# A new approach to fuzzy sets: Application to the design of nonlinear time-series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations

## TL;DR

This paper introduces a novel fuzzy set framework using the $_{
}$-function, enabling applications in nonlinear time-series, pattern design, and limit-cycle oscillations, with the fuzziness parameter controlling symmetry breaking and pattern shaping.

## Contribution

The paper presents a new fuzzy set approach with the $_{
}$-function, providing general formulas for switching functions, pattern design, and a theorem on shaping limit cycles in nonlinear systems.

## Key findings

- Fuzziness parameter $
$ controls deviation from crisp sets.
- New formulas for switching functions and pattern design.
- Theorem on shaping limit cycles far from bifurcations.

## Abstract

It is shown that characteristic functions of sets can be made fuzzy by means of the $\mathcal{B}_{\kappa}$-function, recently introduced by the author, where the fuzziness parameter $\kappa \in \mathbb{R}$ controls how much a fuzzy set deviates from the crisp set obtained in the limit $\kappa \to 0$. As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time-series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter $\kappa$ plays an analogous role to temperature in physical applications and may be used to break the symmetry of spatial patterns. As a further, important application, we establish a theorem on the shaping of limit cycle oscillations far from bifurcations in smooth deterministic nonlinear dynamical systems governed by differential equations. Following this application, we briefly discuss a generalization of the Stuart-Landau equation to non-sinusoidal oscillators obtained as a consequence of our theorem.