Draft:Shukur Hyperchaotic Map

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Declined by Theroadislong 19 hours ago. Last edited by Citation bot 2 seconds ago. Reviewer: Inform author.

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Submission declined on 2 June 2025 by Sophisticatedevening (talk).

This draft's references do not show that the subject qualifies for a Wikipedia article. In summary, the draft needs multiple published sources that are:

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Declined by Sophisticatedevening 38 hours ago.

Comment: independent sources? Theroadislong (talk) 16:58, 3 June 2025 (UTC)

Shukur Hyperchaotic Map

The Shukur Hyperchaotic Map is a three-dimensional discrete-time mathematical map that exhibits hyperchaos, meaning it has more than one positive Lyapunov exponent. The map was introduced by Ali A. Shukur in a 2025 study for use in secure color image encryption schemes due to its high sensitivity to initial conditions and complex dynamic behavior.

Mathematical Definition

The map is defined as follows:

SH_{a,l,i}={\begin{cases}s_{n+1}=s_{n}+a\cdot \sin(u_{n})\mod 2\pi \\u_{n+1}=u_{n}+l\cdot s_{n}\sin(v_{n})\mod 2\pi \\v_{n+1}=v_{n}+i\cdot u_{n}\tanh(s_{n})\mod 2\pi \end{cases}}

where $a$ , $l$ , $i$ are real-valued parameters.

Dynamic Properties

The map exhibits:

Hyperchaotic behavior for a wide range of parameters.
Multiple positive Lyapunov exponents.
High sensitivity to initial conditions.
Dense and unpredictable phase space trajectories.

Stability Analysis

The system has four fixed points:

E_{1}=(0,0,0),\quad E_{2}=(0,0,\pi ),\quad E_{3}=(0,\pi ,0),\quad E_{4}=(0,\pi ,\pi )

At these points, the Jacobian matrix has all eigenvalues equal to 1, indicating marginal stability. This means the system neither converges nor diverges under small perturbations, but remains on a bounded trajectory due to the modulo operation.

Applications

The function was originally developed for image encryption. Its complex dynamics and extreme sensitivity make it suitable for generating encryption keys and scrambling pixel data in color images. It has been demonstrated to resist common cryptanalytic attacks.

Publication

Obaid, Mohammed Jabbar; Neamah, Ammar Ali; Shukur, Ali A.; Pham, Viet-Thanh; Grassi, Giuseppe (May 2025). "A Reliable Color Image Encryption Scheme Based on a Novel Dual-Wing Hyperchaotic Map". Expert Systems with Applications. 289: 128237. doi:10.1016/j.eswa.2025.128237.