Universal Orbits: Unveiling the Connection between Chaotic Dynamics, Normal Numbers, and Neurochaos Learning

Year 2025, Volume: 7 Issue: 1, 61 - 69

Akhila Henry , Nithin Nagaraj , Rajan Sundaravaradhan

https://doi.org/10.51537/chaos.1560943

Abstract

This study explores the realm of chaotic dynamics, Neurochaos Learning (a brain-inspired machine learning paradigm) and Normal numbers, focusing on the introduction of a novel chaotic trajectory termed the Universal Orbit. The study investigates the characteristics and generation of universal orbits within two prominent chaotic maps: the Decimal Shift Map and the Gauss Map. It explores the set of points capable of forming such orbits, revealing connections with normal numbers and continued fractions. Points within the interval (0, 1) can produce universal orbits under specific conditions, highlighting the intricate relationship between machine learning, chaotic dynamics and number theory. While not all points forming universal orbits are normal numbers, the trajectory of a normal number may represent a universal orbit (under certain conditions). When employing the universal orbit for feature extraction in Neurochaos Learning, the firing time feature can be interpreted by establishing an upper bound and examining its trend. Future research aims to identify sets of points producing universal orbits under various chaotic maps, intending to enhance the performance of algorithms like the Neurochaos Learning algorithm. This study contributes to advancing our understanding of chaotic systems and their applications in artificial intelligence.

Keywords

Universal orbit, Decimal shift map, Gauss map, Normal numbers, Neurochaos learning, Brain-inspired machine learning

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