Nonlinear Stability and Bifurcation Theory

1 Introduction.- 2 Representation of systems.- 2.1 Dynamical systems.- 2.1.1 Time continuous system.- 2.1.2 Time discrete system.- 2.2 Statical systems.- 2.3 Definitions of stability.- 2.3.1 Stability in the sense of Ljapunov.- 2.3.2 Structural stability (robustness, coarseness).- 3 Reduction process, bifurcation equations.- 3.1 Finite-dimensional dynamical systems.- 3.1.1 Steady states.- 3.1.2 Periodic motions.- 3.2 Infinite-dimensional statical and dynamical systems..- 3.2.1 Statical systems.- 3.2.2 Dynamical systems.- 4 Application of the reduction process.- 4.1 Equilibria of finite-dimensional systems.- 4.1.1 Double pendulum with axially elastic rods and follower force loading.- 4.1.2 Double pendulum with elastic end support and follower force loading.- 4.1.3 Double pendulum under aerodynamic excitation..- 4.1.4 Loss of stability of the straight line motion of a tractor-semitrailer.- 4.1.5 Loss of stability of the straight line motion of a railway vehicle.- 4.1.6 Summary of Section 4.1.- 4.2 Periodic solutions of finite-dimensional systems.- 4.2.1 Mechanical model and equations of motion.- 4.2.2 Calculation of the power series expansion of the Poincare mapping.- 4.2.3 Stability boundary in parameter space.- 4.2.4 Center manifold reduction.- 4.3 Finite- and infinite-dimensional statical systems.- 4.3.1 Buckling of a rod: discrete model.- 4.3.2 Buckling of a rod: continuous model.- 4.3.3 Buckling of a circular ring.- 4.3.4 Buckling at a double eigenvalue: rectangular plate.- 4.3.5 The pattern formation problem: buckling of complete spherical shells.- 5 Bifurcations under symmetries.- 5.1 Introduction.- 5.2 Finite dimensional dynamical systems.- 5.2.1 Two zero roots.- 5.2.2 Two purely imaginary pairs.- 5.3 Infinite dimensional statical systems.- 5.4 Infinite dimensional dynamical systems.- 6 Discussion of the bifurcation equations.- 6.1 Transformation to normal form.- 6.1.1 Time-continuous dynamical systems.- 6.1.2 Time-discrete dynamical systems.- 6.1.3 Statical systems.- 6.2 Codimension.- 6.2.1 Static bifurcation.- 6.2.2 Dynamic bifurcation.- 6.3 Determinacy.- 6.4 Unfolding.- 6.5 Classification.- 6.5.1 Dynamic bifurcation.- 6.5.2 Static bifurcation: elementary catastrophe theory.- 6.5.3 The unfolding theory of Golubitsky and Schaeffer.- 6.5.4 Restricted generic bifurcation.- 6.6 Bifurcation diagrams.- 6.6.1 Statical systems.- 6.6.2 Time-continuous dynamical systems.- 6.6.3 Time-discrete dynamical systems.- 6.6.4 Symmetric dynamical systems.- 6.6.5 Symmetric statical systems.- 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.