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

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