A Simple Nonlinear Circuit Contains an Infinite Number of Functions

Abstract

The complex dynamics of a simple nonlinear circuit contains an infinite number of functions. Specifically, this paper shows that the number of different functions that a nonlinear or chaotic circuit can implement exponentially increases as the circuit evolves in time, and this exponential increase is quantified with an exponent that is named the computing exponent. This brief argues that a simple nonlinear circuit that illustrates a rich, complex dynamics can embody infinitely many different functions, each of which can be dynamically selected. In practice not all of these functions may be accessible due to factors such as noise or instability of the functions. However, these infinitely many functions do exist within the dynamics of the nonlinear circuit regardless of accessibility or inaccessibility of the functions in practice. This nonlinear-dynamics-based approach to computation opens the door for implementing extremely slim, low-power circuits that are capable of performing many different types of functions.

Implementing Boolean Functions in Hybrid Digital-Analog Systems

“…Moreover, a large number of iterations are needed to obtain these functions. For example, for 4 inputs bits and 16 control signals, we get about 16 distinct functions after 6 iterations and 56 functions after 10 iterations [5]. This number saturates at about 22000 after 30 iterations which are about one−third of all possible functions (65536).…”

“…where τ is threshold, and T is the number of iterations of the nonlinear circuit. The number of functions that can be obtained depends on the computing exponent [5] of the nonlinear circuit, and the maximum number of functions can be obtained in the chaotic region. The actual number of functions that we obtain is less than 2 2 m as some of the control functions yield the same function even in the chaotic region [5] and all possible functions can be obtained only if n 2 m .…”

“…The impending failure of Moore's law has led to the development of novel approaches like nonlinear or chaos computing (CC) [1][2][3][4] to get more computation out of a limited number of transistors. It was shown recently that a simple nonlinear circuit can theoretically implement an infinite number of functions [5] which in practice are limited by amplification of ambient noise with each iteration. Researchers have described an integrated circuit implementation of CC with a nonlinear circuit [6] as well as other transistor-based implementations of chaotic circuits [7,8].…”

“…The ambient noise is amplified exponentially when the circuits are operating in the chaotic regime thereby limiting the number of iterations for which we can use the nonlinear circuit. The number of implementable functions increases exponentially with the number of iterations [5], so it is crucial that we can maximize the iterations for which output can be obtained successfully. One method to achieve this objective is to suppress the noise evolution through coupled redundant nonlinear circuits as the local noise in different circuits averages out and the overall deviation in the output is reduced [9][10][11].…”

Nonlinear dynamics as an engine of computation

“…Even in non-chaotic regimes, the nonlinear convolutions of an iterated map, as in figure 3, can scramble orbits among partitions of the invariant set and thereby create diverse functions. To motivate the computing exponent definition, consider a dynamics-based computing example [5,8]. If the dynamical system is the nonlinear map of the unit interval…”