4 Boyutlu Modelde Spinör Tipi İnstanton Çözümlerinin Dalgacık Entropisinin İncelenmesi

Yıl 2022, Sayı: 38, 329 - 334, 31.08.2022

Mine Ak

https://doi.org/10.31590/ejosat.1099184

Öz

Son yıllarda, doğrusal olmayan dinamik sistemlerin yörüngelerinin özelliklerini araştırmak için çeşitli alanlarda birçok yöntem önerilmiştir. Bu çalışmada, Heisenberg anzatı aracılığıyla bulunan dört boyutlu Gürsey modeldeki spinör tipi instanton çözümlerinin yörüngelerinin karakteristiği incelenmiştir. Spinör tipi instanton çözümlerinin yörüngeleri, Shannon dalgacık entropisi – wavelet entropy (WE) yöntemiyle analiz edilmektedir. Spinör tipi instanton çözümlerinin yörüngelerinin düzenli veya düzensiz durumlarını analiz etmek için faz uzayında WE ve WE spektrumu üzerine çalışılmaktadır. Spinör tipi instanton çözümlerinin kararlı nokta etrafında düzenli yörüngelere ve diğer noktalar için düzensiz yörüngelere sahip olduğu gözlemlenmiştir. Bu çalışmaya göre, faz uzayında WE doğrusal olmayan dinamik sistemlerin entropi evrimini gözlemlemek için kullanılabilir.

Anahtar Kelimeler

Doğrusal olmayan dinamik sistemler, spinör tipi instanton, dalgacık entropisi, faz uzayı

Teşekkür

Bu makaleyi hazırlarken verdiği destek için değerli hocam Prof. Dr. K.Gediz AKDENİZ’e teşekkür ederim.

Investigation of Wavelet Entropy of Spinor Type Instanton Solutions in a 4-Dimensional Model

Öz

In recent years, many methods have been proposed in various fields to investigate the properties of orbits of nonlinear dynamical systems. In this study, the characteristics of the orbits of spinor-type instanton solutions in the four-dimensional Gursey model via Heisenberg ansatz is investigated. The orbits of spinor-type instanton solutions are analyzed by Shannon wavelet entropy (WE) method. In order to analyze the regular or irregular states of the orbits of spinor-type instanton solutions, WE spectrum and WE in phase space are studied. It is observed that spinor-type instanton solutions have regular orbits around the fixed point and irregular orbits for other points. According to this study, WE can be used to observe the entropy evolution of nonlinear dynamical systems in phase space.

Anahtar Kelimeler

Nonlinear dynamical systems, spinor-type instanton, wavelet entropy, phase space.

Kaynakça

• Akdeniz, K. G. (1982). On classical solutions of Gursey’s conformal-invariant spinor model. Lettere al Nuovo Cimento, 33(2), 40–44.
• Akdeniz, K. G., Arik, M., Durgut, M., Hortaçsu, M., Kaptanoğlu, S., & Pak, N. K. (1982). The quantization of the Gürsey model. Physics Letters B, 116(1), 34–36.
• Aldroubi, A., Unser, M. (1996). Wavelets in Medicine and Biology. Boca Raton: CRC Press.
• Aydogmus, F., Canbaz, B., Onem, C., Akdeniz, K. G. (2013). The behaviours of Gursey instantons in phase space. Acta Physica Polonica B, 44(9), 1837–1845.
• Bandt, C., Pompe, B. (2002). Permutation entropy: a natural complexity measure for time series. Physical Review Letters, 88(17), 174102.
• Baumert, M., Javorka, M., Seeck, A., Faber, R., Sanders, P., Voss, A. (2012). Multiscale entropy and detrended fluctuation analysis of QT interval and heart rate variability during normal pregnancy, Comput Biol Med, 42(3), 347-352.
• Blanco, S., Figliola, A., Quian-Quiroga R., Rosso, O. A., Serrano, E. (1998). Time–frequency analysis of electroencephalogram series (III): wavelet packets and information cost function. Physical Review E, 57, 932-940.
• Boltzmann, L. (1871). Einige allgemeine Satze über Warmegleichgewicht unter Gas-molekulen, Sitzungsber. Akad. Wiss. Wien, 63, 679–711.
• Brin, M., Stuck, G. (2015). Introduction to Dynamical Systems. Illustrated edition, Cambridge University Press.
• Camarena-Martinez, D., Valtierra-Rodriguez, M., Amezquita-Sanchez, J. P., Granados-Lieberman, D., Romero-Troncoso, R. J., Garcia-Perez, A. (2016). Shannon Entropy and K-Means Method for Automatic Diagnosis of Broken Rotor Bars in Induction Motors Using Vibration Signals. Shock and Vibration, 2016, 1-10.
• Canbaz, B., Onem, C., Aydogmus, F., Akdeniz, K. G. (2012). From Heisenberg ansatz to attractor of Thirring Instanton. Chaos, Solitons & Fractals, 45(2), 188–191.
• Chen, W., Wang, Z., Xie, H., Yu, W. (2007). Characterization of Surface EMG Signal Based on Fuzzy Entropy. IEEE Trans Neural Syst Rehabil Eng, 15(2), 266–272.
• Clausius, R. (1850). On the motive power of heat & on the laws which may be deduced from it for the theory of heat. Annalen der Physik, 79, 368-500.
• Costa, M., Goldberger, A. L., Peng, C.-K. (2002). Multiscale entropy analysis of complex physiologic time series. Physical Review Letters, 89(6), 068102.
• Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM.
• Dunajski, M. (2010). Solitons, Instantons, and Twistors. Illustrated edition, Oxford University Press.
• Gursey, F. (1956). On a conform-invariant spinor wave equation. Il Nuovo Cimento, 3(5), 988–1006.
• Heisenberg, W. (1954). Zur quantentheorie nichtrenormierbarer wellengleichungen. Zeitschrift für Naturforschung A, 9, 292–303.
• Hortacsu, M., Lutfuoglu, B. C., & Taskin, F. (2007). Gauged system mimicking the Gürsey model. Modern Physics Letters A, 22, 2521–2532.
• Hortacsu, M., Lutfuoglu, B. C. (2007a). Renormalization group analysis of a Gursey model inspired field theory. Physical Review D, 76, 025013.
• Kortel, F. (1956). On some solutions of Gursey’s conformal-invariant spinor wave equation. Il Nuovo Cimento, 4, 210–215.
• Li, P., Liu, C., Li, K., Zheng, D., Liu, C., Hou, Y. (2015). Assessing the complexity of short-term heartbeat interval series by distribution entropy. Med Biol Eng Comput, 53(1), 77–87.
• Mallat, S. (1999). A Wavelet Tour of Signal Processing. Second edition, San Diego: Academic Press.
• Nicolis, O., Mateu, J., Contreras-Reyes J. E. (2020). Wavelet-Based Entropy Measures to Characterize Two-Dimensional Fractional Brownian Fields. Entropy, 22(2), 196.
• Pan, S., Han, T., Tan, A. C., Lin, T. R. (2016). Fault diagnosis system of induction motors based on multiscale entropy and support vector machine with mutual information algorithm. Shock and Vibration, 2016, 1-12.
• Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA, 88(6), 2297-2301.
• Rajaraman, R. (1987). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. 1st edition, North Holland.
• Richman, J. S., Randall, M. J. (2000). Physiological time-series analysis using approximate entropy and sample Entropy. Am J Physiol Heart Circ Physiol, 278(6), H2039–H2049.
• Riedl, M., Müller, A., Wessel, N. (2013). Practical considerations of permutation entropy: A tutorial review. The European Physical Journal Special Topics, 222, 249–262.
• Rosso, O. A., Blanco, S., Yordanova, J., Kolev, V., Figliola, A., Schürmann, M., Başar, E. (2001). Wavelet entropy: a new tool for analysis of short duration brain electrical signals. Journal of Neuroscience Methods, 105(1), 65-75.
• Rosso, O. A., Mairal, M. L. (2002). Characterization of time dynamical evolution of electroencephalographic records. Physica A, 312, 469–504.
• Shannon, C. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423.
• Truebner, S., Cygankiewicz, I., Schroeder, R., Baumert, M., Vallverdu, M., Caminal, P., Vazquez, R., Luna, A. B., Voss, A. (2006). Compression entropy contributes to risk stratification in patients with cardiomyopathy. Biomed Tech (Berl), 51(2), 77-82.
• Wehrl, A. (1978). General properties of entropy. Reviews of Modern Physics, 50, 221-260.