Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations

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Bringing together 18 chapters written by leading experts indynamical systems, operator theory, partial differential equations,and solid and fluid mechanics, this book presents state-of-the-artapproaches to a wide spectrum of new and challenging stabilityproblems.
Nonlinear Physical Systems: Spectral Analysis, Stability andBifurcations focuses on problems of spectral analysis, stabilityand bifurcations arising in the nonlinear partial differentialequations of modern physics. Bifurcations and stability of solitarywaves, geometrical optics stability analysis in hydro- andmagnetohydrodynamics, and dissipation-induced instabilities aretreated with the use of the theory of Krein and Pontryagin space,index theory, the theory of multi-parameter eigenvalue problems andmodern asymptotic and perturbative approaches.
Each chapter contains mechanical and physical examples, and thecombination of advanced material and more tutorial elements makesthis book attractive for both experts and non-specialists keen toexpand their knowledge on modern methods and trends in stabilitytheory.

Contents

1. Surprising Instabilities of Simple Elastic Structures, DavideBigoni, Diego Misseroni, Giovanni Noselli and DanieleZaccaria.
2. WKB Solutions Near an Unstable Equilibrium and Applications,Jean-François Bony, Setsuro Fujiié, Thierry Ramond andMaher Zerzeri, partially supported by French ANR projectNOSEVOL.
3. The Sign Exchange Bifurcation in a Family of Linear HamiltonianSystems, Richard Cushman, Johnathan Robbins and DimitriiSadovskii.
4. Dissipation Effect on Local and Global Fluid-ElasticInstabilities, Olivier Doaré.
5. Tunneling, Librations and Normal Forms in a Quantum Double Wellwith a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu.Anikin.
6. Stability of Dipole Gap Solitons in Two-Dimensional LatticePotentials, Nir Dror and Boris A. Malomed.
7. Representation of Wave Energy of a Rotating Flow in Terms of theDispersion Relation, Yasuhide Fukumoto, Makoto Hirota and YouichiMie.
8. Determining the Stability Domain of Perturbed Four-DimensionalSystems in 1:1 Resonance, Igor Hoveijn and Oleg N. Kirillov.
9. Index Theorems for Polynomial Pencils, Richard Kollár andRadomír Bosák.
10. Investigating Stability and Finding New Solutions inConservative Fluid Flows Through Bifurcation Approaches, PaoloLuzzatto-Fegiz and Charles H.K. Williamson.
11. Evolution Equations for Finite Amplitude Waves in ParallelShear Flows, Sherwin A. Maslowe.
12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrisonand George I. Hagstrom.
13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstromand Philip J. Morrison.
14. Energy Stability Analysis for a Hybrid Fluid-Kinetic PlasmaModel, Philip J. Morrison, Emanuele Tassi and Cesare Tronci.
15. Accurate Estimates for the Exponential Decay of Semigroups withNon-Self-Adjoint Generators, Francis Nier.
16. Stability Optimization for Polynomials and Matrices, Michael L.Overton.
17. Spectral Stability of Nonlinear Waves in KdV-Type EvolutionEquations, Dmitry E. Pelinovsky.
18. Unfreezing Casimir Invariants: Singular Perturbations GivingRise to Forbidden Instabilities, Zensho Yoshida and Philip J.Morrison.

About the Authors

Oleg N. Kirillov has been a Research Fellow at theMagneto-Hydrodynamics Division of the Helmholtz-ZentrumDresden-Rossendorf in Germany since 2011. His research interestsinclude non-conservative stability problems of structural mechanicsand physics, perturbation theory of non-self-adjoint boundaryeigenvalue problems, magnetohydrodynamics, friction-inducedoscillations, dissipation-induced instabilities and non-Hermitianproblems of optics and microwave physics. Since 2013 he has servedas an Associate Editor for the journal Frontiers in MathematicalPhysics.
Dmitry E. Pelinovsky has been Professor at McMaster University inCanada since 2000. His research profile includes work withnonlinear partial differential equations, discrete dynamicalsystems, spectral theory, integrable systems, and numericalanalysis. He served as the guest editor of the special issue of thejournals Chaos in 2005 and Applicable Analysis in 2010. He is anAssociate Editor of the journal Communications in Nonlinear Scienceand Numerical Simulations.

This book is devoted to the problems of spectral analysis,stability and bifurcations arising from the nonlinear partialdifferential equations of modern physics. Leading experts indynamical systems, operator theory, partial differential equations,and solid and fluid mechanics present state-of-the-art approachesto a wide spectrum of new challenging stability problems.Bifurcations and stability of solitary waves, geometrical opticsstability analysis in hydro- and magnetohydrodynamics anddissipation-induced instabilities will be treated with the use ofthe theory of Krein and Pontryagin space, index theory, the theoryof multi-parameter eigenvalue problems and modern asymptotic andperturbative approaches. All chapters contain mechanical andphysical examples and combine both tutorial and advanced sections,making them attractive both to experts in the field andnon-specialists interested in knowing more about modern methods andtrends in stability theory.

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Additional Information

Publisher
John Wiley & Sons
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Published on
Dec 11, 2013
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Pages
448
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ISBN
9781118577547
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Language
English
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Genres
Mathematics / General
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Content Protection
This content is DRM protected.
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This work gives a complete overview on the subject of nonconservative stability from the modern point of view. Relevant mathematical concepts are presented, as well as rigorous stability results and numerous classical and contemporary examples from mechanics and physics.

It deals with both finite- and infinite-dimensional nonconservative systems and covers the fundamentals of the theory, including such topics as Lyapunov stability and linear stability analysis, Hamiltonian and gyroscopic systems, reversible and circulatory systems, influence of structure of forces on stability, and dissipation-induced instabilities, as well as concrete physical problems, including perturbative techniques for nonself-adjoint boundary eigenvalue problems, theory of the destabilization paradox due to small damping in continuous circulatory systems, Krein-space related perturbation theory for the MHD kinematic mean field α2-dynamo, analysis of Campbell diagrams and friction-induced flutter in gyroscopic continua, non-Hermitian perturbation of Hermitian matrices with applications to optics, and magnetorotational instability and the Velikhov-Chandrasekhar paradox.

The book serves present and prospective specialists providing the current state of knowledge in the actively developing field of nonconservative stability theory. Its understanding is vital for many areas of technology, ranging from such traditional ones as rotor dynamics, aeroelasticity and structural mechanics to modern problems of hydro- and magnetohydrodynamics and celestial mechanics.

For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.
This work gives a complete overview on the subject of nonconservative stability from the modern point of view. Relevant mathematical concepts are presented, as well as rigorous stability results and numerous classical and contemporary examples from mechanics and physics.

It deals with both finite- and infinite-dimensional nonconservative systems and covers the fundamentals of the theory, including such topics as Lyapunov stability and linear stability analysis, Hamiltonian and gyroscopic systems, reversible and circulatory systems, influence of structure of forces on stability, and dissipation-induced instabilities, as well as concrete physical problems, including perturbative techniques for nonself-adjoint boundary eigenvalue problems, theory of the destabilization paradox due to small damping in continuous circulatory systems, Krein-space related perturbation theory for the MHD kinematic mean field α2-dynamo, analysis of Campbell diagrams and friction-induced flutter in gyroscopic continua, non-Hermitian perturbation of Hermitian matrices with applications to optics, and magnetorotational instability and the Velikhov-Chandrasekhar paradox.

The book serves present and prospective specialists providing the current state of knowledge in the actively developing field of nonconservative stability theory. Its understanding is vital for many areas of technology, ranging from such traditional ones as rotor dynamics, aeroelasticity and structural mechanics to modern problems of hydro- and magnetohydrodynamics and celestial mechanics.

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