Complex Dynamics

Readers learn to predict and control low- and high-dimensional as well as continuous- and discrete-time nonlinear systems dynamics in complex variables. In the final chapter, all the previously developed methods are used to present the "Holy Grail" of modern physical and cosmological science, the search for the "theory of everything" and the "true" cosmological dynamics. Complex Dynamics: Advanced System Dynamics in Complex Variables is a graduate-level monographic textbook. It is designed as a comprehensive introduction into methods and techniques of modern complex-valued nonlinear dynamics with its various physical and non-physical applications. This book is a complex-valued continuation of our previous two monographs, Geometrical Dynamics of Complex Systems and High-Dimensional Chaotic and Attractor Systems, Volumes 31 and 32 in the Springer book series Intelligent Systems, Control and Automation: Science and Engineering, where we had developed the most powerful mathematical machinery to deal with high-dimensional nonlinear, attractor and chaotic real-valued dynamics. The present monograph is devoted to understanding, prediction and control of both low- and high-dimensional, as well as both continuous- and discrete-time, nonlinear systems dynamics in complex variables. Its objective is to provide a serious reader with a serious scientific tool that will enable him/her to actually perform a competitive research in modern complex-valued nonlinear dynamics.§This book has seven Chapters. The first, introductory Chapter explains in plain English the objective of the book and provides the preliminaries in complex numbers and variables; it also gives a soft introduction to quantum dynamics. The second Chapter develops low-dimensional dynamics in the complex plane, theoretical and computational, continuous- and discrete-time. The third Chapter presents a modern introduction to quantum dynamics, mainly following Dirac's notation. The fourth Chapter develops geometrical machinery of complex manifolds, essential for the further text. The fifth Chapter develops high-dimensional complex continuous dynamics, which takes place on complex manifolds. The sixth Chapter develops the formalism of complex path integrals, which extends the continuous dynamics to the general high-dimensional dynamics, which can be both discrete and stochastic. In the last, seventh Chapter, all previously developed methods are employed to present the Holy Grail of modern physical and cosmological science, the search for the theory of everything and the true cosmological dynamics.