Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras

Chaouki Aouiti; Farah Dridi

doi:10.62762/JNDA.2025.838385

Volume 2, Issue 1, Journal of Nonlinear Dynamics and Applications

Volume 2, Issue 1, 2026

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Journal of Nonlinear Dynamics and Applications, Volume 2, Issue 1, 2026: 1-12

Free to Read | Research Article | 21 January 2026

Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras

Chaouki Aouiti 1 *

Farah Dridi 1

1 GAMA Laboratory (LR21ES10), Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Bizerte 7021, Tunisia

* Corresponding Author: Chaouki Aouiti, [email protected]

DOI: 10.62762/JNDA.2025.838385

ARK: ark:/57805/jnda.2025.838385

Received: 23 October 2025, Accepted: 09 January 2026, Published: 21 January 2026

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Abstract

The primary objective of this work is to establish the existence, uniqueness, and exponential stability of piecewise weighted pseudo–almost automorphic solutions for impulsive high-order Hopfield neural networks formulated within Clifford algebras. Using the Banach fixed-point principle together with a suitably adapted Gronwall–Bellman inequality, we derive novel and verifiable sufficient conditions that ensure these qualitative properties. The main contributions are as follows: (i) this study is the first to analyze weighted pseudo–almost automorphic (WPAA) dynamics for impulsive high-order Hopfield neural networks directly in the Clifford algebra setting, without reducing the model to real-valued components; (ii) it offers a unified framework that accommodates both first- and second-order synaptic interactions under impulsive perturbations and mixed delays; and (iii) the resulting conditions explicitly capture the geometric structure of Clifford-valued states, providing a broader and algebraically consistent formulation compared to real or quaternion-valued models. The theoretical findings are further supported by a numerical example demonstrating their applicability and effectiveness.

Graphical Abstract

Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras

Keywords

impulsive systems

WPAA-functions

HOHNNs

Gronwall–Bellman inequality

exponential stability

clifford algebra

Data Availability Statement

No datasets were generated or analyzed during the current study.

Funding

This work was supported without any funding.

Conflicts of Interest

The authors declare no conflicts of interest.

AI Use Statement

The authors declare that no generative AI was used in the preparation of this manuscript.

Ethical Approval and Consent to Participate

Not applicable.

References

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APA Style

Aouiti, C.& Dridi, F. (2025). Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras. Journal of Nonlinear Dynamics and Applications, 2(1), 1–12. https://doi.org/10.62762/JNDA.2025.838385

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TY  - JOUR
AU  - Aouiti, Chaouki
AU  - Dridi, Farah
PY  - 2026
DA  - 2026/01/21
TI  - Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras
JO  - Journal of Nonlinear Dynamics and Applications
T2  - Journal of Nonlinear Dynamics and Applications
JF  - Journal of Nonlinear Dynamics and Applications
VL  - 2
IS  - 1
SP  - 1
EP  - 12
DO  - 10.62762/JNDA.2025.838385
UR  - https://www.icck.org/article/abs/JNDA.2025.838385
KW  - impulsive systems
KW  - WPAA-functions
KW  - HOHNNs
KW  - Gronwall–Bellman inequality
KW  - exponential stability
KW  - clifford algebra
AB  - The primary objective of this work is to establish the existence, uniqueness, and exponential stability of piecewise weighted pseudo–almost automorphic solutions for impulsive high-order Hopfield neural networks formulated within Clifford algebras. Using the Banach fixed-point principle together with a suitably adapted Gronwall–Bellman inequality, we derive novel and verifiable sufficient conditions that ensure these qualitative properties. The main contributions are as follows: (i) this study is the first to analyze weighted pseudo–almost automorphic (WPAA) dynamics for impulsive high-order Hopfield neural networks directly in the Clifford algebra setting, without reducing the model to real-valued components; (ii) it offers a unified framework that accommodates both first- and second-order synaptic interactions under impulsive perturbations and mixed delays; and (iii) the resulting conditions explicitly capture the geometric structure of Clifford-valued states, providing a broader and algebraically consistent formulation compared to real or quaternion-valued models. The theoretical findings are further supported by a numerical example demonstrating their applicability and effectiveness.
SN  - 3069-6313
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  -

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@article{Aouiti2026Advances,
  author = {Chaouki Aouiti and Farah Dridi},
  title = {Advances in the Mathematical Theory of WPAA Dynamics for Impulsive High Order Neural Systems in Clifford Algebras},
  journal = {Journal of Nonlinear Dynamics and Applications},
  year = {2026},
  volume = {2},
  number = {1},
  pages = {1-12},
  doi = {10.62762/JNDA.2025.838385},
  url = {https://www.icck.org/article/abs/JNDA.2025.838385},
  abstract = {The primary objective of this work is to establish the existence, uniqueness, and exponential stability of piecewise weighted pseudo–almost automorphic solutions for impulsive high-order Hopfield neural networks formulated within Clifford algebras. Using the Banach fixed-point principle together with a suitably adapted Gronwall–Bellman inequality, we derive novel and verifiable sufficient conditions that ensure these qualitative properties. The main contributions are as follows: (i) this study is the first to analyze weighted pseudo–almost automorphic (WPAA) dynamics for impulsive high-order Hopfield neural networks directly in the Clifford algebra setting, without reducing the model to real-valued components; (ii) it offers a unified framework that accommodates both first- and second-order synaptic interactions under impulsive perturbations and mixed delays; and (iii) the resulting conditions explicitly capture the geometric structure of Clifford-valued states, providing a broader and algebraically consistent formulation compared to real or quaternion-valued models. The theoretical findings are further supported by a numerical example demonstrating their applicability and effectiveness.},
  keywords = {impulsive systems, WPAA-functions, HOHNNs, Gronwall–Bellman inequality, exponential stability, clifford algebra},
  issn = {3069-6313},
  publisher = {Institute of Central Computation and Knowledge}
}

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