This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, offers a detailed account of fractional calculus tools, signatures of complex systems, hidden connections to fractional calculus, and applications and case studies involving fractional calculus in complex signal analysis and complex system modelling, analysis and control (MAD). The authors document both the foundational concepts of fractional calculus in complexity science as well as their applications to, and role in, the optimization of, complex engineered systems. 

Fractional calculus is about differentiation and integration of non-integer orders. Convenience has driven the use of integer-order models and controllers for complex natural or man-made systems, but these traditional models and tools for the control of dynamic systems may result in suboptimum performance and even "anomalous" behavior. In contrast, a growing literature documents "more optimal" performance when fractional order calculus tools are applied. From an engineering point of view, new and beneficial uses of this versatile mathematical tool are both possible and important, and may become an enabler of new scientific discoveries.

Topic areas (ToC in preparation):

Fractional calculus:.-  definitions, history, basic properties.- Fractional order dynamic systems.- Fractional noises.- Signatures of complex systems and its fractional calculus connection.- Power law.- Long range dependence.- Long memory.- Long range interaction.- Heavytailedness.- Complex signal analysis using fractional calculus.- Complex system modeling using fractional calculus.- Complex system control using fractional calculus.

YangQuan Chen received his B.S. in industrial automation from the University of Science and Technology of Beijing, China, in 1985, his M.S. in automatic control from the Beijing Institute of Technology, in 1989, and his Ph.D. degree in advanced control and instrumentation from the Nanyang Technological University, Singapore, in 1998. He was a member of the control systems/mechatronics faculty of ECE Dept. at Utah State University, Logan, and the Director of the Center for Self-Organizing and Intelligent Systems (CSOIS), before joining the University of California, Merced in Fall 2012. Prof. Chen's research interests include mechatronics, control systems, unmanned aircraft systems, cyber-physical systems, and applied fractional calculus.