The Unreasonable Effectiveness of the Chaotic Tent Map in Engineering Applications

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 (and their skewed cousins) are instances of a larger family of maps namely Generalized Luröth Series (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 (GLS maps) in a number of applications in electronics, communication and computer engineering. To list a few of these applications: (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; (b) GLS maps are used as neurons in the recently proposed Neurochaos Learning architecture which delivers state-of-the-art performance in classification tasks; (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; (d) Noise-resistant versions of GLS maps are useful for signal multiplexing in the presence of noise and error detection; (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 (GLS maps) that enable these wide variety of engineering applications will be investigated. A list of open problems are indicated as well.

Chaotic Maps with Tunable Mean Value—Application to a UAV Surveillance Mission

“…For this family, simple rules can be followed for appropriately tuning the control parameters in order to affect the statistical mean of the resulting time series. Piecewise maps are chosen here, as they are very prominent in applications [3][4][5][18][19][20][21][22][23], and their shape is easy to control. They can also yield chaotic behavior for a wide range of parameter values.…”

“…Five maps are proposed and studied based on this general model. One map is a generalization of the skewed tent map [20], two are modifications of the maps proposed in [18], and the last two are combinations of the previous ones.…”