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

This paper introduces a class of $q$ -deformed fractional order maps that unify memoryless and classical fractional dynamics by replacing binomial kernels with Gaussian coefficients, revealing a continuous transition of stability regions from circles to cardioids via crescents and demonstrating how deformation parameters control the shift from power-law to exponential memory decay.

Original authors: Sachin Bhalekar, Prashant M. Gade

Published 2026-07-20✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Sachin Bhalekar, Prashant M. Gade

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future of a bouncing ball, a growing population, or even the stock market. In the real world, things don't just react to what is happening right now; they carry a memory of what happened yesterday, last week, or even years ago. This "memory" changes how the system behaves. In the world of math and physics, scientists use special tools called "maps" to model these systems. Some maps are simple and forgetful, like a ball bouncing in a vacuum where only the current bounce matters. Others are "fractional," meaning they have a long, heavy memory that stretches far back in time, following a slow, fading rule called a "power law." But what if the memory wasn't just a simple fading line? What if the memory could be stretched, squeezed, or reshaped into something entirely new? This is the playground of "q-deformation," a mathematical trick that lets scientists tweak the rules of memory to see how stability and chaos dance together.

This paper introduces a new family of mathematical maps that act like a shape-shifting bridge between a forgetful system and one with a deep, long-range memory. The authors, Sachin Bhalekar and Prashant M. Gade, take the standard "memory kernel" (the mathematical recipe for how much the past influences the present) and replace the usual numbers with "Gaussian" or "q-binomial" coefficients. Think of this as taking a standard recipe for a cake and swapping the flour for a magical, stretchy dough. By adjusting a single dial, called the parameter $q$ , they can morph the system's behavior. When the dial is set to zero, the system is forgetful and its stability looks like a perfect circle. When the dial is set to one, it becomes a classic fractional system with a long memory, and its stability shape transforms into a heart-like "cardioid." But the real magic happens in between: as they turn the dial, the stability shape morphs into a crescent moon, a shape that had never been seen in this specific context before.

The researchers didn't just guess this; they used advanced math tools like the Z-transform to derive exact equations describing these shapes. They found that the "memory" itself changes character. In the old fractional maps, the memory fades slowly like a whisper that never quite dies out (a power-law decay). In their new q-deformed maps, the memory fades quickly and then settles into a steady, finite value, like a sponge that soaks up water until it's full and stops absorbing. They also expanded this idea to include a second dial, $p$ , creating a $(p, q)$ -deformed framework. This allows them to create memories that decay exponentially fast, or even grow, depending on how the dials are set. Through computer simulations of a famous "logistic map" (a model often used for population growth), they showed that these new shapes accurately predict when a system will stay calm and when it will spiral into chaos. The paper suggests that by tuning these deformation parameters, we can design systems with very specific memory profiles, offering a new geometric way to understand how memory influences stability in complex, nonlinear worlds.

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

Problem Statement
Fractional order maps offer a compact framework for modeling discrete-time systems with memory, yet their classical formulations rely on binomial memory kernels that exhibit power-law decay. This long-range memory results in stability regions (typically cardioid-shaped) that differ significantly from the circular stability regions of memoryless discrete maps. A gap exists in understanding how the memory structure and associated stability properties change when the underlying binomial coefficients are deformed. Specifically, there is a need for a unified framework that interpolates between memoryless dynamics and classical fractional dynamics, and a method to analyze how deformation parameters influence the geometry of stability regions and the asymptotic behavior of memory kernels.

Methodology
The authors introduce a class of $q$ -deformed fractional order maps by replacing the classical binomial memory kernel with Gaussian ( $q$ -) binomial coefficients. The study proceeds through the following analytical steps:

Mathematical Formulation: The classical fractional map is generalized by substituting the standard binomial coefficient $\binom{t-j+\alpha-1}{t-j}$ with its $q$ -analogue $\binom{t-j+\alpha-1}{t-j}_q$ . This is expressed using $q$ -Pochhammer symbols.
Stability Analysis: The Z-transform is applied to the generalized map to derive the characteristic equation. The stability of the zero solution is determined by analyzing the roots of this equation in the complex plane.
Boundary Derivation: The boundary of the stable region is derived parametrically by setting $z = e^{i\theta}$ . The authors examine the limits $q \to 0$ (memoryless) and $q \to 1$ (classical fractional) to recover known results.
Extension to (p, q)-Deformation: The framework is extended to $(p, q)$ -binomial coefficients, introducing a second deformation parameter $p$ (where $p = q^\rho$ ). This allows for the investigation of memory kernels that can exhibit exponential decay or growth, depending on the relationship between $p$ and $q$ .
Nonlinear Analysis: The linear stability conditions are applied to nonlinear logistic-type maps by linearizing around equilibrium points.
Numerical Validation: Theoretical stability boundaries and asymptotic behaviors are verified through numerical simulations of the maps for various parameter sets ( $\alpha, q, p, a$ ).

Key Contributions

Unified Framework: The paper establishes a continuous interpolation between memoryless discrete maps ( $q=0$ ) and classical fractional order maps ( $q=1$ ) via $q$ -deformed fractional maps.
Geometric Transition of Stability Regions: The authors demonstrate that the stability region in the complex plane undergoes a continuous geometric deformation as the parameter $q$ varies. The region transitions from a circle (at $q=0$ ) to a crescent shape (for intermediate $0 < q < 1$ ) and finally to a cardioid (at $q=1$ ).
Asymptotic Behavior of Kernels: The study characterizes the asymptotic decay of the memory kernels:
- Classical fractional kernels exhibit algebraic power-law decay.
- $q$ -deformed kernels approach a finite, non-zero asymptotic value (asymptotic saturation) with exponential corrections.
- $(p, q)$ -deformed kernels (with $p > 1$ ) exhibit exponential decay, while those with $q < p < 1$ exhibit exponential growth.
Generalization to (p, q)-Calculus: The work extends the theory to a broader $(p, q)$ -deformed framework, showing that the $q$ -deformed case is a special instance ( $p=1$ ) and providing a hierarchy of memory structures ranging from power-law to exponential.

Results

Characteristic Equations: Explicit characteristic equations involving $q$ -Pochhammer symbols and $(p, q)$ -hypergeometric functions were derived. For the $q$ -deformed case, the stability condition for real parameter $a$ is $1 - \Lambda_{q,\alpha} < a < 1$ , where $\Lambda_{q,\alpha}$ is a specific ratio of infinite products.
Stability Geometry: Numerical plots confirm that varying $q$ smoothly deforms the stability boundary. For $\alpha = 0.082, 0.3, 0.6, 0.8$ , the transition from circle to crescent to cardioid is clearly visualized.
Nonlinear Dynamics: In the context of the logistic map $f(x) = rx(1-x)$, the stability of fixed points is shown to depend on the derivative $f'(x^*)$ falling within the derived stability bounds. Simulations demonstrate stable convergence, periodic oscillations, and chaotic behavior consistent with the theoretical stability regions.
Kernel Decay: Numerical plots of the memory kernels confirm the theoretical predictions: $q$ -deformed kernels saturate to a constant, while $(p, q)$ -deformed kernels (with $\rho < 0$ ) decay exponentially on a semi-logarithmic scale.

Significance
The paper claims to provide a geometric and analytic framework for understanding how deformed memory kernels influence stability in discrete nonlinear dynamics. By introducing $q$ -deformation directly into the memory kernel rather than the map function itself, the work offers a new perspective on the interplay between memory effects and stability geometry. The results unify previously distinct stability regions (circles and cardioids) into a single continuum mediated by crescent-shaped domains. Furthermore, the extension to $(p, q)$ -deformations enriches the memory structure, allowing for the modeling of systems with exponentially localized memory, thereby broadening the applicability of fractional calculus concepts to systems with different memory profiles. The work highlights a deep interplay between special functions (specifically $q$ - and $(p, q)$ -hypergeometric structures), combinatorial analysis, and the geometric structures of stability in dynamical systems.

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