Nonlinear Physical Systems : Spectral Analysis, Stability and Bifurcations.
Title:
Nonlinear Physical Systems : Spectral Analysis, Stability and Bifurcations.
Author:
Kirillov, Oleg N.
ISBN:
9781118577578
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (449 pages)
Contents:
Cover -- Title Page -- Table of Contents -- Preface -- Chapter 1. Surprising Instabilities of Simple Elastic Structures -- 1.1. Introduction -- 1.2. Buckling in tension -- 1.3. The effect of constraint's curvature -- 1.4. The Ziegler pendulum made unstable by Coulomb friction -- 1.5. Conclusions -- 1.6. Acknowledgments -- 1.7. Bibliography -- Chapter 2. WKB Solutions Near an Unstable Equilibrium and Applications -- 2.1. Introduction -- 2.2. Connection of microlocal solutions near a hyperbolic fixed point -- 2.2.1. A model in one dimension -- 2.2.2. Classical mechanics -- 2.2.3. Review of semi-classical microlocal analysis -- 2.2.4. The microlocal Cauchy problem - uniqueness -- 2.2.5. The microlocal Cauchy problem - transition operator -- 2.3. Applications to semi-classical resonances -- 2.3.1. Spectral projection and Schrödinger group -- 2.3.2. Resonance-free zone for homoclinic trajectories -- 2.4. Acknowledgment -- 2.5. Bibliography -- Chapter 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems -- 3.1. Statement of problem -- 3.2. Bifurcation values of γ -- 3.3. Versal normal forms near the bifurcation values -- 3.3.1. Normal forms -- 3.3.2. Linear Hamiltonian Hopf bifurcation γ± -- 3.3.3. The Switch twist bifurcation at γ+ -- 3.3.4. Sign exchange bifurcation -- 3.4. Infinitesimally symplectic normal form -- 3.4.1. Normal form of Xγ at γ± -- 3.4.2. Normal form of Xγ at γ± -- 3.5. Global issues -- 3.5.1. Invariant Lagrange planes -- 3.5.2. Symplectic signs -- 3.6. Bibliography -- Chapter 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities -- 4.1. Introduction -- 4.2. Local and global stability analyses -- 4.2.1. Local analysis -- 4.2.2. Global analysis -- 4.3. The fluid-conveying pipe: a model problem -- 4.4. Effect of damping on the local and global stability of the fluid-conveying pipe.
4.4.1. Local stability -- 4.4.2. Global stability -- 4.5. Application to energy harvesting -- 4.6. Conclusion -- 4.7. Bibliography -- Chapter 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field -- 5.1. Introduction -- 5.2. 1D Landau-Lifshitz splitting formula and its analog for the ground states -- 5.3. The splitting formula in multi-dimensional case -- 5.4. Normal forms and complex Lagrangian manifolds -- 5.4.1. Normal form in the classically allowed and forbidden regions -- 5.4.2. Complex continuation of integrals -- 5.4.3. Almost invariant complex Lagrangian manifolds -- 5.5. Constructing the asymptotics for the eigenfunctions in tunnel problems -- 5.5.1. Complex WKB-method -- 5.5.2. WKB-methods with real and pure imaginary phases -- 5.5.3. Variational methods -- 5.6. Splitting of the eigenvalues in the presence of magnetic field -- 5.7. Proof of main theorem (a sketch) -- 5.7.1. Lifshitz-Herring formula -- 5.7.2. Instanton splitting formula -- 5.7.3. Asymptotic behavior of the libration action -- 5.7.4. Reduction to the 1D splitting problem -- 5.7.5. Asymptotic behavior of the Floquet exponents -- 5.7.6. Finishing the proof -- 5.8. Conclusion -- 5.9. Acknowledgments -- 5.10. Bibliography -- Chapter 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials -- 6.1. Introduction -- 6.2. The model -- 6.3. Solitons in the first bandgap: the SF nonlinearity -- 6.3.1. Solution families -- 6.3.2. Stability of solitons in the first finite bandgap -- 6.3.3. Bound states of solitons in the first bandgap -- 6.4. Stability GSs in the second bandgap -- 6.5. Conclusions -- 6.6. Bibliography -- Chapter 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation -- 7.1. Introduction -- 7.2. Lagrangian approach to wave energy -- 7.3. Kelvin waves.
7.4. Wave energy in terms of the dispersion relation -- 7.5. Conclusion -- 7.6. Bibliography -- Chapter 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance -- 8.1. Introduction -- 8.1.1. Physical motivation -- 8.1.2. Setting -- 8.1.3. Main question and examples -- 8.2. Methods -- 8.2.1. Centralizer unfolding -- 8.2.2. Stability domain -- 8.2.3. Mapping into the centralizer unfolding -- 8.3. Examples -- 8.3.1. Modulation instability -- 8.3.2. Non-conservative gyroscopic system -- 8.4. Conclusions -- 8.5. Bibliography -- Chapter 9. Index Theorems for Polynomial Pencils -- 9.1. Introduction -- 9.2. Krein signature -- 9.3. Index theorems for linear pencils and linearized Hamiltonians -- 9.4. Graphical interpretation of index theorems -- 9.4.1. Algebraic calculation of Z and Z -- 9.5. Conclusions -- 9.6. Acknowledgments -- 9.7. Bibliography -- Chapter 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches -- 10.1. Introduction -- 10.2. Counting positive-energy modes from IVI diagrams -- 10.3. An approximate prediction for the onset of resonance in 2D vortices -- 10.4. An example: three corotating vortices -- 10.4.1. Building a family of solutions from vorticity-preserving rearrangements -- 10.4.2. Computing signatures for one member of the family -- 10.4.3. The velocity-impulse diagram -- 10.4.4. Uncovering bifurcations by introducing imperfections -- 10.4.5. Counting positive-energy modes from turning points in impulse -- 10.4.6. Recovering the underlying bifurcation structure -- 10.4.7. An approximate prediction for resonance -- 10.5. Comparison with exact eigenvalues and discussion -- 10.6. Conclusions -- 10.7. Bibliography -- Chapter 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows -- 11.1. Introduction -- 11.2. Wave packets.
11.2.1. Conservative systems -- 11.2.2. Applications to hydrodynamic stability -- 11.2.3. The Ginzburg-Landau equation -- 11.3. Critical layer theory -- 11.3.1. Asymptotic theory of the Orr-Sommerfeld equation -- 11.3.2. Nonlinear critical layers -- 11.3.3. The wave packet critical layer -- 11.4. Nonlinear instabilities governed by integro-differential equations -- 11.4.1. The zonal wave packet critical layer -- 11.5. Concluding remarks -- 11.6. Bibliography -- Chapter 12. Continuum Hamiltonian Hopf Bifurcation I -- 12.1. Introduction -- 12.2. Discrete Hamiltonian bifurcations -- 12.2.1. A class of 1 + 1 Hamiltonian multifluid theories -- 12.2.2. Examples -- 12.2.3. Comparison and commentary -- 12.3. Continuum Hamiltonian bifurcations -- 12.3.1. A class of 2 + 1 Hamiltonian mean field theories -- 12.3.2. Example of the CHH bifurcation -- 12.4. Summary and conclusions -- 12.5. Acknowledgments -- 12.6. Bibliography -- Chapter 13. Continuum Hamiltonian Hopf Bifurcation II -- 13.1. Introduction -- 13.2. Mathematical aspects of the continuum Hamiltonian Hopf bifurcation -- 13.2.1. Structural stability -- 13.2.2. Normal forms and signature -- 13.3. Application to Vlasov-Poisson -- 13.3.1. Structural stability in the space Cn(R) L1(R) -- 13.3.2. Structural stability in W 1,1 -- 13.3.3. Dynamical accessibility and structural stability -- 13.4. Canonical infinite-dimensional case -- 13.4.1. Negative energy oscillator coupled to a heat bath -- 13.5. Commentary: degeneracy and nonlinearity -- 13.6. Summary and conclusions -- 13.7. Acknowledgments -- 13.8. Bibliography -- Chapter 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model -- 14.1. Introduction -- 14.2. Stability and the energy-Casimir method -- 14.3. Planar Hamiltonian hybrid model -- 14.3.1. Planar hybrid model equations of motion -- 14.3.2. Hamiltonian structure.
14.3.3. Casimir invariants -- 14.4. Energy-Casimir stability analysis -- 14.4.1. Equilibrium variational principle -- 14.4.2. Stability conditions -- 14.5. Conclusions -- 14.6. Acknowledgments -- 14.7. Appendix A: derivation of hybrid Hamiltonian structure -- 14.8. Appendix B: Casimir verification -- 14.9. Bibliography -- Chapter 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators -- 15.1. Introduction -- 15.2. Relevant quantities for sectorial operators -- 15.3. Natural examples -- 15.3.1. An example related to linearized equations of fluid mechanics -- 15.3.2. Kramers-Fokker-Planck operators -- 15.4. Artificial examples -- 15.4.1. Adiabatic evolution of quantum resonances in the one-dimensional case -- 15.4.2. Optimizing the sampling of equilibrium distributions -- 15.5. Conclusion -- 15.6. Bibliography -- Chapter 16. Stability Optimization for Polynomials and Matrices -- 16.1. Optimization of roots of polynomials -- 16.1.1. Root optimization over a polynomial family with a single affine constraint -- 16.1.2. The root radius -- 16.1.3. The root abscissa -- 16.1.4. Examples -- 16.1.5. Polynomial root optimization with several affine constraints -- 16.1.6. Variational analysis of the root radius and abscissa -- 16.1.7. Computing the root radius and abscissa -- 16.2. Optimization of eigenvalues of matrices -- 16.2.1. Static output feedback -- 16.2.2. Numerical methods for non-smooth optimization -- 16.2.3. Numerical results for some SOF problems -- 16.2.4. The Diaconis-Holmes-Neal Markov chain -- 16.2.5. Active derogatory eigenvalues -- 16.3. Concluding remarks -- 16.4. Acknowledgments -- 16.5. Bibliography -- Chapter 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations -- 17.1. Introduction -- 17.2. Historical remarks and examples -- 17.3. Proof of theorem 17.1.
17.4. Generalization of theorem 17.1 for a periodic nonlinear wave.
Abstract:
Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics, and dissipation-induced instabilities are treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. Each chapter contains mechanical and physical examples, and the combination of advanced material and more tutorial elements makes this book attractive for both experts and non-specialists keen to expand their knowledge on modern methods and trends in stability theory. Contents 1. Surprising Instabilities of Simple Elastic Structures, Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria. 2. WKB Solutions Near an Unstable Equilibrium and Applications, Jean-François Bony, Setsuro Fujiié, Thierry Ramond and Maher Zerzeri, partially supported by French ANR project NOSEVOL. 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems, Richard Cushman, Johnathan Robbins and Dimitrii Sadovskii. 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities, Olivier Doaré. 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu. Anikin. 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials, Nir Dror and Boris A.
Malomed. 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation, Yasuhide Fukumoto, Makoto Hirota and Youichi Mie. 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance, Igor Hoveijn and Oleg N. Kirillov. 9. Index Theorems for Polynomial Pencils, Richard Kollár and Radomír Bosák. 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches, Paolo Luzzatto-Fegiz and Charles H.K. Williamson. 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows, Sherwin A. Maslowe. 12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrison and George I. Hagstrom. 13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstrom and Philip J. Morrison. 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model, Philip J. Morrison, Emanuele Tassi and Cesare Tronci. 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators, Francis Nier. 16. Stability Optimization for Polynomials and Matrices, Michael L. Overton. 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations, Dmitry E. Pelinovsky. 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities, Zensho Yoshida and Philip J. Morrison. About the Authors Oleg N. Kirillov has been a Research Fellow at the Magneto-Hydrodynamics Division of the Helmholtz-Zentrum Dresden-Rossendorf in Germany since 2011. His research interests include non-conservative stability problems of structural mechanics and physics, perturbation theory of non-self-adjoint boundary eigenvalue problems, magnetohydrodynamics, friction-induced oscillations, dissipation-induced instabilities and non-Hermitian problems of optics and microwave physics. Since 2013 he has served as an Associate Editor fo.
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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