The Fractional View of Complexity

Abstract

Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity. There is a general agreement about the relation between the two perspectives, but the formal mathematical arguments supporting their relation are still being developed.The fractional derivative of real order appears as the degree of structural heterogeneity between homogeneous and inhomogeneous domains. A purely real derivative order would imply a system with no characteristic scale, where a given property would hold regardless of the scale of the observations. However, in real-world systems, physical cut-offs may prevent the invariance spreading over all scales and, therefore, complex-order derivatives could yield more realistic models.Information theory addresses the quantification and communication of information. Entropy and complexity are concepts that often emerge in relation to systems composed of many elements that interact with each other, which appear intrinsically difficult to model. This Special Issue focuses on the synergies of fractals or fractional calculus and information theory tools, such as entropy, when modeling complex phenomena in engineering, physics, life, and social sciences. It includes 16 manuscripts addressing novel issues and specific topics that illustrate the role of entropy-based techniques in fractality, fractionality, and complexity. In the follow-up the selected manuscripts are presented in alphabetic order.The manuscript "A Fractional-Order Partially Non-Linear Model of a Laboratory Prototype of Hydraulic Canal System", by Saddam Gharab, Vicente Feliu-Batlle and Raul Rivas-Perez, addresses the identification of the nonlinear dynamics of the main pool of a laboratory hydraulic canal installed in the University of Castilla La Mancha. A new dynamic model is developed by taking into account the measurement errors caused by the different parts of the experimental setup. Fractional and integer order plus time delay models are used to approximate the responses of the main pool of the canal in its different flow regimes [1].In the paper "Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag-Leffler Stability", Licai Liu, Chuanhong Du, Xiefu Zhang, Jian Li and Shuaishuai Shi investigate two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with high complexity. Based on the fractional Mittag-Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is derived [2].In "An Entropy Formulation Based on the Generalized Liouville Fractional Derivative", Rui A. C. Ferreira and J. Tenreiro Machado present a new entropy formula based in the Liouville fractional derivative. The new concept is illustrated when applied to the Dow Jones Industrial Average time series. The Jensen-Shannon divergence is also generalized and its variation with the fractional ord...

Persistence of economic complexity in OECD countries

“…In order to successfully compute economic complexity, there is a need to determine both the complexity of locations and the economic activities taking place in such locations. This is because of the sophisticated nature of the subject-matter (Tenreiro Machado and Lopes, 2015;Lopes et al 2019). The general notion is that the activities taking place, exported or produced from a place, convey information about such location's complexity.…”

Computational Complexity-based Fractional-Order Neural Network Models for the Diagnostic Treatments and Predictive Transdifferentiability of Heterogeneous Cancer Cell Propensity

“…Fractional differential equations are also beneficial means to characterize and show the dynamics of complex phenomena with spatial heterogeneous characteristics and long memory. The fractional derivative of real order is seen as the degree of structural heterogeneity between the homogeneous and also in homogeneous spheres in which complexity usually arise with respect to systems made up of elements interacting with one another which may be intrinsically hard in terms of modeling (Lopes and Tenreiro Machado 2019).…”