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

Lyapunov Exponents of Thirring Instantons

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.

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