This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomotion, Condensed Matter among others, while all other contributions have been revised and updated. Coverage includes the theory of (Poincare'-Birkhoff) Normal Forms, aspects of PT in specific mathematical settings (Hamiltonian, KAM theory, Nekhoroshev theory, and symmetric systems), technical problems arising in PT with solutions, convergence of series expansions, diagrammatic methods, parametric resonance, systems with nilpotent real part, PT for non-smooth systems, and on PT for PDEs [write out this acronym partial differential equations]. Another group of papers is focused specifically on applications to Celestial Mechanics, Quantum Mechanics and the related semiclassical PT, Quantum Bifurcations, Molecular Dynamics, the so-called choreographies in the N-body problem, as well as Evolutionary Theory. Overall, this unique volume serves to demonstrate the wide utility of PT, while creating a foundation for innovations from a new generation of graduate students and professionals in Physics, Mathematics, Mechanics, Engineering and the Biological Sciences.

The Encyclopedia of Complexity and System Science Series is a multivolume authoritative source for understanding and applying the basic tenets of complexity and systems theory as well as the tools and measures for analyzing complex systems in science, engineering, and many areas of social, financial, and business interactions. It is written for an audience of advanced university undergraduate and graduate students, professors, and professionals in a wide range of fields who must manage complexity on scales ranging from the atomic and molecular to the societal and global.
Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through selforganization, e.g., the spontaneous formation of temporal, spatial, or functional structures. They are therefore adaptive as they evolve and may contain selfdriving feedback loops. Thus, complex systems are much more than a sum of their parts. Complex systems are often characterized as having extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The conclusion is that a reductionist (bottom-up) approach is often an incomplete description of a phenomenon. This recognition that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components has led to many new concepts and sophisticated mathematical and modeling tools for application to many scientific, engineering, and societal issues that can be adequately described only in terms of complexity and complex systems.
Examples of Grand Scientific Challenges which can be approached through complexity and systems science include: the structure, history, and future of the universe; the biological basis of consciousness; the true complexity of the genetic makeup and molecular functioning of humans (genetics and epigenetics) and other life forms; human longevity limits; unification of the laws of physics; the dynamics and extent of climate change and the effects of climate change; extending the boundaries of and understanding the theoretical limits of computing; sustainability of life on the earth; workings of the interior of the earth; predictability, dynamics, and extent of earthquakes, tsunamis, and other natural disasters; dynamics of turbulent flows and the motion of granular materials; the structure of atoms as expressed in the Standard Model and the formulation of the Standard Model and gravity into a Unified Theory; the structure of water; control of global infectious diseases; and also evolution and quantification of (ultimately) human cooperative behavior in politics, economics, business systems, and social interactions. In fact, most of these issues have identified nonlinearities and are beginning to be addressed with nonlinear techniques, e.g., human longevity limits, the Standard Model, climate change, earthquake prediction, workings of the earth's interior, natural disaster prediction, etc.
The individual complex systems mathematical and modeling tools and scientific and engineering applications that comprised the Encyclopedia of Complexity and Systems Science are being completely updated and the majority will be published as individual books edited by experts in each field who are eminent university faculty members.

Perturbation Theory 1

Hamiltonian Perturbation Theory (and Transition to Chaos) 15

Perturbation Theory in Quantum Mechanics 47

Normal Forms in Perturbation Theory 79

Convergence of Perturbative Expansions 105

Diagrammatic Methods in Classical Perturbation Theory 119

Perturbation Theory and the Method of Detuning 141

Computational Methods in Perturbation Theory 153

Perturbation Analysis of Parametric Resonance 167

Symmetry and Perturbation Theory in Non-linear Dynamics 185

Perturbation of Systems with Nilpotent Real Part 211

Perturbation Theory for PDEs 229

Kolmogorov-Arnold-Moser (KAM) Theory for Finite and Infinite Dimensional Systems 247

Nekhoroshev Theory 291

Perturbation of Superintegrable Hamiltonian Systems 307

Perturbation Theory in Celestial Mechanics 339

n-Body Problem and Choreographies 357

Semiclassical Perturbation Theory 391

Perturbation Theory and Molecular Dynamics 409

Quantum Adiabatic Theorem 419

Quantum Bifurcations 433

Convergent Perturbative Expansion in Condensed Matter and Quantum Field Theory 457

Correlation Corrections as a Perturbation to the Quasi-free Approximation in Many-Body Quantum Systems 465

Perturbation of Equilibria in the Mathematical Theory of Evolution 489

Perturbation Theory for Non-smooth Systems 503

Exact and Perturbation Methods in the Dynamics of Legged Locomotion 519

Perturbation Theory for Water Waves 541

Periodic Rogue Waves and Perturbation Theory 565

Index 585

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