Characterization of Keçeci Number Systems as Chaotic and Hyperchaotic Maps

Mehmet Keçeci

ORCID:  https://orcid.org/0000-0001-9937-9839

Received: 08.26.2025

Abstract:

This paper provides a theoretical characterisation of Keçeci Number Systems as discrete-time dynamical systems, investigating their chaotic and hyperchaotic behaviours. Keçeci Numbers are generated through an iterative process based on addition, conditional division, and a primality test, analogous to the Collatz Conjecture. The most significant feature of this process is its definition across 20 distinct algebraic structures, ranging from real and complex numbers to higher-dimensional and non-standard systems such as quaternions, neutrosophic, and bicomplex numbers. The study posits that the Keçeci Number generator can be modelled as a map of the form x_n+1 = f(x_n). Here, the function f represents the set of rules that define the non-linear and conditional nature of the system. Specifically, the application of different mathematical operations based on a number's divisibility by 2 or 3, or its primality, introduces a strong non-linearity. This structure has the potential to lead to extreme sensitivity to initial conditions—a hallmark of chaotic systems, often referred to as the "butterfly effect". The aperiodic and bounded nature of the system's trajectories are further indicators of chaotic behaviour. More importantly, this work highlights the potential for "hyperchaos" in number systems with four or more dimensions, such as quaternions, bicomplex, and neutrosophic-complex numbers. Hyperchaotic systems are characterised by having more than one positive Lyapunov exponent, implying that the system's dynamics expand in multiple directions simultaneously, leading to more complex and unpredictable behaviour. The dynamics of Keçeci Numbers in a multi-dimensional space provides a natural framework for such hyperchaotic dynamics. The Keçeci Conjecture—which asserts that sequences converge to a periodic structure or a repeating prime representation (Keçeci Prime Number, KPN) after a finite number of steps—is reinterpreted within this chaotic framework. This conjecture may point towards the existence of "strange attractors" or limit cycles underlying the chaotic dynamics, which pull the system's trajectories towards specific regions. By positioning Keçeci Numbers at the intersection of number theory and dynamical systems, this study offers a novel pathway for analysing the complexity and richness of Collatz-like problems through the lens of modern chaos theory.

Keywords: Keçeci Numbers, Chaos Theory, Chaos, Chaotic Map, Hyperchaos, Hyperchaotic Dynamics, Hyperchaotic Map, Dynamical Number System, Dynamical Systems, Non-linear Dynamics, Collatz Conjecture, Keçeci Conjecture, Number Theory, Quaternions, Neutrosophic Numbers, Discrete Maps, Butterfly Effect, Keçeci Prime Number, KPN, Keçeci Map, Keçeci System.