Computational Methods in Bifurcation Theory and Dissipative Structures

1. Introduction.- 1.1 General Introduction.- 1.2 Dissipative Structures in Physical, Chemical, and Biological Systems.- 1.3 Basic Concepts and Properties of Nonlinear Systems.- 1.4 Examples.- 2. Multiplicity and Stability in Lumped-Parameter Systems (LPS).- 2.1 Steady-State Solutions.- 2.2 Dependence of Steady-State Solutions on a Parameter-Solution Diagram.- 2.3 Stability of Steady-State Solutions.- 2.4 Branch Points-Real Bifurcation.- 2.5 Branch Points-Complex Bifurcations.- 2.6 Bifurcation Diagram.- 2.7 Transient Behavior of LPS-Numerical Methods.- 2.8 Computation of Periodic Solutions.- 2.9 Chaotic Attractors.- 3. Multiplicity and Stability in Distributed-Parameter Systems (DPS).- 3.1 Steady-State Solutions-Methods for Solving Nonlinear Boundary-Value Problems.- 3.2 Dependence of Steady-State Solutions on a Parameter.- 3.3 Branch Points-Methods for Evaluating Real and Complex Bifurcation Points.- 3.4 Methods for Transient Simulation of Parabolic Equations-Finite-Difference Methods.- 4. Development of Quasi-stationary Patterns with Changing Parameter.- 4.1 Quasi-stationary Behavior in LPS-Examples.- 4.2 Quasi-stationary Behavior in DPS-Examples.- 5. Perspectives.- Appendix A DERPAR-A Continuation Algorithm.- Appendix B SHOOT-An Algorithm for Solving Nonlinear Boundary-Value Problems by the Shooting Method.- Appendix C Bifurcation and Stability Theory.- C. 1 Invariant Manifolds and the Center-Manifold Theorem (Reduction of Dimension).- C.2 Normal Forms.- C.3 Bifurcation of Singular Points of Vector Fields.- C.4 Codimension of a Vector Field. Unfolding of a Vector Field.- C.5 Construction of a Versal Deformation.- C.6 Bifurcations of Codimension 2.- C.7 Bifurcations from Limit Cycles.- References.