Evolutionary Mathematical Science, Fractional Modeling and Artificial Intelligence of Nonlinear Dynamics in Complex Systems

Year 2022, Volume: 4 Issue: 3, 111 - 118, 30.11.2022

Yeliz Karaca Dumitru Baleanu

Abstract

Complex problems in nonlinear dynamics foreground the critical support of artificial phenomena so that each domain of complex systems can generate applicable answers and solutions to the pressing challenges. This sort of view is capable of serving the needs of different aspects of complexity by minimizing the problems of complexity whose solutions are based on advanced mathematical foundations and analogous algorithmic models consisting of numerous applied aspects of complexity. Evolutionary processes, nonlinearity and all the other dimensions of complexity lie at the pedestal of time, reveal time and occur within time. In the ever-evolving landscape and variations, with causality breaking down, the idea of complexity can be stated to be a part of unifying and revolutionary scientific framework to expound complex systems whose behavior is perplexing to predict and control with the ultimate goal of attaining a global understanding related to many branches of possible states as well as high-dimensional manifolds, while at the same time keeping abreast with actuality along the evolutionary and historical path, which itself, has also been through different critical points on the manifold. In view of these, we put forth the features of complexity of varying phenomena, properties of evolution and adaptation, memory effects, nonlinear dynamic system qualities, the importance of chaos theory and applications of related aspects in this study. In addition, processes of fractional dynamics, differentiation and systems in complex systems as well as the dynamical processes and dynamical systems of fractional order with respect to natural and artificial phenomena are discussed in terms of their mathematical modeling. Fractional calculus and fractional-order calculus approach to provide novel models with fractional-order calculus as employed in machine learning algorithms to be able to attain optimized solutions are also set forth besides the justification of the need to develop analytical and numerical methods. Subsequently, algorithmic complexity and its goal towards ensuring a more effective handling of efficient algorithms in computational sciences is stated with regard to the classification of computational problems. We further point out the neural networks, as descriptive models, for providing the means to gather, store and use experiential knowledge as well as Artificial Neural Networks (ANNs) in relation to their employment for handling experimental data in different complex domains. Furthermore, the importance of generating applicable solutions to problems for various engineering areas, medicine, biology, mathematical science, applied disciplines and data science, among many others, is discussed in detail along with an emphasis on power of predictability, relying on mathematical sciences, with Artificial Intelligence (AI) and machine learning being at the pedestal and intersection with different fields which are characterized by complex, chaotic, nonlinear, dynamic and transient components to validate the significance of optimized approaches both in real systems and in related realms.

Keywords

Complex systems, Chaos theory, Chaos-order and complexity, Computational complexity, Fractal operators, Fractional calculus, Fractional dynamics, Evolutionary models, Fractional-order algorithms, Complex-valued neural networks, Data science, Mathematical biology and medicine, Prediction of changes, Fractional integro-differentiation, Artificial Intelligence

References

• Baleanu, D. and Y. Karaca, 2022 Mittag-leffler functions with heavy-tailed distributions’ algorithm based on different biology datasets to be fit for optimum mathematical models’ strategies. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 117–132, Elsevier.
• Baleanu, D., R. L. Magin, S. Bhalekar, and V. Daftardar-Gejji, 2015 Chaos in the fractional order nonlinear bloch equation with delay. Communications in Nonlinear Science and Numerical Simulation 25: 41–49.
• Clegg, B., 2020 Everyday Chaos: The Mathematics of Unpredictability, from the Weather to the Stock Market. MIT Press.
• Diethelm, K., V. Kiryakova, Y. Luchko, J. Machado, and V. E. Tarasov, 2022 Trends, directions for further research, and some open problems of fractional calculus. Nonlinear Dynamics pp. 1–26.
• Goldenfeld, N. and L. P. Kadanoff, 1999 Simple lessons from complexity. Science 284: 87–89.
• Karaca, Y., 2022a Global attractivity, asymptotic stability and blowup points for nonlinear functional-integral equations’solutions and applications in banach space BC (R+) with computational complexity. Fractals 30: 2240188.
• Karaca, Y., 2022b Multi-chaos, fractal and multi-fractional AI in different complex systems. In Multi-Chaos, Fractal and Multifractional Artificial Intelligence of Different Complex Systems, pp. 21–54, Elsevier.
• Karaca, Y., 2022c Theory of complexity, origin and complex systems. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 9–20, Elsevier.
• Karaca, Y. and D. Baleanu, 2022a Artificial neural network modeling of systems biology datasets fit based on mittag-leffler functions with heavy-tailed distributions for diagnostic and predictive precision medicine. In Multi-Chaos, Fractal and Multifractional Artificial Intelligence of Different Complex Systems, pp. 133–148, Elsevier.
• Karaca, Y. and D. Baleanu, 2022b Computational fractional-order calculus and classical calculus AI for comparative differentiability prediction analyses of complex-systems-grounded paradigm. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 149–168, Elsevier.
• Karaca, Y., Y.-D. Zhang, A. D. Dursun, and S.-H.Wang, 2022 Multifractal complexity analysis-based dynamic media text categorization models by natural language processing with BERT. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 95–115, Elsevier.
• Kia, B., J. F. Lindner, and W. L. Ditto, 2017 Nonlinear Dynamics as an engine of computation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375: 20160222.
• Kia, B., A. Mendes, A. Parnami, R. George, K. Mobley, et al., 2020 Nonlinear dynamics based machine learning: Utilizing dynamics-based flexibility of nonlinear circuits to implement different functions. Plos one 15: e0228534.
• Kober, H., 1940 On fractional integrals and derivatives. The quarterly journal of mathematics pp. 193–211.
• Mainzer, K. and K. Mainzer, 1997 Thinking in complexity: The complex dynamics of matter, mind, and mankind, volume 3. Springer.
• Mateos, J. L., 2009 Complex systems and non-linear dynamics. Fundamentals of Physıcs-Volume I p. 356.
• Mitchell, M., 2009 Complexity: A guided tour. Oxford university press.
• Monje, C. A., Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu-Batlle, 2010 Fractional-order systems and controls: fundamentals and applications. Springer Science & Business Media.
• Sanjuán, M. A., 2021 Artificial intelligence, chaos, prediction and understanding in science. International Journal of Bifurcation and Chaos 31: 2150173.
• Schueler, G. J., 1996 The unpredictability of complex systems. Journal of the Washington Academy of Sciences pp. 3–12.
• Sun, K., S. He, and H. Wang, 2022 Dynamics of fractional order chaotic systems. In Solution and Characteristic Analysis of Fractional-Order Chaotic Systems, pp. 77–115, Springer.
• Tang, Y., J. Kurths,W. Lin, E. Ott, and L. Kocarev, 2020 Introduction to focus issue: When machine learning meets complex systems: Networks, chaos, and nonlinear dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 063151.
• Waldrop, M. M., 1993 Complexity: The emerging science at the edge of order and chaos. Simon and Schuster.
• West, B. J., 2022 Fractional calculus and the future of science.