**About the Editors**

**Marius-F. Danca** received an MSc degree in Mathematics in 1980 from Babes-Bolyai University of Cluj-Napoca, Romania, Faculty of Mathematics; an MSc in Electrotechnics in 1986 from the Technical University of Cluj-Napoca, Romania, Faculty of Electrotechnics; a PhD degree in Automation in 1997 from the Technical University of Cluj-Napoca, Romania, Faculty of Automation and Computer Science; and a PhD degree in Mathematics in 2002 from Babes-Bolyai University of Cluj-Napoca, Romania, Faculty of Mathematics.

Professor Marius-F. Danca was Guest Associate Editor at Discrete and Continuous Dynamical Systems—Series S (DCDS-S) (2007–2017), Associate Editor at Dynamics of Continuous, Discrete & Impulsive Systems—Series B (DCDIS-B) (2006–present), Associate Editor at Journal of Nonlinear Systems and Applications (JNSA) (2009–present), and Guest Editor of the Special Issue "Research Frontier in Chaos Theory and Complex Networks" (2018).

Professor Marius-F. Danca has 85 ISI papers, an h-index of 18, and more than 200 publications in Romanian and foreign dissemination journals, and was a reviewer for more than 40 ISI journals (at Springer, Elsevier, Taylor & Francis, AIMS, Willey, IET, Hindawi, World Scientific, MDPI, AIP, John Willey & Sons, Hacettepe, IOP, and Sage).

He is the author of two books, entitled "Functia logistica: dinamica, bifurcatie si haos" Seria MATEMATICA APLICATA SI INDUSTRIALA 7, Editura Universitatii din Pitesti, 2001; and "Sisteme dinamice discontinue" Seria MATEMATICA APLICATA SI INDUSTRIALA 14, Editura Universitatii din Pitesti, 2004 (in Romanian); and one chapter in "Complex Systems and Networks: Dynamics, Controls and Applications" in Understanding Complex Systems, Springer 2016.

**Guanrong Chen** received an MSc degree in Computer Science from Sun Yat-sen University, Guangzhou, China, in 1981 and a PhD degree in Applied Mathematics from Texas A&M University, USA, in 1987. Since 2000, he has been a Chair Professor and the founding director of the "Centre for Chaos and Complex Networks" at the City University of Hong Kong.

Professor Chen was elected a Fellow of the IEEE in 1997, awarded the 2011 Euler Gold Medal from Russia, and conferred Honorary Doctor Degrees by the Saint Petersburg State University, Russia, in 2011 and by the University of Normandy, France, in 2014. He is a Member of the Academy of Europe (since 2014) and a Fellow of The World Academy of Sciences (since 2015).

Professor Chen's research interests are in the fields of complex networks, nonlinear dynamics, and control systems. He is a Highly Cited Researcher in Engineering (since 2009), according to Thomson Reuters.

## **Preface to "Bifurcation and Chaos in Fractional-Order Systems"**

The concept of fractional-order differentiation first emerged in 1965 regarding a historical correspondence between the Marquise de L'Hospital and the mathematician Leibnitz. In the sequel, famous mathematicians such as Euler, Laplace, Abel, Liouville, and Riemann further developed fundamental technical details. It was realized recently that many scientific phenomena with complex dynamics cannot be well modeled by differential equations using integer-order derivatives. Fractional calculus (calculus of non-integer order) became a rapidly developing topic in science and engineering, which has been attracting a grea<sup>t</sup> deal of attention recently in the academic and industrial world. While exponential laws represent a classical approach to studying dynamical systems, there are systems where faster or slower dynamics are better and more accurately described by Mittag–Leffler functions. As a result, there has been an increasing interest to merge the mathematical fundamentals of fractional calculus into scientific and engineering applications as an interdisciplinary approach, which has started to demonstrate some advantages over conventional integer-order differential systems. Although the last decade had witnessed significant development in this research area, many theoretical and technical problems remain to be further explored, including particularly chaotic fractional-order systems. On the other hand, finding hidden attractors in continuous-time and discrete-time chaotic fractional-order systems represents a new trend of research, as an exciting and challenging direction in the fields of complex dynamics. Of particular interest are those systems with symmetry, in which bifurcations can lead to a family of conjugate attractors, all connected by symmetry. Therefore, this research direction of bifurcation and chaos in fractional-order dynamical systems opens up a corpus of opportunities with encouraging promises in such scientific fields as complex dynamics, systems and networks, and signal processing, to name just a few.

The book consists of seven contributed papers written by active leading experts on various topics.

The first paper, entitled "Fractional Dynamics in Soccer Leagues", addresses the dynamics of four European soccer teams over the 2018–2019 season, with modeling based on fractional calculus and power law. The new model embeds implicit details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees, and a multitude of other effects. Two approaches are taken to evaluate the teams' progress along the league by models fitting real-world data and to analyze statistical information by using entropy.

The second paper, entitled "Puu System of Fractional Order and Its Chaos Suppression", studies the fractional-order variant of Puu's system and compares it with the integer-order counterpart. Also, an impulsive chaos control algorithm is applied to suppress chaos in the system.

The third paper, entitled "Dynamical Properties of Fractional-Order Memristor", investigates the properties of a fractional-order memristor, revealing the influences of parameters such as the fraction order, frequency, switch resistor ratio, average mobility on the system dynamics, and so on. The fractional-order memristor is implemented by circuits.

The forth paper, entitled "Fractional Levy Stable and Maximum Lyapunov Exponent for Wind Speed Prediction", proposes a wind speed prediction method based on the maximum Lyapunov exponent and the fractional Levy stable motion in an iterative prediction model. Theoretical analysis and numerical simulations are performed with comparisons, showing some advantages of the new method.

The fifth paper, entitled "On Two-Dimensional Fractional Chaotic Maps with Symmetries", discusses two new two-dimensional chaotic maps with closed curve fixed points. It analyzes the chaotic behavior of the two maps by the 0–1 test and explores it numerically using Lyapunov exponents and bifurcation diagrams, showing that chaos exists in both fractional maps and that the fractional-order map has coexisting attractors.

The sixth paper, entitled "Bifurcations, Hidden Chaos and Control in Fractional Maps", introduces, based on discrete fractional calculus, simple two-dimensional and three-dimensional fractional maps; both are chaotic and have a unique equilibrium point, with coexisting attractors. Moreover, control schemes are introduced to stabilize the chaotic trajectories of the two systems.

The seventh paper, entitled "Generalized Bessel Polynomial for Multi-Order Fractional Differential Equations", defines a simple but effective method for approximating solutions of multi-order fractional differential equations relying on the Caputo fractional derivative under some conditions. Basis functions used are generalized Bessel polynomials satisfying many properties shared by the classical orthogonal polynomials of Hermit, Laguerre, and Jacobi. The new method has good performance in accuracy and simplicity. Some practical test problems with symmetries are used to verify the proposed technique with comparisons.

We thank all the authors for their excellent research and fine contributions to this edited book.

> **Marius-F. Danca, Guanrong Chen**

> > *Editors*
