The Unreasonable Effectiveness of the Chaotic Tent Map in Engineering Applications

Authors : Nithin NAGARAJ

Pages : 197-204

Doi:10.51537/chaos.1196653

View : 16 | Download : 3

Publication Date : 2022-12-31

Article Type : Research Paper

Abstract :From decimal expansion of real numbers to complex behaviour in physical, biological and human-made systems, deterministic chaos is ubiquitous. One of the simplest examples of a nonlinear dynamical system that exhibits chaos is the well known 1-dimensional piecewise linear Tent map. The Tent map insert ignore into journalissuearticles values(and their skewed cousins); are instances of a larger family of maps namely Generalized Luröth Series insert ignore into journalissuearticles values(GLS); which are studied for their rich number theoretic and ergodic properties. In this work, we discuss the unreasonable effectiveness of the Tent map and their generalizations insert ignore into journalissuearticles values(GLS maps); in a number of applications in electronics, communication and computer engineering. To list a few of these applications: insert ignore into journalissuearticles values(a); GLS-coding: a lossless data compression algorithm for i.i.d sources is Shannon optimal and is in fact a generalization of the popular Arithmetic Coding algorithm used in the image compression standard JPEG2000; insert ignore into journalissuearticles values(b); GLS maps are used as neurons in the recently proposed Neurochaos Learning architecture which delivers state-of-the-art performance in classification tasks; insert ignore into journalissuearticles values(c); GLS maps are ideal candidates for chaos-based computing since they can simulate XOR, NAND and other gates and for dense storage of information for efficient search and retrieval; insert ignore into journalissuearticles values(d); Noise-resistant versions of GLS maps are useful for signal multiplexing in the presence of noise and error detection; insert ignore into journalissuearticles values(e); GLS maps are shown to be useful in a number of cryptographic protocols - for joint compression and encryption and also for generating pseudo-random numbers. The unique properties and rich features of the Tent Map insert ignore into journalissuearticles values(GLS maps); that enable these wide variety of engineering applications will be investigated. A list of open problems are indicated as well.
Keywords : Tent map, Chaos, Generalized Luröth Series, Compression, Coding, Cryptography, Neurochaos Learning, Ergodicity