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

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-spread believe that no straightforward solution procedures for nonlinear problems are available prevails even today in engineering cir­ cles. Though in some courses it is indicated that in principle nonlinear problems are solveable by numerical methods the treatment of nonlinear problems, more or less, is considered to be an art or an intellectual game. A good example for this statement was the search for Ljapunov functions for nonlinear stability problems in the seventies. However things have changed. At the beginning of the seventies, start­ ing with the work of V.1. Arnold, R. Thom and many others, new ideas which, however, have their origin in the work of H. Poincare and A. A. Andronov, in the treatment of nonlinear problems appeared. These ideas gave birth to the term Bifurcation Theory. Bifurcation theory allows to solve a great class of nonlinear problems under variation of parameters in a straightforward manner.

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 Poincaré 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.