Advanced Fractional Mathematics, Fractional Calculus, Algorithms and Artificial Intelligence with Applications in Complex Chaotic Systems

Year 2023, Volume: 5 Issue: 4, 257 - 266, 31.12.2023

Yeliz Karaca , Dumitru Baleanu

Abstract

Chaos, comprehended characteristically, is the mathematical property of a dynamical system which is a deterministic mathematical model in which time can be either continuous or discrete as a variable. These respective models are investigated as mathematical objects or can be employed for describing a target system. As a long-term aperiodic and random-like behavior manifested by many nonlinear complex dynamic systems, chaos induces that the system itself is inherently unstable and disordered, which requires the revealing of representative and accessible paths towards affluence of complexity and experimental processes so that novelty, diversity and robustness can be generated. Hence, complexity theory focuses on non-deterministic systems, whereas chaos theory rests on deterministic systems. These entailments demonstrate that chaos and complexity theory provide a synthesis of emerging wholes of individual components rather than the orientation of analyzing systems in isolation. Therefore, mathematical modeling and scientific computing are among the chief tools to solve the challenges and problems related to complex and chaotic systems through innovative ways ascribed to data science with a precisely tailored approach which can examine the data applied. The complexity definitions need to be weighed over different data offering a highly extensive applicability spectrum with more practicality and convenience owing to the fact that the respective processes lie in the concrete mathematical foundations, which all may as well indicate that the methods are required to be examined thoroughly regarding their mathematical foundation along with the related methods to be applied. Furthermore, making use of chaos theory can be considered to be a way to better understand the internal machinations of neural networks, and the amalgamation of chaos theory as well as Artificial Intelligence (AI) can open up stimulating possibilities acting instrumental to tackle diverse challenges, with AI algorithms providing improvements in the predictive capabilities via the introduction of adaptability, enabling chaos theory to respond to even slight changes in the input data, which results in a higher level of predictive accuracy. Therefore, chaos-based algorithms are employed for the optimization of neural network architectures and training processes. Fractional mathematics, with the application of fractional calculus techniques geared towards the problems’ solutions, describes the existence characteristics of complex natural, applied sciences, scientific, engineering related and medical systems more accurately to reflect the actual state properties co-evolving entities and patterns of the systems concerning nonlinear dynamic systems and modeling complexity evolution with fractional chaotic and complex systems. Complexity entails holistic understanding of various processes through multi-stage integrative models across spanning scales for expounding complex systems while following actuality across evolutionary path. Moreover, Fractional Calculus (FC), related to the dynamics of complicated real-world problems, ensures emerging processes adopting fractional dynamics rather than the ordinary integer-ordered ones, which means the related differential equations feature non integer valued derivatives. Given that slight perturbation leads to a significantly divergent future concatenation of events, pinning down the state of different systems precisely can enable one to unveil uncertainty to some extent. Predicting the future evolution of chaotic systems can screen the direction towards distant horizons with extensive applications in order to understand the internal machinations of neural and chaotic complex systems. Even though many problems are solvable and have been solved, they remain to be open constantly under transient circumstances. Thus, fields with a broad range of spectrum range from mathematics, physics, biology, fluid mechanics, medicine, engineering, image analysis, based on differing perspectives in our special issue which presents a compilation of recent research elaborating on the related advances in foundations, theory, methodology and topic-based implementations regarding fractals, fractal methodology, fractal spline, non-differentiable fractal functions, fractional calculus, fractional mathematics, fractional differential equations, differential equations (PDEs, ODEs), chaos, bifurcation, Lie symmetry, stability, sensitivity, deep learning approaches, machine learning, and so forth through advanced fractional mathematics, fractional calculus, data intensive schemes, algorithms and machine learning applications surrounding complex chaotic systems.

Keywords

Fractional mathematics, Complexity, Fractional Calculus, Deep learning, Computational complexity, Fractal methodology, Fractalization, Complex versus Chaotic Systems and Chaos, Bifurcation, Control and optimization, Strange attractors, Approximation theory, Lie symmetry, Complex Chaotic Systems, Complex systems, Data-intensive computational application processes, Real data interpolation and applications, Differential equations, Machine Learning, Deep neural network

Thanks

Acknowledgements We, as the Editors of our special issue, would like to extend our sincere thanks to Professor Akif Akgül, the Editor-in-Chief, the Editorial Board members and the staff of the Chaos Theory and Applications Journal for enabling the publication of our special issue where significant contributions from the authors from diverse fields have been included. We would also like to extend our many sincere thanks to all the referees for their rigorous reviewing processes during the related course of time. Last but not least, much earnest appreciation is conveyed to all the authors who have contributed to our special issue with their papers. Special Acknowledgements Yeliz Karaca would like to extend her genuine gratitude to late Professor Abul Hasan Siddiqi (1943-2020), a highly prominent and respectable Indian mathematician and Professor of Applied Mathematics, whose academic positions include being the President of the Indian Society of Industrial and Applied Mathematics (ISIAM) and Editor-in-Chief of a series of Industrial and Applied Mathematics of Springer Nature. Some of the papers accepted in this special issue have been through the International Conference on Applied and Industrial Mathematics (ICAIM) held by the A.H. Siddiqi Centre for Advanced Research in Applied Mathematics & Physics (CARAMP). Dr. Karaca would like to extend her sincere thanks to all the members of CARAMP for their efforts and continual projects which were initiated by the dedicated work of Professor Abul Hasan Siddiqi and his team.

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