Chaos to fractals
Abstract
In undergraduate classrooms, while teaching chaos and fractals, it is taught as if there is no relation between these two. By using some non linear oscillators we demonstrate that there is a connection between chaos and fractals. By plotting the phase space diagrams of four nonlinear oscillators and using box counting method of finding the fractal dimension we established the chaotic nature of the nonlinear oscillators. The awareness that all chaotic systems are good fractals will add more insights to the concept of chaotic systems.
References
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Andronov, I. L. (2020). Advanced Time Series Analysis of Generally Irregularly Spaced Signals: Beyond the Oversimplified Methods. In Knowledge Discovery in Big Data from Astronomy and Earth Observation: Astrogeoinformatics. https://doi.org/10.1016/B978-0-12-819154-5.00022-9
Appleton, E. V., & van der Pol, B. (1922). XVI. On a type of oscillation-hysteresis in a simple triode generator . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 43(253). https://doi.org/10.1080/14786442208633861
Arfken, G. B., Weber, H. J., & Harris, F. E. (2013). Mathematical Methods for Physicists. In Mathematical Methods for Physicists. https://doi.org/10.1016/C2009-0-30629-7
Atakan, C., Dağalp, R., Potas, N., & Öztürk, F. (2019). Randomness and Chaos (pp. 621–646). https://doi.org/10.1007/978-3-319-89875-9_51
Bannur, V. M. (1998). Dynamical temperatures of quartic and Henon-Heiles oscillators.
Bannur, V. M., Kaw, P. K., & Parikh, J. C. (1997). Statistical mechanics of quartic oscillators.
Beale, P. D., & Pathria, R. K. (2011). Statistical Mechanics. 745.
Biswas, H. R., Hasan, M. M., & Kumar Bala, S. (2018). Chaos Theory And Its Applications In Our Real Life. Barishal University Journal Part 1, 5(1&2), 123–140.
Boeing, G. (2016). Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. https://doi.org/10.3390/systems4040037
Cattani, M., Caldas, I. L., de Souza, S. L., & Iarosz, K. C. (2017). Deterministic chaos theory: Basic concepts. Revista Brasileira de Ensino de Fisica, 39(1). https://doi.org/10.1590/1806-9126-RBEF-2016-0185
Forgues, B., & Thietart, R.-A. (2016). Chaos Theory. In The Palgrave Encyclopedia of Strategic Management (pp. 1–5). Palgrave Macmillan UK. https://doi.org/10.1057/978-1-349-94848-2_384-1
Fusic, S., & Kufner, A. (2014). Nonlinear Differential Equations. Elsevier.
Ginoux, J. M., & Letellier, C. (2012). Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept. Chaos, 22(2). https://doi.org/10.1063/1.3670008
Greiner, W. (2010). Lyapunov Exponents and Chaos. In Classical Mechanics. https://doi.org/10.1007/978-3-642-03434-3_26
Henon, M., & Heiles, C. (1964). The Applicability of the Third Integral Of Motion: Some Numerical Experiments. In The Astronomical Journal (Vol. 69, Issue 1).
Humberto, A., Salas, S., Castillo Hernándezhern´hernández, J. E., & Julio Martínez Hernández, L. (2021). The Duffing Oscillator Equation and Its Applications in Physics. https://doi.org/10.1155/2021/9994967
Korolj, A., Wu, H. T., & Radisic, M. (2019). A healthy dose of chaos: Using fractal frameworks for engineering higher-fidelity biomedical systems. In Biomaterials (Vol. 219). Elsevier Ltd. https://doi.org/10.1016/j.biomaterials.2019.119363
Kovacic, I., & Brennan, M. J. (2011). The Duffing Equation: Nonlinear Oscillators and their Behaviour. In The Duffing Equation: Nonlinear Oscillators and their Behaviour. https://doi.org/10.1002/9780470977859
Mandelbrot, B. B. (1982). The fractal geometry of nature. W.H. Freeman, San Francisco.
Motter, A. E., & Campbell, D. K. (2013). Chaos at Fifty.
Mou, D., Geophysics, Z. W.-N. P. in, & 2016, undefined. (2016). Comparison of box counting and correlation dimension methods in well logging data analysis associate with the texture of volcanic rocks. Npg.Copernicus.Org. https://doi.org/10.5194/npg-2014-85
Özer, A. B., & Akin, E. (2005). Tools for Detecting Chaos.
Öztürk, F. (2020). Some Conceptual and Measurement Aspects of Complexity, Chaos, and Randomness from Mathematical Point of View (pp. 33–66). https://doi.org/10.1007/978-3-030-27672-0_4
Palmore, J. (1991). A review of nonlinear dynamics, chaos and fractals. Journal of Geological Education, 39(5). https://doi.org/10.5408/0022-1368-39.5.393
Rosenberg, E. (2020). Fractal Dimensions of Networks. Fractal Dimensions of Networks. https://doi.org/10.1007/978-3-030-43169-3
Shukla, J. (1998). Predictability in the Midst of Chaos: A Scientific Basis for Climate Forecasting. Science, 282(5389), 728–731. https://doi.org/10.1126/science.282.5389.728
Struble, R. A. (2018). Nonlinear differential equations.
Tarnopolski, M. (2013). On the Fractal Dimension of the Duffing Attractor.
Thompson, J. M. T. (2016). Chaos, fractals and their applications. International Journal of Bifurcation and Chaos, 26(13). https://doi.org/10.1142/S0218127416300354
van der Pol, B. (1920). A theory of the amplitude of free and forced triode vibrations. Radio Review, 701–710.
van der Pol, B. (1926). LXXXVIII. On “relaxation-oscillations” . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 978–992. https://doi.org/10.1080/14786442608564127
Zhu, Z., Wei, N., Yan, B., Shen, B., Gao, J., Sun, S., Xie, H., Xiong, H., Zhang, C., Zhang, R., Qian, W., Fu, S., Peng, L., & Wei, F. (2021). Monochromatic Carbon Nanotube Tangles Grown by Microfluidic Switching between Chaos and Fractals. ACS Nano, 15(3), 5129–5137. https://doi.org/10.1021/acsnano.0c10300
Authors
Pulinchery, P., Pothyiodath, N., & Murkoth, U. K. (2023). Chaos to fractals. Momentum: Physics Education Journal, 7(1), 17–32. https://doi.org/10.21067/mpej.v7i1.7502
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