Maps of q-deformed fractional order: From circle to cardioid via crescent

We introduce a class of \(q\)-deformed fractional order maps by replacing the classical binomial memory kernel in discrete fractional dynamics with Gaussian (\(q\)-) binomial coefficients.

The proposed framework interpolates between memoryless discrete maps and classical fractional order maps, unifying circle- and cardioid-shaped stability regions through intermediate crescent geometries.

Using the \(Z\)-transform and the \(q\)-binomial theorem, we derive characteristic equations and determine the associated stability regions in the complex plane.

We further analyze the asymptotic behavior of the memory kernels, showing that the classical fractional kernel exhibits power-law decay, whereas the \(q\)- and \((p,q)\)-deformed kernels exhibit exponential-type localization.

The theory is extended to nonlinear logistic-type maps and to a broader \((p,q)\)-deformed framework, where the regime \(p>q\) yields decaying memory kernels and stable dynamics.

Numerical simulations illustrate the theoretical results and the interplay between the deformation parameters, memory effects, and stability geometry.