Functions and Analysis


Ramanujan Theta Functions: The Route to Chaos

Authors: Sai Venkatesh Balasubramanian

Motivated by the call for a more flexible, easy-to-tune means of generating chaos, the present work elaborates upon the development of the Ramanujan Theta Function as a source of chaos. By setting the variables of this function to sinusoids of competing frequencies the chaotic output signal is generated, with the ratio of the input frequencies serving as control parameter. The characteristics of such chaos generated are studied using the iterative map and bifurcation plots, and the presence of chaos is ascertained using Lyapunov Exponents and Kolmogorov Entropy. Following this, the route from order to chaos of the proposed system are studied using three techniques – phase portrait, Fourier Spectra and wavelet analysis. In the phase portraits, it is seen that for non-chaotic regimes, phase portraits are orderly with definite number of loops, whereas for chaotic regimes, trajectories are spread all over the phase space, suggesting ergodicity and giving the phase portrait a rich, ornamental look. The Fourier spectra highlighted the discrete frequency components in non-chaotic regimes, with well formed sidebands, whereas in chaotic regimes, a lot of new frequency components are seen, giving the spectral profile a ‘grassy’ appearance. Finally, a hyperbolic wavelet, termed the Solitary Wavelet seen to possess vanishing higher order moments with a negative logarithmic slope, is used as the basis to perform wavelet analysis. The results reveal that the rhythmic periodicity observed in large scale values for non-chaotic regimes is significantly absent for chaotic regimes, with variations in the trends of new pulses emerging alongside the main pulse train. It is seen that the wavelet analyses combine the best features of phase portrait analyses (ergodicity detected by peak sporadicity and pulse variance), and Fourier Spectral analyses (new frequency component generation seen by observing dominance at various scales, and new peaks at lower scales corresponding to new pulses), while revealing additional features such as fractal nature, not seen in the other two analysis tools. In summary, the present article ushers in a novel perspective pertaining to signal oriented chaos, calling for a change in the way bifurcation plots and iterative maps are perceived, as also the means to generate and control such chaos. It is hoped that the wavelet analysis, highlighting both spectral and temporal aspects of the signal, emerges as a reliable and assertive qualitative means to identify, detect and to an extent, characterize the nature of chaos, either stand-alone, or in conjunction with tools such as phase portraits. Progress in such an area will eventually drive chaos analyses away from Lyapunov Exponents, which are most useful in system based chaos where initial conditions are well known, and are at best computed with approximations from output chaotic signals, using methods such as the Rosenstein Algorithm.

Comments: 12 Pages.

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Submission history

[v1] 2016-03-30 21:39:45

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