Dynamical systems theory as the foundation of rule-based models of complex systems takes into consideration not only the static properties of observations but also the way systems evolve over time in response to problems of organized complexity. From population growth to swinging pendula through the movement of celestial bodies in the universe, examples of dynamical and complex systems are abundant in the form of a system whose state can exclusively specified by a set of variables and whose behaviors can be described through pre-defined rules. Conventional linear modeling paradigms may fall short in describing subtle, nonlocal as well as memory-dependent behaviors which are characteristic of multiple natural, applied, medical, biological, physical, engineered phenomena, among others. Advanced mathematical frameworks including fractals, fractional calculus, fractional dynamics, Itô calculus, wavelet, entropy, network systems, network theory, adaptative control systems, pattern formation, among others permeate in most aspects of life in integration with Artificial Intelligence (AI) techniques, machine learning, deep learning and quantum computing can be employed for enhancing decision-making processes, optimization and prediction within complex contexts. Network structures provide the abstraction of elements in a complex system as in the case of nodes and the interactions between elements as edges. Complex network theory concurrently procures methods for modeling complex systems with its methods affording significant advantages owing to its enabling of direct display of interactions between the elements of complex systems elements along with the investigation of their features, including nonlinear dynamics, self-organization, emergence, and so forth. While manifesting the attributes, behaviors and patterns of complex systems, they are acknowledged to be one of the most fundamental methods opted for to model and analyze complex systems. Network motifs are recurring, significant patterns of interconnections found within networks, and assortativity describes the tendency of nodes to connect to others that are similar in some way, such as degree. Thus, structural networks focus on the arrangement and connectivity of nodes and edges. The number of observations and amount of data pertinent to characterize complex systems as well as their dynamics along with the capabilities of processing voluminous data have provided a cross-section of current research on complex, chaotic, dynamical and nonlinear systems by bringing the focal lens onto novel experimental, observational, empirical, simulation-related, algorithmic and data-driven approaches to such systems so that patterns and behaviors thereof can be modeled appropriately. Mathematics bolsters fundamental theories like quantum mechanics, quantum field theory and general relativity. Its applicability in cellular biology, genomics and molecular biology is noted to be limited. AI has hitherto provided a transformation in science, engineering and technology with the forceful potential of discovering rules of life albeit the challenges of AI-based discovery due to nonlinearity, complexity, high dimensionality and multiscale nature of biological systems. In addition to these, fractional differential equations, as one extension of the fractional calculus theory associated with nonlocal phenomena in physics, have manifested themselves as robust mathematical means and research frameworks. Fractional differential equations can, thus, yield the evolution of various systems properly, which underpins its position in mathematics, engineering and science while establishing the stage for the description of dynamic, complex, nonlinear and chaotic phenomena. These suggest the importance of deriving explicit fractional differential equations to understand both physical mechanisms and mathematical properties of these systems in generating capable algorithms in terms of extracting symbolic representations. Furthermore, fractional dynamics provides the necessary mathematical mechanisms and machineries for the precise modeling and analysis of characteristics which in particular involve memory, nonlocality as well as fractal-like behaviors so that more accurate predictions and estimates can be attained in complex biological, physical, chemical, medical, neurological phenomena, to name a few. Nonlinear dynamics involve complex adaptive systems that exhibit nonlinear behavior, where minor alterations may cause disproportionately major effects. This also points toward chaos theory which posits that small variations in initial conditions can change system trajectories in a dramatic way. The challenges arising out of such transient, unpredictable and variational conditions require an exploration of patterns for medical, neurological, physical, biological, natural, financial, engineered, ecological, social systems and many more by considering interconnections across theoretical, modeling-based and computational aspects in line with the data at hand and state in question. This sort of holistic integrative view can provide a flexible approach and attention-grabbing highlight for modeling hereditary and memory effects in complex systems toward organizing assurance for applicable solutions. As a consequence, the theoretical reflections on how all these processes, patterns and configurations are modeled, merging all together the advanced mathematical modeling, methods, optimization, analyses, computational, emerging technologies and technology-mediated frameworks are expounded in our special issue which elaborates on, showcases and exemplifies the implications of solution-oriented and applicable approaches in real-world systems as well as other related domains. Hence, our special issue aims at extrapolating new mathematical theoretical directions in view of the sound groundwork of pure (theoretical) mathematics in conjunction with mathematical modeling, analyses and applications that could be justified from a decision-theoretic viewpoint of constructively based solution and proof procedures enhanced by exponential growth of data volume, algorithmizing, computer affordances, hardware and storage capacities in science, medicine, biology, engineering through applied sciences, and many more realms pondering on the computability of the universe.
KARACA et al. (Thu,) studied this question.