Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state master equation based decision making model has been shown to generate phase transitions, to be topologically complex and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy

Foundations of Fractional Calculus on Time Scales

Bakalářská práce pojednává o zlomkovém kalkule na časových škálach, přesněji - zavádí zlomkový kalkulus na časových škálach a taktéž vyšetřuje jednoznačnost axiomatické definice zavádějící mocninné funkce. Po zavedení základních pojmů je předmětem diskuze hlavně zobecněná Laplaceova transformace a důkaz jednoznačnosti zobecněné Laplaceovy transformace, která je použita jako nástroj pro dokázání jednoznačnosti zlomkových mocniných funkcií na časových škálach.The bachelor thesis concerns fractional calculus on time scales, more precisely, it introduces fractional calculus on time scales and also investigates the property of uniqueness of the axiomatic definition of the power functions. After introducing basic concepts, the subject of discussion is mostly generalized Laplace transform as well as proof of uniqueness of generalized Laplace transform, which is used as a tool to proving the uniqueness of fractional power functions on time scales.

Recent Advances and Applications of Fractional-Order Neural Networks

This paper focuses on the growth, development, and future of various forms of fractional-order neural networks. Multiple advances in structure, learning algorithms, and methods have been critically investigated and summarized. This also includes the recent trends in the dynamics of various fractional-order neural networks. The multiple forms of fractional-order neural networks considered in this study are Hopfield, cellular, memristive, complex, and quaternion-valued based networks. Further, the application of fractional-order neural networks in various computational fields such as system identification, control, optimization, and stability have been critically analyzed and discussed

Enhanced Fractional Adaptive Processing Paradigm for Power Signal Estimation

Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order)

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

FSCSCOOT: Functional Calculus Competitive Swarm Coot Optimization-based CNN transfer learning for Parkinson’s disease classification

Parkinson's disease (PD) is a neurological disorder of the central nervous system that causes difficulty in movement, often including tremors and rigidity. Early detection of PD can prevent symptoms up to a certain age and increase life expectancy. For this purpose, we have used brain images from magnetic resonance imaging (MRI) technique. Generally dementia can be either classified as Alzheimer’s or Parkinson’s or sometimes may be due to tumor in brain. Therefore, effectual methods such as Competitive Swarm Coot Optimization_ Convolutional Neural Network (CSCOOT_CNN) with transfer learning and Fractional CSCOOT_ deep neuro-fuzzy network (FCSCOOT_DNFN are newly introduced for classification of brain diseases. At first, input images are acquired from particular datasets, and then input images are given to the pre-processing stage.  In a pre-processing module, median filter is utilized for the elimination of noises. Afterward, pre-processed image is then subjected to feature extraction in which CNN features are extracted. In the level of classification, the images are classified into Parkinson by DNFN that is trained utilizing the introduced FCSCOOT algorithm. Furthermore, the FCSCOOT algorithm is newly designed by combination of Fractional Calculus (FC) with CSCOOT algorithm