Primers in Complex Systems, 7), Princeton University Press, 2019.
• [12] C. Skokos, “The Lyapunov Characteristic Exponents and Their Computation,” in Dynamics of Small Solar System Bodies and Exoplanets, Springer: Berlin/Heidelberg, Germany, pp. 63–135, 2010.
• [13] W. Heisenberg, Zs. Naturforsch., 9a, 292, 1954.
• [14] W. Siegert, 2009. Local Lyapunov Exponents, Springer, Berlin, pp. 143-229, 1954.
• [15] A. M. Lyapunov, “General problem of stability of motion. Annals of the Faculty of Sciences of Toulouse” Mathematics, Series 2, vol. 9, pp. 203-474, 1947.
• [16] V. I. Oseledec, “A multiplicative ergodic theorem. Lyapunov characteristic number for dynamical systems,” Trans. Moscow Math. Soc., vol. 19, pp. 197-231, 1968.
• [17] G. Benettin, L. Galgani, A. Giorgilli and J. Strelcyn, “Lyapunov Characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1,” Theory. Meccanica, vol. 15, pp. 9–20, 1980.
• [18] G. Benettin, L. Galgani, A. Giorgilli and J. Strelcyn, “Lyapunov Characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: Numerical application,” Meccanica, vol. 15 pp. 21–30, 1980.
• [19] J. P. Singh and B. K. Roy, “The nature of Lyapunov exponents is (+,+,−,−). Is it a hyperchaotic system?,” Chaos, Solitons Fractals, vol. 92, pp. 73–85, 2016.
• [20] J. H. Verner, “Numerically optimal Runge–Kutta pairs with interpolants,” Numer Algor, vol. 53, pp. 383–396, 2010.
• [21] G. Datseris, 2018 “Dynamical Systems. jl: A Julia software library for chaos and non- linear dynamics,” J. Open Source Softw. vol. 3, pp. 598, 2010.
• [22] C. Rackauckas and Q. Nie, 2017. “Differential Equations. jl—A performant and feature-rich ecosystem for solving differential equations Julia,” J. Open Res. Softw., vol. 5, pp. 15, 2010.
• [23] C. Skokos, G. A. Gottwald and J. Laskar, Chaos Detection and Predictability, Springer, 2016.
• [24] M. Ak, “Investigation of Chaos in 4D Fermionic Model by the Generalized Alignment Index Method,” Journal of the Institute of Science and Technology, vol. 12, no. 2, pp. 726-734, 2022.
• [25] F. Aydogmus, B. Canbaz, C. Onem and K. G. Akdeniz, “The Behaviours of Gursey Instantons in Phase Space,” Acta Physica Polonica B, vol. 44, pp. 1837-1845, 2013.