Chaos to fractals

Prasanth Pulinchery; Nishanth Pothyiodath; Udayanandan Kandoth Murkoth

doi:10.21067/mpej.v7i1.7502

Authors

Prasanth Pulinchery Government Engineering College, India
Nishanth Pothyiodath PRNSS College, Mattannur, India
Udayanandan Kandoth Murkoth Gurudev Arts And Science College, India

DOI:

https://doi.org/10.21067/mpej.v7i1.7502

Keywords:

chaos, fractals, nonlinear oscillators

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.

Downloads

Download data is not yet available.

References

Adams, H. M., & Russ, J. C. (1992). Chaos in the classroom: Exposing gifted elementary school children to chaos and fractals. Journal of Science Education and Technology, 1(3). https://doi.org/10.1007/BF00701363

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., DaÄŸalp, R., Potas, N., & Ã–ztÃ¼rk, F. (2019). Randomness and Chaos (pp. 621â€“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), 123â€“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. 1â€“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 HernÃ¡ndezhernÂ´hernÃ¡ndez, J. E., & Julio MartÃnez HernÃ¡ndez, 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

Ã–zer, A. B., & Akin, E. (2005). Tools for Detecting Chaos.

Ã–ztÃ¼rk, F. (2020). Some Conceptual and Measurement Aspects of Complexity, Chaos, and Randomness from Mathematical Point of View (pp. 33â€“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), 728â€“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, 701â€“710.

van der Pol, B. (1926). LXXXVIII. On â€œrelaxation-oscillationsâ€ . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 978â€“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), 5129â€“5137. https://doi.org/10.1021/acsnano.0c10300