Nonlinear Stability and Bifurcation Theory An Introduction for Engineers and Applied Scientists

• Main Authors: Troger, Hans, Steindl, Alois
• Format: eBook
• Language: English
• Published: Vienna Springer Wien 1991; Springer Vienna
• Edition: 1
• Subjects: Bifurcation theory; Civil Engineering; Classical Mechanics; Engineering mathematics; Mathematical and Computational Engineering; Mechanics; Physics; Physics and Astronomy

Every student in engineering or in other fields of the applied sciences who has passed through his curriculum knows that the treatment of nonlin­ ear problems has been either avoided completely or is confined to special courses where a great number of different ad-hoc methods are presented. The wide...

Table of Contents:

• Intro -- Nonlinear Stability and Bifurcation Theory -- Copyright -- Preface -- Contents -- Chapter 1 Introduction -- Chapter 2 Representation of systems and definition of stability -- Chapter 3 Reduction process, bifurcation equations -- Chapter 4 Application of the reduction process -- Chapter 5 Bifurcations under symmetries -- Chapter 6 Discussion of the bifurcation equations -- Appendix A Linear spaces and linear operators -- Appendix B Transformation of matrices to diagonal or Jordan form -- Appendix C Adjoint and self-adjoint linear differential operators -- Appendix D Projection operators -- Appendix E Spectral decomposition of the inverse of linear differential operators -- Appendix F Some formulas concerning the shell equations on the complete sphere -- Appendix G Some properties of groups -- Appendix H Stability boundaries in parameter space -- Appendix I Differential equation of an elastic ring -- Appendix J Static shallow shell and plate equations for moderately large deformations -- Appendix K Shell equations for axisymmetric deformations -- Appendix L Equations of motion of a fluid conveying tube -- Appendix M Various concepts of equivalences -- Appendix N Slowly varying parameter -- Appendix O Transformation of dynamical systems into standard form -- Bibliography -- Index
• 1 Introduction -- 2 Representation of systems -- 2.1 Dynamical systems -- 2.2 Statical systems -- 2.3 Definitions of stability -- 3 Reduction process, bifurcation equations -- 3.1 Finite-dimensional dynamical systems -- 3.2 Infinite-dimensional statical and dynamical systems. -- 4 Application of the reduction process -- 4.1 Equilibria of finite-dimensional systems -- 4.2 Periodic solutions of finite-dimensional systems -- 4.3 Finite- and infinite-dimensional statical systems -- 5 Bifurcations under symmetries -- 5.1 Introduction -- 5.2 Finite dimensional dynamical systems -- 5.3 Infinite dimensional statical systems -- 5.4 Infinite dimensional dynamical systems -- 6 Discussion of the bifurcation equations -- 6.1 Transformation to normal form -- 6.2 Codimension -- 6.3 Determinacy -- 6.4 Unfolding -- 6.5 Classification -- 6.6 Bifurcation diagrams -- A Linear spaces and linear operators -- A.1 Linear spaces -- A.2 Linear operators -- B Transformation of matrices to Jordan form -- C Adjoint and self-adjoint linear differential operators -- C.1 Calculation of the adjoint operator -- C.2 Self-adjoint differential operators -- D Projection operators -- D.1 General considerations -- D.2 Projection for non-self-adjoint operators -- D.3 Application to the Galerkin reduction -- E Spectral decomposition -- E.1 Derivation of an inversion formula -- E.2 Three examples -- F Shell equations on the complete sphere -- F.1 Tensor notations in curvilinear coordinates -- F.2 Spherical harmonics -- G Some properties of groups -- G.1 Naive definition of a group -- G.2 Symmetry groups -- G.3 Representation of groups by matrices -- G.4 Transformation of functions and operators -- G.5 Examples of invariant functions and operators -- G.6 Abstract definition of a group -- H Stability boundaries in parameter space -- I Differential equation of an elastic ring -- I.1 Equilibrium equations and bending -- I.2 Ring equations -- J Shallow shell and plate equations -- J.1 Deformation of the shell -- J.2 Constitutive law -- J.3 Equations of equilibrium -- J.4 Special cases -- J.4.1 Plate -- J.4.2 Sphere -- J.4.3 Cylinder -- K Shell equations for axisymmetric deformations -- K.1 Geometrical relations -- K.2 Stress resultants, couples and equilibrium equations -- K.3 Stress strain relations -- K.4 Spherical shell -- L Equations of motion of a fluid conveying tube -- L.1 Geometry of tube deformation -- L.2 Stress-strain relationship -- L.3 Linear and angular momentum -- L.4 Tube equations and boundary conditions -- M Various concepts of equivalences -- M.1 Right-equivalence -- M.2 Contact equivalence -- M.3 Vector field equivalence -- M.4 Bifurcation equivalence -- M.5 Recognition problem -- N Slowly varying parameter -- O Transformation of dynamical systems into standard form -- O.1 Power series expansion -- O.2 Recursive calculation.