The Many Facets of Complexity Science : In Memory of Professor Valentin Afraimovich Chapter 1. Maurice Courbage: The Directional Entropy for Spatially Extended Dynamical Systems Chapter 2. Olivier Bui Xavier Leoncini Detecting Regularity with Complexity Functions Chapter 3. Raul Rechtman Local Complexity Functions of the Ehrenfest’s wind tree model Chapter 4. Bastien Fernandez Selective Chaos of Travelling Waves in Feedforward Chains of Bistable Maps Chapter 5. Yeyin Xu Albert C.J. Luo On Periodic Motions in a van der Pol Oscillator Chapter 6. Jianzhe Huang Fuhong Min Hidden periodic motions for brushless motor with unsteady torque excitation Chapter 7. Stefano Boccaletti H. Bi T. Qui I. Bonamassa Shuguang Guan Chunking Rhythmic Synchronization: Bellerophon States and Quantized Clusters of Globally Coupled Phase Oscillators Chapter 8. Shasha Zheng Xilin Fu Chatter Dynamics and Stability of the Impulsive van der Pol Equation Chapter 9. Xilin Fu Yanyan Zhang Complex Motions in an Inclined Impact Pair With a Periodic Excitation Chapter 10. Dmitry V. Kovalevsky Igor L. Bashmachnikov: Nonlinear Dynamics of Deep Open Ocean Convection: An Analytical Approach Chapter 11. Messoud Efendiev Vitali Vougalter: On the Necessary Conditions for Preserving the Nonnegative Cone: Mixed Diffusion Chapter 12. Dimitri Volchenkov，Veniamin Smirnov: Multi scale Analysis of Urban Spatial Structures acquired from OpenStreetMap. This book explores recent developments in theoretical research and data analysis of real world complex systems organized in three parts namely Entropy information and complexity functions Multistability oscillations and rhythmic synchronization Diffusions rotation and convection in fluids The collection of works devoted to the memory of Professor Valentin Afraimovich provides a deep insight into the recent developments in complexity science by introducing new concepts methods and applications in nonlinear dynamical systems covering physical problems and mathematical modelling relevant to economics genetics engineering vibrations as well as classic problems in physics fluid and climate dynamics and urban dynamics. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. It can be read by mathematicians physicists complex systems scientists IT specialists civil engineers data scientists and urban planners.